#set document(title: "2.7 Solve Linear Inequalities", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 2.7#h(0.6em)Solve Linear Inequalities #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Translate from algebra to English: #math.equation(block: false, alt: "15 greater than x")[$15 > x$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: false, alt: "15")[$15$] is greater than #emph[x]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "n minus 9 equals −42 .")[$n − 9 = −42 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "n equals minus 33")[$n = − 33$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "−5 p equals −23 .")[$−5 p = −23 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "p equals the fraction 23 over 5")[$p = frac(23, 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "3 a minus 12 equals 7 a minus 20 .")[$3 a − 12 = 7 a − 20 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "a equals 2")[$a = 2$] ] ] === Graph Inequalities on the Number Line Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation. What about the solution of an inequality? What number would make the inequality #math.equation(block: false, alt: "x greater than 3")[$x > 3$] true? Are you thinking, ‘#emph[x] could be 4’? That’s correct, but #emph[x] could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality #math.equation(block: false, alt: "x greater than 3")[$x > 3$]. We show the solutions to the inequality #math.equation(block: false, alt: "x greater than 3")[$x > 3$] on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of #math.equation(block: false, alt: "x greater than 3")[$x > 3$] is shown. Please note that the following convention is used: thick arrows point in the positive direction and thin arrows point in the negative direction. #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis.", caption: [The inequality #math.equation(block: false, alt: "x greater than 3")[$x > 3$] is graphed on this number line.]) The graph of the inequality #math.equation(block: false, alt: "x greater than or equal to 3")[$x ≥ 3$] is very much like the graph of #math.equation(block: false, alt: "x greater than 3")[$x > 3$], but now we need to show that 3 is a solution, too. We do that by putting a bracket at #math.equation(block: false, alt: "x equals 3")[$x = 3$], as shown. #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3 is graphed on the number line, with an open bracket at x equals 3, and a red line extending to the right of the bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3 is graphed on the number line, with an open bracket at x equals 3, and a red line extending to the right of the bracket.", caption: [The inequality #math.equation(block: false, alt: "x greater than or equal to 3")[$x ≥ 3$] is graphed on this number line.]) Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, \[, shows that the endpoint is included. #examplebox("Example 1")[][ Graph on the number line: ⓐ #math.equation(block: false, alt: "x less than or equal to 1")[$x ≤ 1$] ⓑ #math.equation(block: false, alt: "x less than 5")[$x < 5$] ⓒ #math.equation(block: false, alt: "x greater than minus 1")[$x > − 1$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "x less than or equal to 1")[$x ≤ 1$] #linebreak() This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at #math.equation(block: false, alt: "x equals 1")[$x = 1$] to show that it is included. #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket.", caption: none) + ⓑ #math.equation(block: false, alt: "x less than 5")[$x < 5$] #linebreak() This means all numbers less than 5, but not including 5. We shade in all the numbers on the number line to the left of 5 and put a parenthesis at #math.equation(block: false, alt: "x equals 5")[$x = 5$] to show it is not included. #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 5 is graphed on the number line, with an open parenthesis at x equals 5, and a red line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 5 is graphed on the number line, with an open parenthesis at x equals 5, and a red line extending to the right of the parenthesis.", caption: none) + ⓒ #math.equation(block: false, alt: "x greater than minus 1")[$x > − 1$] #linebreak() This means all numbers greater than #math.equation(block: false, alt: "−1")[$−1$], but not including #math.equation(block: false, alt: "−1")[$−1$]. We shade in all the numbers on the number line to the right of #math.equation(block: false, alt: "−1")[$−1$], then put a parenthesis at #math.equation(block: false, alt: "x equals −1")[$x = −1$] to show it is not included. #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 1 is graphed on the number line, with an open parenthesis at x equals negative 1, and a red line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 1 is graphed on the number line, with an open parenthesis at x equals negative 1, and a red line extending to the right of the parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph on the number line: ⓐ #math.equation(block: false, alt: "x less than or equal to minus 1")[$x ≤ − 1$] ⓑ #math.equation(block: false, alt: "x greater than 2")[$x > 2$] ⓒ #math.equation(block: false, alt: "x less than 3")[$x < 3$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the left of the bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the left of the bracket.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark line extending to the left of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark line extending to the left of the parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph on the number line: ⓐ #math.equation(block: false, alt: "x greater than minus 2")[$x > − 2$] ⓑ #math.equation(block: false, alt: "x less than minus 3")[$x < − 3$] ⓒ #math.equation(block: false, alt: "x greater than or equal to −1")[$x ≥ −1$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 2 is graphed on the number line, with an open parenthesis at x equals negative 2, and a dark line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 2 is graphed on the number line, with an open parenthesis at x equals negative 2, and a dark line extending to the right of the parenthesis.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the right of the bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the right of the bracket.", caption: none) ] ] We can also represent inequalities using #emph[interval notation.] As we saw above, the inequality #math.equation(block: false, alt: "x greater than 3")[$x > 3$] means all numbers greater than 3. There is no upper end to the solution to this inequality. In #strong[interval notation], we express #math.equation(block: false, alt: "x greater than 3")[$x > 3$] as #math.equation(block: false, alt: "open parenthesis 3 , ∞ close parenthesis .")[$( 3 , ∞ ) .$] The symbol #math.equation(block: false, alt: "∞")[$∞$] is read as ‘infinity’. It is not an actual number. shows both the number line and the interval notation. #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 3 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 3 comma infinity, parenthesis.", caption: [The inequality #math.equation(block: false, alt: "x greater than 3")[$x > 3$] is graphed on this number line and written in interval notation.]) The inequality #math.equation(block: false, alt: "x less than or equal to 1")[$x ≤ 1$] means all numbers less than or equal to 1. There is no lower end to those numbers. We write #math.equation(block: false, alt: "x less than or equal to 1")[$x ≤ 1$] in interval notation as #math.equation(block: false, alt: "open parenthesis − ∞ , 1 ]")[$( "−" ∞ , 1 ]$]. The symbol #math.equation(block: false, alt: "− ∞")[$"−" ∞$] is read as ‘negative infinity’. shows both the number line and interval notation. #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket.", caption: [The inequality #math.equation(block: false, alt: "x less than or equal to 1")[$x ≤ 1$] is graphed on this number line and written in interval notation.]) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Inequalities, Number Lines, and Interval Notation] #figure(figph[This figure show four number lines, all without tick marks. The inequality x is greater than a is graphed on the first number line, with an open parenthesis at x equals a, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, a comma infinity, parenthesis. The inequality x is greater than or equal to a is graphed on the second number line, with an open bracket at x equals a, and a red line extending to the right of the bracket. The inequality is also written in interval notation as bracket, a comma infinity, parenthesis. The inequality x is less than a is graphed on the third number line, with an open parenthesis at x equals a, and a red line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma a, parenthesis. The inequality x is less than or equal to a is graphed on the last number line, with an open bracket at x equals a, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma a, bracket.], alt: "This figure show four number lines, all without tick marks. The inequality x is greater than a is graphed on the first number line, with an open parenthesis at x equals a, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, a comma infinity, parenthesis. The inequality x is greater than or equal to a is graphed on the second number line, with an open bracket at x equals a, and a red line extending to the right of the bracket. The inequality is also written in interval notation as bracket, a comma infinity, parenthesis. The inequality x is less than a is graphed on the third number line, with an open parenthesis at x equals a, and a red line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma a, parenthesis. The inequality x is less than or equal to a is graphed on the last number line, with an open bracket at x equals a, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma a, bracket.", caption: none) ] Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown. #figure(figph[This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads: “Both have a left parenthesis.” The second number line is a graph of x is greater than or equal to a, and the interval notation is bracket, a comma infinity, parenthesis. The text below reads: “Both have a left bracket.” The third number line is a graph of x is less than a, and the interval notation is parenthesis, negative infinity comma a, parenthesis. The text below reads: “Both have a right parenthesis.” The last number line is a graph of x is less than or equal to a, and the interval notation is parenthesis, negative infinity comma a, bracket. The text below reads: “Both have a right bracket.”], alt: "This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads: “Both have a left parenthesis.” The second number line is a graph of x is greater than or equal to a, and the interval notation is bracket, a comma infinity, parenthesis. The text below reads: “Both have a left bracket.” The third number line is a graph of x is less than a, and the interval notation is parenthesis, negative infinity comma a, parenthesis. The text below reads: “Both have a right parenthesis.” The last number line is a graph of x is less than or equal to a, and the interval notation is parenthesis, negative infinity comma a, bracket. The text below reads: “Both have a right bracket.”", caption: [The notation for inequalities on a number line and in interval notation use similar symbols to express the endpoints of intervals.]) #examplebox("Example 2")[][ Graph on the number line and write in interval notation. ⓐ #math.equation(block: false, alt: "x greater than or equal to −3")[$x ≥ −3$] ⓑ #math.equation(block: false, alt: "x less than 2.5")[$x < 2.5$] ⓒ #math.equation(block: false, alt: "x less than or equal to minus the fraction 3 over 5")[$x ≤ − frac(3, 5)$] #solutionbox[ + ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the mathematical inequality 'x is greater than or equal to -3' in black text against a plain white background.], alt: "The image displays the mathematical inequality 'x is greater than or equal to -3' in black text against a plain white background.", caption: none)]), [Shade to the right of #math.equation(block: false, alt: "−3")[$−3$], and put a bracket at #math.equation(block: false, alt: "−3")[$−3$].], [#figure(figph[A number line graph representing the inequality x is greater than or equal to -3. A thick blue line starts at -3 with a closed point and extends to the right, towards positive infinity, indicating all numbers greater than or equal to -3.], alt: "A number line graph representing the inequality x is greater than or equal to -3. A thick blue line starts at -3 with a closed point and extends to the right, towards positive infinity, indicating all numbers greater than or equal to -3.", caption: none)], [Write in interval notation.], [#figure(figph[The image displays the mathematical interval notation \[-3, ', representing all real numbers greater than or equal to -3.], alt: "The image displays the mathematical interval notation [-3, ', representing all real numbers greater than or equal to -3.", caption: none)], )) + ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the mathematical inequality 'x \< 2.5' in black text against a plain white background.], alt: "The image displays the mathematical inequality 'x < 2.5' in black text against a plain white background.", caption: none)]), [Shade to the left of #math.equation(block: false, alt: "2.5")[$2.5$], and put a parenthesis at #math.equation(block: false, alt: "2.5")[$2.5$].], [#figure(figph[A number line showing labeled points at 0, 1, 2, 2.5, and 3.], alt: "A number line showing labeled points at 0, 1, 2, 2.5, and 3.", caption: none)], [Write in interval notation.], [#figure(figph[A mathematical notation showing an open interval from negative infinity to 2.5, represented as (-∞, 2.5), typically indicating a set of real numbers less than 2.5.], alt: "A mathematical notation showing an open interval from negative infinity to 2.5, represented as (-∞, 2.5), typically indicating a set of real numbers less than 2.5.", caption: none)], )) + ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical inequality states 'x is less than or equal to -3/5'.], alt: "A mathematical inequality states 'x is less than or equal to -3/5'.", caption: none)]), [Shade to the left of #math.equation(block: false, alt: "minus the fraction 3 over 5")[$− frac(3, 5)$], and put a bracket at #math.equation(block: false, alt: "minus the fraction 3 over 5")[$− frac(3, 5)$].], [#figure(figph[A number line showing the interval from negative infinity to -3/5, not including -3/5. This is represented by a blue line with an arrow to the left and an open bracket at -3/5.], alt: "A number line showing the interval from negative infinity to -3/5, not including -3/5. This is represented by a blue line with an arrow to the left and an open bracket at -3/5.", caption: none)], [Write in interval notation.], [#figure(figph[A mathematical interval notation is displayed, showing an open parenthesis followed by negative infinity, a comma, then negative three-fifths, and finally a closed square bracket. It represents the interval from negative infinity to -3/5, inclusive of -3/5.], alt: "A mathematical interval notation is displayed, showing an open parenthesis followed by negative infinity, a comma, then negative three-fifths, and finally a closed square bracket. It represents the interval from negative infinity to -3/5, inclusive of -3/5.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph on the number line and write in interval notation: ⓐ #math.equation(block: false, alt: "x greater than 2")[$x > 2$] ⓑ #math.equation(block: false, alt: "x less than or equal to minus 1.5")[$x ≤ − 1.5$] ⓒ #math.equation(block: false, alt: "x greater than or equal to the fraction 3 over 4")[$x ≥ frac(3, 4)$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 2 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 2 comma infinity, parenthesis.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 1.5, bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 1.5, bracket.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3/4 is graphed on the number line, with an open bracket at x equals 3/4, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 3/4 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3/4 is graphed on the number line, with an open bracket at x equals 3/4, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 3/4 comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph on the number line and write in interval notation: ⓐ #math.equation(block: false, alt: "x less than or equal to minus 4")[$x ≤ − 4$] ⓑ #math.equation(block: false, alt: "x greater than or equal to 0.5")[$x ≥ 0.5$] ⓒ #math.equation(block: false, alt: "x less than minus the fraction 2 over 3")[$x < − frac(2, 3)$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 4 is graphed on the number line, with an open bracket at x equals negative 4, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 4, bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 4 is graphed on the number line, with an open bracket at x equals negative 4, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 4, bracket.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 0.5 is graphed on the number line, with an open bracket at x equals 0.5, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, o.5 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 0.5 is graphed on the number line, with an open bracket at x equals 0.5, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, o.5 comma infinity, parenthesis.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 2/3 is graphed on the number line, with an open parenthesis at x equals negative 2/3, and a dark line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 2/3, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 2/3 is graphed on the number line, with an open parenthesis at x equals negative 2/3, and a dark line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 2/3, parenthesis.", caption: none) ] ] === Solve Inequalities using the Subtraction and Addition Properties of Inequality The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Equality] #math.equation(block: true, alt: "Subtraction Property of Equality, Addition Property of Equality; For any numbers a , b , and c ,, For any numbers a , b , and c ,; if, a, equals, b ,; then, a minus c, equals, b minus c ., if, a, equals, b ,; then, a plus c, equals, b plus c .")[$"Subtraction Property of Equality" & & & "Addition Property of Equality" \ "For any numbers" #h(0.2em) a , b , "and" #h(0.2em) c , & & & "For any numbers" #h(0.2em) a , b , "and" #h(0.2em) c , \ "if" & a & = & b , \ "then" & a − c & = & b − c . & & & "if" & a & = & b , \ "then" & a + c & = & b + c .$] ] Similar properties hold true for inequalities. #figure(table( columns: 2, align: left, inset: 6pt, table.header([For example, we know that −4 is less than 2.], [#figure(figph[A mathematical inequality displays the expression '-4 \< 2' against a white background, signifying that negative four is less than two.], alt: "A mathematical inequality displays the expression '-4 < 2' against a white background, signifying that negative four is less than two.", caption: none)]), [If we subtract 5 from both quantities, is the #linebreak() left side still less than the right side?], [#figure(figph[A mathematical sequence showing '-4 - 5 ? 2 - 5', with a red question mark in the middle.], alt: "A mathematical sequence showing '-4 - 5 ? 2 - 5', with a red question mark in the middle.", caption: none)], [We get −9 on the left and −3 on the right.], [#figure(figph[The inequality -9 ? -3 asks what operator goes between the two negative numbers. The answer is \< (less than).], alt: "The inequality -9 ? -3 asks what operator goes between the two negative numbers. The answer is < (less than).", caption: none)], [And we know −9 is less than −3.], [#figure(figph[The image displays a mathematical inequality, -9 \< -3, which correctly states that negative nine is less than negative three.], alt: "The image displays a mathematical inequality, -9 < -3, which correctly states that negative nine is less than negative three.", caption: none)], [], [#strong[The inequality sign stayed the same.]], )) Similarly we could show that the inequality also stays the same for addition. This leads us to the Subtraction and Addition Properties of Inequality. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Inequality] #math.equation(block: true, alt: "Subtraction Property of Inequality, Addition Property of Inequality; For any numbers a , b , and c ,, For any numbers a , b , and c ,; if, a, less than, b; then, a minus c, less than, b minus c .; if, a, greater than, b; then, a minus c, greater than, b minus c ., if, a, less than, b; then, a plus c, less than, b plus c .; if, a, greater than, b; then, a plus c, greater than, b plus c .")[$"Subtraction Property of Inequality" & & & "Addition Property of Inequality" \ "For any numbers" #h(0.2em) a , b , "and" #h(0.2em) c , & & & "For any numbers" #h(0.2em) a , b , "and" #h(0.2em) c , \ & & "if" & a & < & b \ & & "then" & a − c & < & b − c . \ \ & & "if" & a & > & b \ & & "then" & a − c & > & b − c . & & & & & "if" & a & < & b \ & & "then" & a + c & < & b + c . \ \ & & "if" & a & > & b \ & & "then" & a + c & > & b + c .$] ] We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality #math.equation(block: false, alt: "x plus 5 greater than 9")[$x + 5 > 9$], the steps would look like this: #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "x plus 5 greater than 9")[$x + 5 > 9$]]), [Subtract 5 from both sides to isolate #math.equation(block: false, alt: "x")[$x$].], [#math.equation(block: false, alt: "x plus 5 minus 5 greater than 9 minus 5")[$x + 5 − 5 > 9 − 5$]], [Simplify.], [#math.equation(block: false, alt: "x greater than 4")[$x > 4$]], )) Any number greater than 4 is a solution to this inequality. #examplebox("Example 3")[][ Solve the inequality #math.equation(block: false, alt: "n minus the fraction 1 over 2 less than or equal to the fraction 5 over 8")[$n − frac(1, 2) ≤ frac(5, 8)$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical inequality displays 'n - 1/2 is less than or equal to 5/8' centered on a white background.], alt: "A mathematical inequality displays 'n - 1/2 is less than or equal to 5/8' centered on a white background.", caption: none)]), [Add #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] to both sides of the inequality.], [#figure(figph[A mathematical inequality is shown where 'n minus 1/2 plus 1/2 is less than or equal to 5/8 plus 1/2' is written.], alt: "A mathematical inequality is shown where 'n minus 1/2 plus 1/2 is less than or equal to 5/8 plus 1/2' is written.", caption: none)], [Simplify.], [#figure(figph[A mathematical inequality shows 'n' is less than or equal to the fraction 9/8, presented in black font on a white background.], alt: "A mathematical inequality shows 'n' is less than or equal to the fraction 9/8, presented in black font on a white background.", caption: none)], [Graph the solution on the number line.], [#figure(figph[A number line shows the interval x is less than or equal to 2 1/8. The line is shaded from negative infinity up to and including the point 2 1/8 (labeled as 1 9/8).], alt: "A number line shows the interval x is less than or equal to 2 1/8. The line is shaded from negative infinity up to and including the point 2 1/8 (labeled as 1 9/8).", caption: none)], [Write the solution in interval notation.], [#math.equation(block: false, alt: "open parenthesis minus infinity , the fraction 9 over 8 ]")[$( − ∞ , frac(9, 8) ]$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "p minus the fraction 3 over 4 greater than or equal to the fraction 1 over 6")[$p − frac(3, 4) ≥ frac(1, 6)$] #solutionbox[ #figure(figph[This figure shows the inequality p is greater than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at p equals 11/12, and a dark line extends to the right from 11/12. Below the number line is the solution written in interval notation: bracket, 11/12 comma infinity, parenthesis.], alt: "This figure shows the inequality p is greater than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at p equals 11/12, and a dark line extends to the right from 11/12. Below the number line is the solution written in interval notation: bracket, 11/12 comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "r minus the fraction 1 over 3 less than or equal to the fraction 7 over 12")[$r − frac(1, 3) ≤ frac(7, 12)$] #solutionbox[ #figure(figph[This figure shows the inequality r is less than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at r equals 11/12, and a dark line extends to the left from 11/12. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/12, bracket.], alt: "This figure shows the inequality r is less than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at r equals 11/12, and a dark line extends to the left from 11/12. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/12, bracket.", caption: none) ] ] === Solve Inequalities using the Division and Multiplication Properties of Inequality The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0). #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Equality] #math.equation(block: true, alt: "Division Property of Equality, Multiplication Property of Equality; For any numbers a , b , c , and c not equal to 0 ,, For any real numbers a , b , c ,; if, a, equals, b ,; then, the fraction a over c, equals, the fraction b over c ., if, a, equals, b ,; then, a c, equals, b c .")[$"Division Property of Equality" & & & "Multiplication Property of Equality" \ "For any numbers" #h(0.2em) a , b , c , "and" #h(0.2em) c ≠ 0 , & & & "For any real numbers" #h(0.2em) a , b , c , \ #h(1em) "if" & a & = & b , \ "then" & frac(a, c) & = & frac(b, c) . & & & #h(1em) "if" & a & = & b , \ "then" & a c & = & b c .$] ] Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant? Consider some numerical examples. #figure(table( columns: 4, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays a simple mathematical inequality: 10 is less than 15, written as '10 \< 15' in a clear, digital font against a white background.], alt: "The image displays a simple mathematical inequality: 10 is less than 15, written as '10 < 15' in a clear, digital font against a white background.", caption: none)], [], [#figure(figph[The image displays a mathematical inequality '10 \< 15', indicating that the number 10 is less than the number 15. The text is in a clear, dark gray font against a white background.], alt: "The image displays a mathematical inequality '10 < 15', indicating that the number 10 is less than the number 15. The text is in a clear, dark gray font against a white background.", caption: none)]), [Divide both sides by 5.], [#figure(figph[A math problem showing two fractions, 10/5 and 15/5, with a red question mark between them, implying a comparison or operation needs to be determined.], alt: "A math problem showing two fractions, 10/5 and 15/5, with a red question mark between them, implying a comparison or operation needs to be determined.", caption: none)], [Multiply both sides by 5.], [#figure(figph[A mathematical equation shows 10(5) ? 15(5), questioning the relationship between 50 and 75.], alt: "A mathematical equation shows 10(5) ? 15(5), questioning the relationship between 50 and 75.", caption: none)], [Simplify.], [#figure(figph[The image displays the numbers '2' and '3' with a red question mark between them, suggesting an unknown operation or value in a numerical sequence or problem.], alt: "The image displays the numbers '2' and '3' with a red question mark between them, suggesting an unknown operation or value in a numerical sequence or problem.", caption: none)], [], [#figure(figph[A question about the relationship between 50 and 75, with a red question mark emphasizing the query.], alt: "A question about the relationship between 50 and 75, with a red question mark emphasizing the query.", caption: none)], [Fill in the inequality signs.], [#figure(figph[The numbers 2 and 3 are displayed on a white background, with a red less-than symbol (\<) between them, visually representing the inequality 2 \< 3.], alt: "The numbers 2 and 3 are displayed on a white background, with a red less-than symbol (<) between them, visually representing the inequality 2 < 3.", caption: none)], [], [#figure(figph[The image displays the mathematical inequality '50 \< 75' in black font against a white background, with the less-than symbol appearing in a slightly reddish hue.], alt: "The image displays the mathematical inequality '50 < 75' in black font against a white background, with the less-than symbol appearing in a slightly reddish hue.", caption: none)], )) #math.equation(block: true, alt: "The inequality signs stayed the same.")[$"The inequality signs stayed the same."$]Does the inequality stay the same when we divide or multiply by a negative number? #figure(table( columns: 4, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the mathematical inequality '10 \< 15' in black font against a white background, indicating that ten is less than fifteen.], alt: "The image displays the mathematical inequality '10 < 15' in black font against a white background, indicating that ten is less than fifteen.", caption: none)], [], [#figure(figph[A mathematical expression '10 \< 15' (ten is less than fifteen) is displayed in black text against a white background.], alt: "A mathematical expression '10 < 15' (ten is less than fifteen) is displayed in black text against a white background.", caption: none)]), [Divide both sides by −5.], [#figure(figph[A mathematical problem comparing two fractions, 10/-5 and 15/-5, with a red question mark positioned between them, indicating the need for a relational operator.], alt: "A mathematical problem comparing two fractions, 10/-5 and 15/-5, with a red question mark positioned between them, indicating the need for a relational operator.", caption: none)], [Multiply both sides by −5.], [#figure(figph[A math problem comparing 10 multiplied by -5 with 15 multiplied by -5, featuring a red question mark between them. The correct comparison is 10(-5) \> 15(-5) because -50 is greater than -75.], alt: "A math problem comparing 10 multiplied by -5 with 15 multiplied by -5, featuring a red question mark between them. The correct comparison is 10(-5) > 15(-5) because -50 is greater than -75.", caption: none)], [Simplify.], [#figure(figph[A math problem asking to compare -2 and -3, with a red question mark between them. The correct comparison is -2 \> -3.], alt: "A math problem asking to compare -2 and -3, with a red question mark between them. The correct comparison is -2 > -3.", caption: none)], [], [#figure(figph[The image displays the numbers -50 and -75 separated by a red question mark, indicating a comparison needs to be made. The correct operator to place in the question mark's position is '\>' (greater than).], alt: "The image displays the numbers -50 and -75 separated by a red question mark, indicating a comparison needs to be made. The correct operator to place in the question mark's position is '>' (greater than).", caption: none)], [Fill in the inequality signs.], [#figure(figph[A mathematical inequality displays '-2 \> -3' in black text with a red greater-than sign, against a white background, demonstrating that negative two is greater than negative three.], alt: "A mathematical inequality displays '-2 > -3' in black text with a red greater-than sign, against a white background, demonstrating that negative two is greater than negative three.", caption: none)], [], [#figure(figph[A mathematical inequality is shown, stating that -50 is greater than -75. The numbers -50 and -75 are in gray, and the greater than symbol (\>) is in red, all against a white background.], alt: "A mathematical inequality is shown, stating that -50 is greater than -75. The numbers -50 and -75 are in gray, and the greater than symbol (>) is in red, all against a white background.", caption: none)], )) #math.equation(block: true, alt: "The inequality signs reversed their direction.")[$"The inequality signs reversed their direction."$]When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses. Here are the Division and Multiplication Properties of Inequality for easy reference. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Division and Multiplication Properties of Inequality] #math.equation(block: true, alt: "For any real numbers a , b , c; if a less than b and c greater than 0 , then the fraction a over c less than the fraction b over c and a c less than b c .; if a greater than b and c greater than 0 , then the fraction a over c greater than the fraction b over c and a c greater than b c .; if a less than b and c less than 0 , then the fraction a over c greater than the fraction b over c and a c greater than b c .; if a greater than b and c less than 0 , then the fraction a over c less than the fraction b over c and a c less than b c .")[$"For any real numbers" #h(0.2em) a , b , c \ \ \ \ #h(1em) "if" #h(0.2em) a < b #h(0.2em) "and" #h(0.2em) c > 0 , "then" #h(1em) frac(a, c) < frac(b, c) #h(0.3em) "and" #h(0.2em) a c < b c . \ #h(1em) "if" #h(0.2em) a > b #h(0.2em) "and" #h(0.2em) c > 0 , "then" #h(1em) frac(a, c) > frac(b, c) #h(0.3em) "and" #h(0.2em) a c > b c . \ #h(1em) "if" #h(0.2em) a < b #h(0.2em) "and" #h(0.2em) c < 0 , "then" #h(1em) frac(a, c) > frac(b, c) #h(0.3em) "and" #h(0.2em) a c > b c . \ #h(1em) "if" #h(0.2em) a > b #h(0.2em) "and" #h(0.2em) c < 0 , "then" #h(1em) frac(a, c) < frac(b, c) #h(0.3em) "and" #h(0.2em) a c < b c .$] ] When we #strong[divide or multiply] an inequality by a: - #strong[positive] number, the inequality stays the #strong[same]. - #strong[negative] number, the inequality #strong[reverses]. #examplebox("Example 4")[][ Solve the inequality #math.equation(block: false, alt: "7 y less than ​ ​ 42")[$7 y < "​" "​" 42$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical inequality, '7y \< 42', is shown centered on a plain white background.], alt: "A mathematical inequality, '7y < 42', is shown centered on a plain white background.", caption: none)]), [Divide both sides of the inequality by 7. #linebreak() Since #math.equation(block: false, alt: "7 greater than 0")[$7 > 0$], the inequality stays the same.], [#figure(figph[An image displays the mathematical inequality 7y/7 \< 42/7. Simplifying both sides, the inequality becomes y \< 6. This represents a linear inequality where the variable y is less than 6.], alt: "An image displays the mathematical inequality 7y/7 < 42/7. Simplifying both sides, the inequality becomes y < 6. This represents a linear inequality where the variable y is less than 6.", caption: none)], [Simplify.], [#figure(figph[The image displays the mathematical inequality y \< 6, indicating that the variable y is less than 6. The inequality is written in black text on a plain white background.], alt: "The image displays the mathematical inequality y < 6, indicating that the variable y is less than 6. The inequality is written in black text on a plain white background.", caption: none)], [Graph the solution on the number line.], [#figure(figph[A number line illustrates the inequality x \< 6, with an open circle at 6 and a bold line extending to the left, indicating all numbers less than 6. The line is marked with integers 4, 5, 6, and 7.], alt: "A number line illustrates the inequality x < 6, with an open circle at 6 and a bold line extending to the left, indicating all numbers less than 6. The line is marked with integers 4, 5, 6, and 7.", caption: none)], [Write the solution in interval notation.], [#figure(figph[A mathematical interval notation is displayed, showing '(-∞, 6)' on a white background, representing all real numbers less than 6, excluding 6 itself.], alt: "A mathematical interval notation is displayed, showing '(-∞, 6)' on a white background, representing all real numbers less than 6, excluding 6 itself.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "c minus 8 greater than 0")[$c − 8 > 0$] #solutionbox[ #math.equation(block: true, alt: "c greater than 8")[$c > 8$] #linebreak() #figure(figph[This figure is a number line ranging from 6 to 10 with tick marks for each integer. The inequality c is greater than 8 is graphed on the number line, with an open parenthesis at c equals 8, and a dark line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from 6 to 10 with tick marks for each integer. The inequality c is greater than 8 is graphed on the number line, with an open parenthesis at c equals 8, and a dark line extending to the right of the parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "12 d less than or equal to ​ 60")[$12 d ≤ "​" 60$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis -∞ , 5 ]")[$( -∞ , 5 ]$] #linebreak() #figure(figph[This figure is a number line ranging from 3 to 7 with tick marks for each integer. The inequality d is less than or equal to 5 is graphed on the number line, with an open bracket at d equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.], alt: "This figure is a number line ranging from 3 to 7 with tick marks for each integer. The inequality d is less than or equal to 5 is graphed on the number line, with an open bracket at d equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.", caption: none) ] ] #examplebox("Example 5")[][ Solve the inequality #math.equation(block: false, alt: "−10 a greater than or equal to 50")[$−10 a ≥ 50$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical inequality is shown in black text on a white background, which reads '-10a '\>=], alt: "A mathematical inequality is shown in black text on a white background, which reads '-10a '>=", caption: none)]), [Divide both sides of the inequality by −10. #linebreak() Since #math.equation(block: false, alt: "−10 less than 0")[$−10 < 0$], the inequality reverses.], [#figure(figph[Mathematical inequality: -10a/-10 \<= 50/-10, demonstrating division by a negative number on both sides.], alt: "Mathematical inequality: -10a/-10 <= 50/-10, demonstrating division by a negative number on both sides.", caption: none)], [Simplify.], [#figure(figph[The mathematical inequality a 'less than or equal to' -5 is displayed on a white background.], alt: "The mathematical inequality a 'less than or equal to' -5 is displayed on a white background.", caption: none)], [Graph the solution on the number line.], [#figure(figph[A number line shows an inequality. A solid line starts at -5 and extends infinitely to the left, indicated by an arrow, representing all numbers less than or equal to -5. The numbers -7, -6, -5, and -4 are marked on the line.], alt: "A number line shows an inequality. A solid line starts at -5 and extends infinitely to the left, indicated by an arrow, representing all numbers less than or equal to -5. The numbers -7, -6, -5, and -4 are marked on the line.", caption: none)], [Write the solution in interval notation.], [#figure(figph[The image displays the mathematical interval notation '(-∞, -5\]', indicating all real numbers from negative infinity up to and including -5.], alt: "The image displays the mathematical interval notation '(-∞, -5]', indicating all real numbers from negative infinity up to and including -5.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "−8 q less than 32")[$−8 q < 32$] #solutionbox[ #math.equation(block: true, alt: "q greater than −4")[$q > −4$] #linebreak() #figure(figph[This figure is a number line ranging from negative 6 to negative 3 with tick marks for each integer. The inequality q is greater than negative 4 is graphed on the number line, with an open parenthesis at q equals negative 4, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 4 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 6 to negative 3 with tick marks for each integer. The inequality q is greater than negative 4 is graphed on the number line, with an open parenthesis at q equals negative 4, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 4 comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "−7 r less than or equal to ​ minus 70")[$−7 r ≤ "​" − 70$] #solutionbox[ #figure(figph[This figure is a number line ranging from 9 to 13 with tick marks for each integer. The inequality r is greater than or equal to 10 is graphed on the number line, with an open bracket at r equals 10, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 10 comma infinity, parenthesis.], alt: "This figure is a number line ranging from 9 to 13 with tick marks for each integer. The inequality r is greater than or equal to 10 is graphed on the number line, with an open bracket at r equals 10, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 10 comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solving Inequalities] Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left. #math.equation(block: true, alt: "x greater than a has the same meaning as a less than x")[$& & x > a #h(0.2em) "has the same meaning as" #h(0.2em) a < x$]Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.” ] #examplebox("Example 6")[][ Solve the inequality #math.equation(block: false, alt: "−20 less than the fraction 4 over 5 u")[$−20 < frac(4, 5) u$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical inequality is shown, displaying -20 is less than (4/5)u.], alt: "A mathematical inequality is shown, displaying -20 is less than (4/5)u.", caption: none)]), [Multiply both sides of the inequality by #math.equation(block: false, alt: "the fraction 5 over 4")[$frac(5, 4)$]. #linebreak() Since #math.equation(block: false, alt: "the fraction 5 over 4 greater than 0")[$frac(5, 4) > 0$], the inequality stays the same.], [#figure(figph[A mathematical inequality shows five-fourths times negative twenty is less than five-fourths times the product of four-fifths and u.], alt: "A mathematical inequality shows five-fourths times negative twenty is less than five-fourths times the product of four-fifths and u.", caption: none)], [Simplify.], [#figure(figph[A mathematical inequality, '-25 \< u', is displayed in black text on a plain white background.], alt: "A mathematical inequality, '-25 < u', is displayed in black text on a plain white background.", caption: none)], [Rewrite the variable on the left.], [#figure(figph[The image displays the mathematical inequality u \> -25 in a simple, clear font on a white background.], alt: "The image displays the mathematical inequality u > -25 in a simple, clear font on a white background.", caption: none)], [Graph the solution on the number line.], [#figure(figph[A number line graph shows an open circle at -25 and a thick line extending to the right with an arrow, indicating all numbers greater than -25.], alt: "A number line graph shows an open circle at -25 and a thick line extending to the right with an arrow, indicating all numbers greater than -25.", caption: none)], [Write the solution in interval notation.], [#figure(figph[A mathematical interval notation showing all real numbers greater than -25, extending to positive infinity: (-25, ).], alt: "A mathematical interval notation showing all real numbers greater than -25, extending to positive infinity: (-25, ).", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "24 less than or equal to the fraction 3 over 8 m")[$24 ≤ frac(3, 8) m$] #solutionbox[ #figure(figph[This figure shows the inequality m is greater than or equal to 64. Below this inequality is a number line ranging from 63 to 67 with tick marks for each integer. The inequality m is greater than or equal to 64 is graphed on the number line, with an open bracket at m equals 64, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 64 comma infinity, parenthesis.], alt: "This figure shows the inequality m is greater than or equal to 64. Below this inequality is a number line ranging from 63 to 67 with tick marks for each integer. The inequality m is greater than or equal to 64 is graphed on the number line, with an open bracket at m equals 64, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 64 comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "−24 less than the fraction 4 over 3 n")[$−24 < frac(4, 3) n$] #solutionbox[ #figure(figph[This figure shows the inequality n is greater than negative 18. Below this inequality is a number line ranging from negative 20 to negative 16 with tick marks for each integer. The inequality n is greater than negative 18 is graphed on the number line, with an open parenthesis at n equals negative 18, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 18 comma infinity, parenthesis.], alt: "This figure shows the inequality n is greater than negative 18. Below this inequality is a number line ranging from negative 20 to negative 16 with tick marks for each integer. The inequality n is greater than negative 18 is graphed on the number line, with an open parenthesis at n equals negative 18, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 18 comma infinity, parenthesis.", caption: none) ] ] #examplebox("Example 7")[][ Solve the inequality #math.equation(block: false, alt: "the fraction t over −2 greater than or equal to 8")[$frac(t, −2) ≥ 8$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical inequality is shown, where the variable 't' divided by -2 is greater than or equal to 8. It reads: t/-2 '\>= 8'.], alt: "A mathematical inequality is shown, where the variable 't' divided by -2 is greater than or equal to 8. It reads: t/-2 '>= 8'.", caption: none)]), [Multiply both sides of the inequality by #math.equation(block: false, alt: "−2")[$−2$]. #linebreak() Since #math.equation(block: false, alt: "−2 less than 0")[$−2 < 0$], the inequality reverses.], [#figure(figph[An algebraic inequality is displayed, showing -2 multiplied by the fraction t/-2, which is less than or equal to -2 multiplied by 8.], alt: "An algebraic inequality is displayed, showing -2 multiplied by the fraction t/-2, which is less than or equal to -2 multiplied by 8.", caption: none)], [Simplify.], [#figure(figph[The mathematical inequality 't ' is displayed, followed by the less than or equal to symbol, and then the number -16.], alt: "The mathematical inequality 't ' is displayed, followed by the less than or equal to symbol, and then the number -16.", caption: none)], [Graph the solution on the number line.], [#figure(figph[A number line representing the inequality x \<= -16, with a dark blue ray extending left from -16 and including -16, marked by a closed bracket.], alt: "A number line representing the inequality x <= -16, with a dark blue ray extending left from -16 and including -16, marked by a closed bracket.", caption: none)], [Write the solution in interval notation.], [#figure(figph[The mathematical notation '(-∞, -16\]' is displayed on a white background, representing an interval from negative infinity up to and including -16.], alt: "The mathematical notation '(-∞, -16]' is displayed on a white background, representing an interval from negative infinity up to and including -16.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "the fraction k over −12 less than or equal to 15")[$frac(k, −12) ≤ 15$] #solutionbox[ #figure(figph[This figure shows the inequality k is greater than or equal to negative 180. Below this inequality is a number line ranging from negative 181 to negative 177 with tick marks for each integer. The inequality k is greater than or equal to negative 180 is graphed on the number line, with an open bracket at n equals negative 180, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, negative 180 comma infinity, parenthesis.], alt: "This figure shows the inequality k is greater than or equal to negative 180. Below this inequality is a number line ranging from negative 181 to negative 177 with tick marks for each integer. The inequality k is greater than or equal to negative 180 is graphed on the number line, with an open bracket at n equals negative 180, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, negative 180 comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "the fraction u over −4 greater than or equal to −16")[$frac(u, −4) ≥ −16$] #solutionbox[ #figure(figph[This figure shows the inequality u is less than or equal to 64. Below this inequality is a number line ranging from 62 to 66 with tick marks for each integer. The inequality u is less than or equal to 64 is graphed on the number line, with an open bracket at u equals 64, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 64, bracket.], alt: "This figure shows the inequality u is less than or equal to 64. Below this inequality is a number line ranging from 62 to 66 with tick marks for each integer. The inequality u is less than or equal to 64 is graphed on the number line, with an open bracket at u equals 64, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 64, bracket.", caption: none) ] ] === Solve Inequalities That Require Simplification Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division. #examplebox("Example 8")[][ Solve the inequality #math.equation(block: false, alt: "4 m less than or equal to 9 m plus 17")[$4 m ≤ 9 m + 17$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical inequality problem showing 4m is less than or equal to 9m + 17, written in black text on a white background. This is a common algebra problem.], alt: "A mathematical inequality problem showing 4m is less than or equal to 9m + 17, written in black text on a white background. This is a common algebra problem.", caption: none)]), [Subtract #math.equation(block: false, alt: "9 m")[$9 m$] from both sides to collect the variables on the left.], [#figure(figph[A mathematical inequality shows '4m minus 9m is less than or equal to 9m minus 9m plus 17'.], alt: "A mathematical inequality shows '4m minus 9m is less than or equal to 9m minus 9m plus 17'.", caption: none)], [Simplify.], [#figure(figph[A mathematical inequality is displayed on a white background, which reads '-5m ××\<= 17'.], alt: "A mathematical inequality is displayed on a white background, which reads '-5m ××<= 17'.", caption: none)], [Divide both sides of the inequality by −5, and reverse the inequality.], [#figure(figph[A mathematical inequality shows -5m divided by -5 is greater than or equal to 17 divided by -5, indicating a step in solving for m.], alt: "A mathematical inequality shows -5m divided by -5 is greater than or equal to 17 divided by -5, indicating a step in solving for m.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing the inequality m is greater than or equal to negative seventeen-fifths.], alt: "A mathematical expression showing the inequality m is greater than or equal to negative seventeen-fifths.", caption: none)], [Graph the solution on the number line.], [#figure(figph[A number line graph showing the interval \[-17/5, infinity), with a closed bracket at -17/5 and a shaded arrow extending to the right.], alt: "A number line graph showing the interval [-17/5, infinity), with a closed bracket at -17/5 and a shaded arrow extending to the right.", caption: none)], [Write the solution in interval notation.], [#figure(figph[The mathematical interval \[-17/5, ∞), denoting all real numbers greater than or equal to -17/5 up to positive infinity. The square bracket indicates inclusion of -17/5, while the parenthesis signifies that infinity is not included.], alt: "The mathematical interval [-17/5, ∞), denoting all real numbers greater than or equal to -17/5 up to positive infinity. The square bracket indicates inclusion of -17/5, while the parenthesis signifies that infinity is not included.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality #math.equation(block: false, alt: "3 q greater than or equal to 7 q minus 23")[$3 q " " ≥ " " 7 q " " − " " 23$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(figph[This figure shows the inequality q is less than or equal to 23/4. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality q is less than or equal to 23/4 is graphed on the number line, with an open bracket at q equals 23/4 (written in), and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 23/4, bracket.], alt: "This figure shows the inequality q is less than or equal to 23/4. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality q is less than or equal to 23/4 is graphed on the number line, with an open bracket at q equals 23/4 (written in), and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 23/4, bracket.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality #math.equation(block: false, alt: "6 x less than 10 x plus 19")[$6 x < 10 x + 19$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(figph[This figure shows the inequality x is greater than negative 19/4. Below this inequality is a number line ranging from negative 7 to negative 3, with tick marks for each integer. The inequality x is greater than negative 19/4 is graphed on the number line, with an open parenthesis at x equals negative 19/4 (written in), and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 19/4 comma infinity, parenthesis.], alt: "This figure shows the inequality x is greater than negative 19/4. Below this inequality is a number line ranging from negative 7 to negative 3, with tick marks for each integer. The inequality x is greater than negative 19/4 is graphed on the number line, with an open parenthesis at x equals negative 19/4 (written in), and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 19/4 comma infinity, parenthesis.", caption: none) ] ] #examplebox("Example 9")[][ Solve the inequality #math.equation(block: false, alt: "8 p plus 3 open parenthesis p minus 12 close parenthesis greater than 7 p minus 28")[$8 p + 3 ( p − 12 ) > 7 p − 28$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Simplify each side as much as possible.], [#math.equation(block: false, alt: "8 p plus 3 open parenthesis p minus 12 close parenthesis greater than 7 p minus 28")[$8 p + 3 ( p − 12 ) > 7 p − 28$]]), [Distribute.], [#math.equation(block: false, alt: "8 p plus 3 p minus 36 greater than 7 p minus 28")[$#h(0.6em) 8 p + 3 p − 36 > 7 p − 28$]], [Combine like terms.], [#math.equation(block: false, alt: "11 p minus 36 greater than 7 p minus 28")[$#h(2.3em) 11 p − 36 > 7 p − 28$]], [Subtract #math.equation(block: false, alt: "7 p")[$7 p$] from both sides to collect the variables on the left.], [#math.equation(block: false, alt: "11 p minus 36 minus 7 p greater than 7 p minus 28 minus 7 p")[$#h(2.2em) 11 p − 36 − 7 p > 7 p − 28 − 7 p$]], [Simplify.], [#math.equation(block: false, alt: "4 p minus 36 greater than −28")[$#h(1.2em) 4 p − 36 > −28$]], [Add 36 to both sides to collect the constants on the right.], [#math.equation(block: false, alt: "4 p minus 36 plus 36 greater than −28 plus 36")[$#h(1.3em) 4 p − 36 + 36 > −28 + 36$]], [Simplify.], [#math.equation(block: false, alt: "4 p greater than 8")[$#h(2em) 4 p > 8$]], [Divide both sides of the inequality by 4; the inequality stays the same.], [#math.equation(block: false, alt: "the fraction 4 p over 4 greater than the fraction 8 over 4")[$#h(2em) frac(4 p, 4) > frac(8, 4)$]], [Simplify.], [#math.equation(block: false, alt: "p greater than 2")[$#h(2.5em) p > 2$]], [Graph the solution on the number line.], [#figure(figph[A number line illustrates the inequality x \> 2, represented by an open circle at 2 and a shaded line extending to the right, indicating all values greater than 2.], alt: "A number line illustrates the inequality x > 2, represented by an open circle at 2 and a shaded line extending to the right, indicating all values greater than 2.", caption: none)], [Write the solution in interal notation.], [#math.equation(block: false, alt: "open parenthesis 2 , ∞ close parenthesis")[$( 2 , ∞ )$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality #math.equation(block: false, alt: "9 y plus 2 open parenthesis y plus 6 close parenthesis greater than 5 y minus 24")[$9 y + 2 ( y + 6 ) > 5 y − 24$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(figph[This figure shows the inequality y is greater than negative 6. Below this inequality is a number line ranging from negative 7 to negative 3 with tick marks for each integer. The inequality y is greater than negative 6 is graphed on the number line, with an open parenthesis at y equals negative 6, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 6 comma infinity, parenthesis.], alt: "This figure shows the inequality y is greater than negative 6. Below this inequality is a number line ranging from negative 7 to negative 3 with tick marks for each integer. The inequality y is greater than negative 6 is graphed on the number line, with an open parenthesis at y equals negative 6, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 6 comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality #math.equation(block: false, alt: "6 u plus 8 open parenthesis u minus 1 close parenthesis greater than 10 u plus 32")[$6 u + 8 ( u − 1 ) > 10 u + 32$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(figph[This figure shows the inequality u is greater than 10. Below this inequality is a number line ranging from 9 to 13 with tick marks for each integer. The inequality u is greater than 10 is graphed on the number line, with an open parenthesis at u equals 10, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 10 comma infinity, parenthesis.], alt: "This figure shows the inequality u is greater than 10. Below this inequality is a number line ranging from 9 to 13 with tick marks for each integer. The inequality u is greater than 10 is graphed on the number line, with an open parenthesis at u equals 10, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 10 comma infinity, parenthesis.", caption: none) ] ] Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction. #examplebox("Example 10")[][ Solve the inequality #math.equation(block: false, alt: "8 x minus 2 open parenthesis 5 minus x close parenthesis less than 4 open parenthesis x plus 9 close parenthesis plus 6 x")[$8 x − 2 ( 5 − x ) < 4 ( x + 9 ) + 6 x$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Simplify each side as much as possible.], [#math.equation(block: false, alt: "8 x minus 2 open parenthesis 5 minus x close parenthesis less than 4 open parenthesis x plus 9 close parenthesis plus 6 x")[$8 x − 2 ( 5 − x ) < 4 ( x + 9 ) + 6 x$]]), [Distribute.], [#math.equation(block: false, alt: "8 x minus 10 plus 2 x less than 4 x plus 36 plus 6 x")[$8 x − 10 + 2 x < 4 x + 36 + 6 x$]], [Combine like terms.], [#math.equation(block: false, alt: "10 x minus 10 less than 10 x plus 36")[$10 x − 10 < 10 x + 36$]], [Subtract #math.equation(block: false, alt: "10 x")[$10 x$] from both sides to collect the variables on the left.], [#math.equation(block: false, alt: "10 x minus 10 minus 10 x less than 10 x plus 36 minus 10 x")[$10 x − 10 − 10 x < 10 x + 36 − 10 x$]], [Simplify.], [#math.equation(block: false, alt: "−10 less than 36")[$−10 < 36 #h(0.6em)$]], [The #math.equation(block: false, alt: "x")[$x$]’s are gone, and we have a true statement.], [The inequality is an identity. #linebreak() The solution is all real numbers.], [Graph the solution on the number line.], [#figure(figph[A number line displaying integers from -1 to 2, with arrows indicating it extends infinitely.], alt: "A number line displaying integers from -1 to 2, with arrows indicating it extends infinitely.", caption: none)], [Write the solution in interval notation.], [#math.equation(block: false, alt: "open parenthesis −∞ , ∞ close parenthesis")[$( −∞ , ∞ )$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality #math.equation(block: false, alt: "4 b minus 3 open parenthesis 3 minus b close parenthesis greater than 5 open parenthesis b minus 6 close parenthesis plus 2 b")[$4 b − 3 ( 3 − b ) > 5 ( b − 6 ) + 2 b$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(figph[This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.], alt: "This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality #math.equation(block: false, alt: "9 h minus 7 open parenthesis 2 minus h close parenthesis less than 8 open parenthesis h plus 11 close parenthesis plus 8 h")[$9 h − 7 ( 2 − h ) < 8 ( h + 11 ) + 8 h$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(figph[This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.], alt: "This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.", caption: none) ] ] #examplebox("Example 11")[][ Solve the inequality #math.equation(block: false, alt: "the fraction 1 over 3 a minus the fraction 1 over 8 a greater than the fraction 5 over 24 a ​ plus the fraction 3 over 4")[$frac(1, 3) a − frac(1, 8) a > frac(5, 24) a "​" + frac(3, 4)$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the inequality (1/3)a - (1/8)a \> (5/24)a + 3/4.], alt: "The image displays the inequality (1/3)a - (1/8)a > (5/24)a + 3/4.", caption: none)]), [Multiply both sides by the LCD, 24, to clear the fractions.], [#figure(figph[A mathematical inequality shows 24 multiplied by the difference of one-third a and one-eighth a, which is greater than 24 multiplied by the sum of five twenty-fourths a and three-fourths.], alt: "A mathematical inequality shows 24 multiplied by the difference of one-third a and one-eighth a, which is greater than 24 multiplied by the sum of five twenty-fourths a and three-fourths.", caption: none)], [Simplify.], [#figure(figph[A mathematical inequality is shown, which reads '8a - 3a \> 5a + 18'. This simplifies to '5a \> 5a + 18', and further to '0 \> 18', indicating that the inequality has no solution.], alt: "A mathematical inequality is shown, which reads '8a - 3a > 5a + 18'. This simplifies to '5a > 5a + 18', and further to '0 > 18', indicating that the inequality has no solution.", caption: none)], [Combine like terms.], [#figure(figph[The image displays the mathematical inequality '5a \> 5a + 18' in black text against a white background.], alt: "The image displays the mathematical inequality '5a > 5a + 18' in black text against a white background.", caption: none)], [Subtract #math.equation(block: false, alt: "5 a")[$5 a$] from both sides to collect the variables on the left.], [#figure(figph[A mathematical inequality, 5a - 5a \> 5a - 5a + 18, which simplifies to 0 \> 18. This represents a false statement, as zero is not greater than eighteen.], alt: "A mathematical inequality, 5a - 5a > 5a - 5a + 18, which simplifies to 0 > 18. This represents a false statement, as zero is not greater than eighteen.", caption: none)], [Simplify.], [#figure(figph[The image displays the mathematically false statement '0 \> 18'.], alt: "The image displays the mathematically false statement '0 > 18'.", caption: none)], [The statement is false!], [The inequality is a contradiction.], [], [There is no solution.], [Graph the solution on the number line.], [#figure(figph[A horizontal number line displays integers -1, 0, 1, and 2 with arrows indicating it extends infinitely in both positive and negative directions.], alt: "A horizontal number line displays integers -1, 0, 1, and 2 with arrows indicating it extends infinitely in both positive and negative directions.", caption: none)], [Write the solution in interval notation.], [There is no solution.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality #math.equation(block: false, alt: "the fraction 1 over 4 x minus the fraction 1 over 12 x greater than the fraction 1 over 6 x plus the fraction 7 over 8")[$frac(1, 4) x − frac(1, 12) x > frac(1, 6) x + frac(7, 8)$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(figph[This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement: “No solution.”], alt: "This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement: “No solution.”", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the inequality #math.equation(block: false, alt: "the fraction 2 over 5 z minus the fraction 1 over 3 z less than the fraction 1 over 15 z ​ plus the fraction 3 over 5")[$frac(2, 5) z − frac(1, 3) z < frac(1, 15) z "​" + frac(3, 5)$], graph the solution on the number line, and write the solution in interval notation. #solutionbox[ #figure(figph[This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement: “No solution.”], alt: "This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement: “No solution.”", caption: none) ] ] === Translate to an Inequality and Solve To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious. Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’. shows some common phrases that indicate inequalities. #figure(table( columns: 4, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "greater than")[$>$]], [#math.equation(block: false, alt: "greater than or equal to")[$≥$]], [#math.equation(block: false, alt: "less than")[$<$]], [#math.equation(block: false, alt: "less than or equal to")[$≤$]]), [is greater than], [is greater than or equal to], [is less than], [is less than or equal to], [is more than], [is at least], [is smaller than], [is at most], [is larger than], [is no less than], [has fewer than], [is no more than], [exceeds], [is the minimum], [is lower than], [is the maximum], )) #examplebox("Example 12")[][ Translate and solve. Then write the solution in interval notation and graph on the number line. Twelve times #emph[c] is no more than 96. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Translate.], [#figure(figph[The image shows how the phrase 'Twelve times c is no more than 96' translates into the mathematical inequality '12c \<= 96'. The phrase 'is no more than' is highlighted as corresponding to the '\<=' symbol.], alt: "The image shows how the phrase 'Twelve times c is no more than 96' translates into the mathematical inequality '12c <= 96'. The phrase 'is no more than' is highlighted as corresponding to the '<=' symbol.", caption: none)]), [Solve—divide both sides by 12.], [#figure(figph[A mathematical inequality showing '12c divided by 12 is less than or equal to 96 divided by 12', which simplifies to 'c is less than or equal to 8'.], alt: "A mathematical inequality showing '12c divided by 12 is less than or equal to 96 divided by 12', which simplifies to 'c is less than or equal to 8'.", caption: none)], [Simplify.], [#figure(figph[A mathematical inequality displaying 'c less than or equal to 8'.], alt: "A mathematical inequality displaying 'c less than or equal to 8'.", caption: none)], [Write in interval notation.], [#figure(figph[A mathematical interval notation is displayed on a white background, showing the set of numbers from negative infinity up to and including eight, written as '(-∞, 8\]'.], alt: "A mathematical interval notation is displayed on a white background, showing the set of numbers from negative infinity up to and including eight, written as '(-∞, 8]'.", caption: none)], [Graph on the number line.], [#figure(figph[A number line from -10 to 10 shows a dark blue shaded region indicating the inequality x \< 8. A thick line with an arrow points left from 8, covering all numbers less than 8.], alt: "A number line from -10 to 10 shows a dark blue shaded region indicating the inequality x < 8. A thick line with an arrow points left from 8, covering all numbers less than 8.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve. Then write the solution in interval notation and graph on the number line. Twenty times #emph[y] is at most 100 #solutionbox[ #figure(figph[This figure shows the inequality 20y is less than or equal to 100, and then its solution: y is less than or equal to 5. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality y is less than or equal to 5 is graphed on the number line, with an open bracket at y equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.], alt: "This figure shows the inequality 20y is less than or equal to 100, and then its solution: y is less than or equal to 5. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality y is less than or equal to 5 is graphed on the number line, with an open bracket at y equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve. Then write the solution in interval notation and graph on the number line. Nine times #emph[z] is no less than 135 #solutionbox[ #figure(figph[This figure shows the inequality 9z is greater than or equal to 135, and then its solution: z is greater than or equal to 15. Below this inequality is a number line ranging from 14 to 18 with tick marks for each integer. The inequality z is greater than or equal to 15 is graphed on the number line, with an open bracket at z equals 15, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 15 comma infinity, parenthesis.], alt: "This figure shows the inequality 9z is greater than or equal to 135, and then its solution: z is greater than or equal to 15. Below this inequality is a number line ranging from 14 to 18 with tick marks for each integer. The inequality z is greater than or equal to 15 is graphed on the number line, with an open bracket at z equals 15, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 15 comma infinity, parenthesis.", caption: none) ] ] #examplebox("Example 13")[][ Translate and solve. Then write the solution in interval notation and graph on the number line. Thirty less than #emph[x] is at least 45. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Translate.], [#figure(figph[The image shows the phrase 'Thirty less than x is at least 45.' and its corresponding algebraic inequality 'x - 30 '\>= ' 45'.], alt: "The image shows the phrase 'Thirty less than x is at least 45.' and its corresponding algebraic inequality 'x - 30 '>= ' 45'.", caption: none)]), [Solve—add 30 to both sides.], [#figure(figph[An algebraic inequality is displayed: x minus 30 plus 30 is greater than or equal to 45 plus 30. The second 30 on the left and the 30 on the right are highlighted in red.], alt: "An algebraic inequality is displayed: x minus 30 plus 30 is greater than or equal to 45 plus 30. The second 30 on the left and the 30 on the right are highlighted in red.", caption: none)], [Simplify.], [#figure(figph[The image displays the mathematical inequality x '\>=' 75, indicating that the variable x is greater than or equal to 75.], alt: "The image displays the mathematical inequality x '>=' 75, indicating that the variable x is greater than or equal to 75.", caption: none)], [Write in interval notation.], [#figure(figph[A mathematical interval notation is displayed, showing a closed bracket before '75' and an open parenthesis before the infinity symbol, representing the interval \[75, oo).], alt: "A mathematical interval notation is displayed, showing a closed bracket before '75' and an open parenthesis before the infinity symbol, representing the interval [75, oo).", caption: none)], [Graph on the number line.], [#figure(figph[Number line showing values greater than or equal to 75. A dark blue arrow starts at 75 and points right, with tick marks for 74, 75, 76, and 77.], alt: "Number line showing values greater than or equal to 75. A dark blue arrow starts at 75 and points right, with tick marks for 74, 75, 76, and 77.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve. Then write the solution in interval notation and graph on the number line. Nineteen less than #emph[p] is no less than 47 #solutionbox[ #figure(figph[This figure shows the inequality p minus 19 is greater than or equal to 47, and then its solution: p is greater than or equal to 66. Below this inequality is a number line ranging from 65 to 69 with tick marks for each integer. The inequality p is greater than or equal to 66 is graphed on the number line, with an open bracket at p equals 66, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 66 comma infinity, parenthesis.], alt: "This figure shows the inequality p minus 19 is greater than or equal to 47, and then its solution: p is greater than or equal to 66. Below this inequality is a number line ranging from 65 to 69 with tick marks for each integer. The inequality p is greater than or equal to 66 is graphed on the number line, with an open bracket at p equals 66, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 66 comma infinity, parenthesis.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve. Then write the solution in interval notation and graph on the number line. Four more than #emph[a] is at most 15. #solutionbox[ #figure(figph[This figure shows the inequality a plus 4 is less than or equal to 15, and then its solution: a is less than or equal to 11. Below this inequality is a number line ranging from 10 to 14 with tick marks for each integer. The inequality a is less than or equal to 11 is graphed on the number line, with an open bracket at a equals 11, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity 11, bracket.], alt: "This figure shows the inequality a plus 4 is less than or equal to 15, and then its solution: a is less than or equal to 11. Below this inequality is a number line ranging from 10 to 14 with tick marks for each integer. The inequality a is less than or equal to 11 is graphed on the number line, with an open bracket at a equals 11, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity 11, bracket.", caption: none) ] ] === Key Concepts - #strong[Subtraction Property of Inequality] #linebreak() For any numbers a, b, and c, #linebreak() if #math.equation(block: false, alt: "a less than b")[$a < b$] then #math.equation(block: false, alt: "a minus c less than b minus c")[$a − c < b − c$] and #linebreak() if #math.equation(block: false, alt: "a greater than b")[$a > b$] then #math.equation(block: false, alt: "a minus c greater than b minus c .")[$a − c > b − c .$] - #strong[Addition Property of Inequality] #linebreak() For any numbers a, b, and c, #linebreak() if #math.equation(block: false, alt: "a less than b")[$a < b$] then #math.equation(block: false, alt: "a plus c less than b plus c")[$a + c < b + c$] and #linebreak() if #math.equation(block: false, alt: "a greater than b")[$a > b$] then #math.equation(block: false, alt: "a plus c greater than b plus c .")[$a + c > b + c .$] - #strong[Division and Multiplication Properties of Inequalit]#strong[y] #linebreak() For any numbers a, b, and c, #linebreak() if #math.equation(block: false, alt: "a less than b")[$a < b$] and #math.equation(block: false, alt: "c greater than 0")[$c > 0$], then #math.equation(block: false, alt: "the fraction a over c less than the fraction b over c")[$frac(a, c) < frac(b, c)$] and #math.equation(block: false, alt: "a c greater than b c")[$a c > b c$]. #linebreak() if #math.equation(block: false, alt: "a greater than b")[$a > b$] and #math.equation(block: false, alt: "c greater than 0")[$c > 0$], then #math.equation(block: false, alt: "the fraction a over c greater than the fraction b over c")[$frac(a, c) > frac(b, c)$] and #math.equation(block: false, alt: "a c greater than b c")[$a c > b c$]. #linebreak() if #math.equation(block: false, alt: "a less than b")[$a < b$] and #math.equation(block: false, alt: "c less than 0")[$c < 0$], then #math.equation(block: false, alt: "the fraction a over c greater than the fraction b over c")[$frac(a, c) > frac(b, c)$] and #math.equation(block: false, alt: "a c greater than b c")[$a c > b c$]. #linebreak() if #math.equation(block: false, alt: "a greater than b")[$a > b$] and #math.equation(block: false, alt: "c less than 0")[$c < 0$], then #math.equation(block: false, alt: "the fraction a over c less than the fraction b over c")[$frac(a, c) < frac(b, c)$] and #math.equation(block: false, alt: "a c less than b c")[$a c < b c$]. #linebreak() - When we #strong[divide or multiply] an inequality by a: - #strong[positive] number, the inequality stays the #strong[same]. - #strong[negative] number, the inequality #strong[reverses]. === Section Exercises ==== Practice Makes Perfect #strong[Graph Inequalities on the Number Line] In the following exercises, graph each inequality on the number line. ⓐ #math.equation(block: false, alt: "x less than or equal to minus 2")[$x ≤ − 2$] #linebreak() ⓑ #math.equation(block: false, alt: "x greater than minus 1")[$x > − 1$]ⓒ #math.equation(block: false, alt: "x less than 0")[$x < 0$] ⓐ #math.equation(block: false, alt: "x greater than 1")[$x > 1$]ⓑ #math.equation(block: false, alt: "x less than minus 2")[$x < − 2$] #linebreak() ⓒ #math.equation(block: false, alt: "x greater than or equal to −3")[$x ≥ −3$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 1 is graphed on the number line, with an open parenthesis at x equals 1, and a dark line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 1 is graphed on the number line, with an open parenthesis at x equals 1, and a dark line extending to the right of the parenthesis.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 2 is graphed on the number line, with an open parenthesis at x equals negative 2, and a dark line extending to the left of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 2 is graphed on the number line, with an open parenthesis at x equals negative 2, and a dark line extending to the left of the parenthesis.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 3 is graphed on the number line, with an open bracket at x equals negative 3, and a dark line extending to the right of the bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 3 is graphed on the number line, with an open bracket at x equals negative 3, and a dark line extending to the right of the bracket.", caption: none) ] ⓐ #math.equation(block: false, alt: "x greater than or equal to −3")[$x ≥ −3$]ⓑ #math.equation(block: false, alt: "x less than 4")[$x < 4$]ⓒ #math.equation(block: false, alt: "x less than or equal to minus 2")[$x ≤ − 2$] ⓐ #math.equation(block: false, alt: "x less than or equal to 0")[$x ≤ 0$]ⓑ #math.equation(block: false, alt: "x greater than minus 4")[$x > − 4$]ⓒ #math.equation(block: false, alt: "x greater than or equal to −1")[$x ≥ −1$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 0 is graphed on the number line, with an open bracket at x equals 0, and a dark line extending to the left of the bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 0 is graphed on the number line, with an open bracket at x equals 0, and a dark line extending to the left of the bracket.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 4 is graphed on the number line, with an open parenthesis at x equals negative 4, and a dark line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 4 is graphed on the number line, with an open parenthesis at x equals negative 4, and a dark line extending to the right of the parenthesis.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the right of the bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the right of the bracket.", caption: none) ] In the following exercises, graph each inequality on the number line and write in interval notation. ⓐ #math.equation(block: false, alt: "x less than minus 2")[$x < − 2$]ⓑ #math.equation(block: false, alt: "x greater than or equal to −3.5")[$x ≥ −3.5$]ⓒ #math.equation(block: false, alt: "x less than or equal to the fraction 2 over 3")[$x ≤ frac(2, 3)$] ⓐ #math.equation(block: false, alt: "x greater than 3")[$x > 3$]ⓑ #math.equation(block: false, alt: "x less than or equal to minus 0.5")[$x ≤ − 0.5$]ⓒ #math.equation(block: false, alt: "x greater than or equal to the fraction 1 over 3")[$x ≥ frac(1, 3)$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark line extending to the right of the parenthsis. Below the number line is the solution written in interval notation: parenthesis, 3 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark line extending to the right of the parenthsis. Below the number line is the solution written in interval notation: parenthesis, 3 comma infinity, parenthesis.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 0.5 is graphed on the number line, with an open bracket at x equals negative 0.5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 0.5, bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 0.5 is graphed on the number line, with an open bracket at x equals negative 0.5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 0.5, bracket.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 1/3 is graphed on the number line, with an open bracket at x equals 1/3 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 1/3 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 1/3 is graphed on the number line, with an open bracket at x equals 1/3 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 1/3 comma infinity, parenthesis.", caption: none) ] ⓐ #math.equation(block: false, alt: "x greater than or equal to −4")[$x ≥ −4$]ⓑ #math.equation(block: false, alt: "x less than 2.5")[$x < 2.5$]ⓒ #math.equation(block: false, alt: "x greater than minus the fraction 3 over 2")[$x > − frac(3, 2)$] ⓐ #math.equation(block: false, alt: "x less than or equal to 5")[$x ≤ 5$]ⓑ #math.equation(block: false, alt: "x greater than or equal to −1.5")[$x ≥ −1.5$]ⓒ #math.equation(block: false, alt: "x less than minus the fraction 7 over 3")[$x < − frac(7, 3)$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line with tick marks. The inequality x is less than or equal to 5 is graphed on the number line, with an open bracket at x equals 5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 5, bracket.], alt: "This figure is a number line with tick marks. The inequality x is less than or equal to 5 is graphed on the number line, with an open bracket at x equals 5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 5, bracket.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 1.5 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 1.5 comma infinity, parenthesis.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 7/3 is graphed on the number line, with an open parenthesis at x equals negative 7/3 (written in), and a dark line extending to the left of the parenthsis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 7/3, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 7/3 is graphed on the number line, with an open parenthesis at x equals negative 7/3 (written in), and a dark line extending to the left of the parenthsis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 7/3, parenthesis.", caption: none) ] #strong[Solve Inequalities using the Subtraction and Addition Properties of Inequality] In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "n minus 11 less than 33")[$n − 11 < 33$] #math.equation(block: true, alt: "m minus 45 less than or equal to 62")[$m − 45 ≤ 62$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: m is less than or equal to 107. Below this is a number line ranging from 105 to 109 with tick marks for each integer. The inequality x is less than or equal to 107 is graphed on the number line, with an open bracket at x equals 107, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 107, bracket.], alt: "At the top of this figure is the solution to the inequality: m is less than or equal to 107. Below this is a number line ranging from 105 to 109 with tick marks for each integer. The inequality x is less than or equal to 107 is graphed on the number line, with an open bracket at x equals 107, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 107, bracket.", caption: none) ] #math.equation(block: true, alt: "u plus 25 greater than 21")[$u + 25 > 21$] #math.equation(block: true, alt: "v plus 12 greater than 3")[$v + 12 > 3$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: v is greater than negative 9. Below this is a number line ranging from negative 11 to negative 7 with tick marks for each integer. The inequality x is greater than negative 9 is graphed on the number line, with an open parenthesis at x equals negative 9, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative 9 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: v is greater than negative 9. Below this is a number line ranging from negative 11 to negative 7 with tick marks for each integer. The inequality x is greater than negative 9 is graphed on the number line, with an open parenthesis at x equals negative 9, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative 9 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "a plus the fraction 3 over 4 greater than or equal to the fraction 7 over 10")[$a + frac(3, 4) ≥ frac(7, 10)$] #math.equation(block: true, alt: "b plus the fraction 7 over 8 greater than or equal to the fraction 1 over 6")[$b + frac(7, 8) ≥ frac(1, 6)$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: b is greater than or equal to negative 17/24. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The inequality b is greater than or equal to negative 17/24 is graphed on the number line, with an open bracket at b equals negative 17/24 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 17/24 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: b is greater than or equal to negative 17/24. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The inequality b is greater than or equal to negative 17/24 is graphed on the number line, with an open bracket at b equals negative 17/24 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 17/24 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "f minus the fraction 13 over 20 less than minus the fraction 5 over 12")[$f − frac(13, 20) < − frac(5, 12)$] #math.equation(block: true, alt: "g minus the fraction 11 over 12 less than minus the fraction 5 over 18")[$g − frac(11, 12) < − frac(5, 18)$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: g is less than 23/26. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The inequality g is less than 23/26 is graphed on the number line, with an open parenthesis at g equals 23/26 (written in), and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 23/26, parenthesis.], alt: "At the top of this figure is the solution to the inequality: g is less than 23/26. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The inequality g is less than 23/26 is graphed on the number line, with an open parenthesis at g equals 23/26 (written in), and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 23/26, parenthesis.", caption: none) ] #strong[Solve Inequalities using the Division and Multiplication Properties of Inequality] In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "8 x greater than 72")[$8 x > 72$] #math.equation(block: true, alt: "6 y less than 48")[$6 y < 48$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: y is less than 8. Below this is a number line ranging from 6 to 10 with tick marks for each integer. The inequality y is less than 8 is graphed on the number line, with an open parenthesis at y equals 8, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 8, parenthesis.], alt: "At the top of this figure is the solution to the inequality: y is less than 8. Below this is a number line ranging from 6 to 10 with tick marks for each integer. The inequality y is less than 8 is graphed on the number line, with an open parenthesis at y equals 8, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 8, parenthesis.", caption: none) ] #math.equation(block: true, alt: "7 r less than or equal to 56")[$7 r ≤ 56$] #math.equation(block: true, alt: "9 s greater than or equal to 81")[$9 s ≥ 81$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: s is greater than or equal to 9. Below this is a number line ranging from 7 to 11 with tick marks for each integer. The inequality s is greater than or equal to 9 is graphed on the number line, with an open bracket at s equals 9, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 9 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: s is greater than or equal to 9. Below this is a number line ranging from 7 to 11 with tick marks for each integer. The inequality s is greater than or equal to 9 is graphed on the number line, with an open bracket at s equals 9, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 9 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "−5 u greater than or equal to 65")[$−5 u ≥ 65$] #math.equation(block: true, alt: "−8 v less than or equal to 96")[$−8 v ≤ 96$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: v is greater than or equal to negative 12. Below this is a number line ranging from negative 14 to negative 10 with tick marks for each integer. The inequality v is greater than or equal to negative 12 is graphed on the number line, with an open bracket at v equals negative 12, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 12 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: v is greater than or equal to negative 12. Below this is a number line ranging from negative 14 to negative 10 with tick marks for each integer. The inequality v is greater than or equal to negative 12 is graphed on the number line, with an open bracket at v equals negative 12, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 12 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "−9 c less than 126")[$−9 c < 126$] #math.equation(block: true, alt: "−7 d greater than 105")[$−7 d > 105$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: d is less than negative 15. Below this is a number line ranging from negative 17 to negative 13 with tick marks for each integer. The inequality d is less than negative 15 is graphed on the number line, with an open parenthesis at d equals negative 15, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 15, parenthesis.], alt: "At the top of this figure is the solution to the inequality: d is less than negative 15. Below this is a number line ranging from negative 17 to negative 13 with tick marks for each integer. The inequality d is less than negative 15 is graphed on the number line, with an open parenthesis at d equals negative 15, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 15, parenthesis.", caption: none) ] #math.equation(block: true, alt: "20 greater than the fraction 2 over 5 h")[$20 > frac(2, 5) h$] #math.equation(block: true, alt: "40 less than the fraction 5 over 8 k")[$40 < frac(5, 8) k$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: k is greater than 64. Below this is a number line ranging from 62 to 66 with tick marks for each integer. The inequality k is greater than 64 is graphed on the number line, with an open parenthesis at k equals 64, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 64, parenthesis.], alt: "At the top of this figure is the solution to the inequality: k is greater than 64. Below this is a number line ranging from 62 to 66 with tick marks for each integer. The inequality k is greater than 64 is graphed on the number line, with an open parenthesis at k equals 64, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 64, parenthesis.", caption: none) ] #math.equation(block: true, alt: "the fraction 7 over 6 j greater than or equal to 42")[$frac(7, 6) j ≥ 42$] #math.equation(block: true, alt: "the fraction 9 over 4 g less than or equal to 36")[$frac(9, 4) g ≤ 36$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: g is less than or equal to 16. Below this is a number line ranging from 14 to 18 with tick marks for each integer. The inequality g is less than or equal to 16 is graphed on the number line, with an open bracket at g equals 16, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 16, bracket.], alt: "At the top of this figure is the solution to the inequality: g is less than or equal to 16. Below this is a number line ranging from 14 to 18 with tick marks for each integer. The inequality g is less than or equal to 16 is graphed on the number line, with an open bracket at g equals 16, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 16, bracket.", caption: none) ] #math.equation(block: true, alt: "the fraction a over −3 less than or equal to 9")[$frac(a, −3) ≤ 9$] #math.equation(block: true, alt: "the fraction b over −10 greater than or equal to 30")[$frac(b, −10) ≥ 30$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: b is less than or equal to negative 300. Below this is a number line ranging from negative 302 to negative 298 with tick marks for each integer. The inequality b is less than or equal to negative 300 is graphed on the number line, with an open bracket at b equals negative 300, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 300, bracket.], alt: "At the top of this figure is the solution to the inequality: b is less than or equal to negative 300. Below this is a number line ranging from negative 302 to negative 298 with tick marks for each integer. The inequality b is less than or equal to negative 300 is graphed on the number line, with an open bracket at b equals negative 300, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 300, bracket.", caption: none) ] #math.equation(block: true, alt: "−25 less than the fraction p over −5")[$−25 < frac(p, −5)$] #math.equation(block: true, alt: "−18 greater than the fraction q over −6")[$−18 > frac(q, −6)$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: q is greater than 108. Below this is a number line ranging from 106 to 110 with tick marks for each integer. The inequality q is greater than 108 is graphed on the number line, with an open parenthesis at q equals 108, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, 108 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: q is greater than 108. Below this is a number line ranging from 106 to 110 with tick marks for each integer. The inequality q is greater than 108 is graphed on the number line, with an open parenthesis at q equals 108, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, 108 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "9 t greater than or equal to −27")[$9 t ≥ −27$] #math.equation(block: true, alt: "7 s less than minus 28")[$7 s < − 28$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: s is less than negative 4. Below this is a number line ranging from negative 6 to negative 2 with tick marks for each integer. The inequality s is less than negative 4 is graphed on the number line, with an open parenthesis at s equals negative 4, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 4, parenthesis.], alt: "At the top of this figure is the solution to the inequality: s is less than negative 4. Below this is a number line ranging from negative 6 to negative 2 with tick marks for each integer. The inequality s is less than negative 4 is graphed on the number line, with an open parenthesis at s equals negative 4, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 4, parenthesis.", caption: none) ] #math.equation(block: true, alt: "the fraction 2 over 3 y greater than minus 36")[$frac(2, 3) y > − 36$] #math.equation(block: true, alt: "the fraction 3 over 5 x less than or equal to minus 45")[$frac(3, 5) x ≤ − 45$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: x is less than or equal to negative 75. Below this is a number line ranging from negative 77 to negative 73 with tick marks for each integer. The inequality x is less than or equal to negative 75 is graphed on the number line, with an open bracket at x equals negative 75, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 75, bracket.], alt: "At the top of this figure is the solution to the inequality: x is less than or equal to negative 75. Below this is a number line ranging from negative 77 to negative 73 with tick marks for each integer. The inequality x is less than or equal to negative 75 is graphed on the number line, with an open bracket at x equals negative 75, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 75, bracket.", caption: none) ] #strong[Solve Inequalities That Require Simplification] In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "4 v greater than or equal to 9 v minus 40")[$4 v ≥ 9 v − 40$] #math.equation(block: true, alt: "5 u less than or equal to 8 u minus 21")[$5 u ≤ 8 u − 21$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: au is greater than or equal to 7. Below this is a number line ranging from 5 to 9 with tick marks for each integer. The inequality u is greater than or equal to 7 is graphed on the number line, with an open bracket at u equals 7, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 7 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: au is greater than or equal to 7. Below this is a number line ranging from 5 to 9 with tick marks for each integer. The inequality u is greater than or equal to 7 is graphed on the number line, with an open bracket at u equals 7, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 7 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "13 q less than 7 q minus 29")[$13 q < 7 q − 29$] #math.equation(block: true, alt: "9 p greater than 14 p minus 18")[$9 p > 14 p − 18$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: p is less than 18/5. Below this is a number line ranging from 2 to 6 with tick marks for each integer. The inequality p is less than 18/5 is graphed on the number line, with an open parenthesis at p equals 18/5 (written in), and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 18/5, parenthesis.], alt: "At the top of this figure is the solution to the inequality: p is less than 18/5. Below this is a number line ranging from 2 to 6 with tick marks for each integer. The inequality p is less than 18/5 is graphed on the number line, with an open parenthesis at p equals 18/5 (written in), and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 18/5, parenthesis.", caption: none) ] #math.equation(block: true, alt: "12 x plus 3 open parenthesis x plus 7 close parenthesis greater than 10 x minus 24")[$12 x + 3 ( x + 7 ) > 10 x − 24$] #math.equation(block: true, alt: "9 y plus 5 open parenthesis y plus 3 close parenthesis less than 4 y minus 35")[$9 y + 5 ( y + 3 ) < 4 y − 35$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: y is less than negative 5. Below this is a number line ranging from negative 6 to negative 2 with tick marks for each integer. The inequality y is less than negative 5 is graphed on the number line, with an open parenthesis at y equals negative 5, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 5, parenthesis.], alt: "At the top of this figure is the solution to the inequality: y is less than negative 5. Below this is a number line ranging from negative 6 to negative 2 with tick marks for each integer. The inequality y is less than negative 5 is graphed on the number line, with an open parenthesis at y equals negative 5, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 5, parenthesis.", caption: none) ] #math.equation(block: true, alt: "6 h minus 4 open parenthesis h minus 1 close parenthesis less than or equal to 7 h minus 11")[$6 h − 4 ( h − 1 ) ≤ 7 h − 11$] #math.equation(block: true, alt: "4 k minus open parenthesis k minus 2 close parenthesis greater than or equal to 7 k minus 26")[$4 k − ( k − 2 ) ≥ 7 k − 26$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: x is less than or equal to 7. Below this is a number line ranging from 5 to 9 with tick marks for each integer. The inequality x is less than or equal to 7 is graphed on the number line, with an open bracket at x equals 7, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 7, bracket.], alt: "At the top of this figure is the solution to the inequality: x is less than or equal to 7. Below this is a number line ranging from 5 to 9 with tick marks for each integer. The inequality x is less than or equal to 7 is graphed on the number line, with an open bracket at x equals 7, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 7, bracket.", caption: none) ] #math.equation(block: true, alt: "8 m minus 2 open parenthesis 14 minus m close parenthesis greater than or equal to ​ 7 open parenthesis m minus 4 close parenthesis plus 3 m")[$8 m − 2 ( 14 − m ) ≥ "​" 7 ( m − 4 ) + 3 m$] #math.equation(block: true, alt: "6 n minus 12 open parenthesis 3 minus n close parenthesis less than or equal to 9 open parenthesis n minus 4 close parenthesis plus 9 n")[$6 n − 12 ( 3 − n ) ≤ 9 ( n − 4 ) + 9 n$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: the inequality is an identity. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: the inequality is an identity. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "the fraction 3 over 4 b minus the fraction 1 over 3 b less than the fraction 5 over 12 b minus the fraction 1 over 2")[$frac(3, 4) b − frac(1, 3) b < frac(5, 12) b − frac(1, 2)$] #math.equation(block: true, alt: "9 u plus 5 open parenthesis 2 u minus 5 close parenthesis greater than or equal to 12 open parenthesis u minus 1 close parenthesis plus 7 u")[$9 u + 5 ( 2 u − 5 ) ≥ 12 ( u − 1 ) + 7 u$] #solutionbox[ #figure(figph[At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement: “No solution”.], alt: "At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement: “No solution”.", caption: none) ] #math.equation(block: true, alt: "the fraction 2 over 3 g minus the fraction 1 over 2 open parenthesis g minus 14 close parenthesis less than or equal to the fraction 1 over 6 open parenthesis g plus 42 close parenthesis")[$frac(2, 3) g − frac(1, 2) ( g − 14 ) ≤ frac(1, 6) ( g + 42 )$] #math.equation(block: true, alt: "the fraction 5 over 6 a minus the fraction 1 over 4 a greater than the fraction 7 over 12 a plus the fraction 2 over 3")[$frac(5, 6) a − frac(1, 4) a > frac(7, 12) a + frac(2, 3)$] #solutionbox[ #figure(figph[At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement: “No solution”.], alt: "At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement: “No solution”.", caption: none) ] #math.equation(block: true, alt: "the fraction 4 over 5 h minus the fraction 2 over 3 open parenthesis h minus 9 close parenthesis greater than or equal to the fraction 1 over 15 open parenthesis 2 h plus 90 close parenthesis")[$frac(4, 5) h − frac(2, 3) ( h − 9 ) ≥ frac(1, 15) ( 2 h + 90 )$] #math.equation(block: true, alt: "12 v plus 3 open parenthesis 4 v minus 1 close parenthesis less than or equal to 19 open parenthesis v minus 2 close parenthesis plus 5 v")[$12 v + 3 ( 4 v − 1 ) ≤ 19 ( v − 2 ) + 5 v$] #solutionbox[ #figure(figph[At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement: “No solution”.], alt: "At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement: “No solution”.", caption: none) ] #strong[Mixed practice] In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "15 k less than or equal to minus 40")[$15 k ≤ − 40$] #math.equation(block: true, alt: "35 k greater than or equal to −77")[$35 k ≥ −77$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: k is greater than or equal to negative 11/5. Below this is a number line ranging from negative 4 to 0 with tick marks for each integer. The inequality k is greater than or equal to negative 11/5 is graphed on the number line, with an open bracket at k equals negative 11/5 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 11/5 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: k is greater than or equal to negative 11/5. Below this is a number line ranging from negative 4 to 0 with tick marks for each integer. The inequality k is greater than or equal to negative 11/5 is graphed on the number line, with an open bracket at k equals negative 11/5 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 11/5 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "23 p minus 2 open parenthesis 6 minus 5 p close parenthesis greater than 3 open parenthesis 11 p minus 4 close parenthesis")[$23 p − 2 ( 6 − 5 p ) > 3 ( 11 p − 4 )$] #math.equation(block: true, alt: "18 q minus 4 open parenthesis 10 minus 3 q close parenthesis less than 5 open parenthesis 6 q minus 8 close parenthesis")[$18 q − 4 ( 10 − 3 q ) < 5 ( 6 q − 8 )$] #solutionbox[ #figure(figph[At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement: “No solution”.], alt: "At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement: “No solution”.", caption: none) ] #math.equation(block: true, alt: "minus the fraction 9 over 4 x greater than or equal to minus the fraction 5 over 12")[$− frac(9, 4) x ≥ − frac(5, 12)$] #math.equation(block: true, alt: "minus the fraction 21 over 8 y less than or equal to minus the fraction 15 over 28")[$− frac(21, 8) y ≤ − frac(15, 28)$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: y is greater than or equal to 10/49. Below this is a number line ranging from negative 1 to 3 with tick marks for each integer. The inequality y is greater than or equal to 10/49 is graphed on the number line, with an open bracket at y equals 10/49 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 10/49 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: y is greater than or equal to 10/49. Below this is a number line ranging from negative 1 to 3 with tick marks for each integer. The inequality y is greater than or equal to 10/49 is graphed on the number line, with an open bracket at y equals 10/49 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 10/49 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "c plus 34 less than minus 99")[$c + 34 < − 99$] #math.equation(block: true, alt: "d plus 29 greater than minus 61")[$d + 29 > − 61$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: d is greater than negative 90. Below this is a number line ranging from negative 92 to negative 88 with tick marks for each integer. The inequality d is greater than negative 90 is graphed on the number line, with an open parenthesis at d equals negative 90, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative 90 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: d is greater than negative 90. Below this is a number line ranging from negative 92 to negative 88 with tick marks for each integer. The inequality d is greater than negative 90 is graphed on the number line, with an open parenthesis at d equals negative 90, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative 90 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "the fraction m over 18 greater than or equal to −4")[$frac(m, 18) ≥ −4$] #math.equation(block: true, alt: "the fraction n over 13 less than or equal to minus 6")[$frac(n, 13) ≤ − 6$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: n is less than or equal to negative 78. Below this is a number line ranging from negative 80 to negative 76 with tick marks for each integer. The inequality n is less than or equal to negative 78 is graphed on the number line, with an open bracket at n equals negative 78, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 78, bracket.], alt: "At the top of this figure is the solution to the inequality: n is less than or equal to negative 78. Below this is a number line ranging from negative 80 to negative 76 with tick marks for each integer. The inequality n is less than or equal to negative 78 is graphed on the number line, with an open bracket at n equals negative 78, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 78, bracket.", caption: none) ] #strong[Translate to an Inequality and Solve] In the following exercises, translate and solve .Then write the solution in interval notation and graph on the number line. Fourteen times #emph[d] is greater than 56. Ninety times #emph[c] is less than 450. #solutionbox[ #figure(figph[At the top of this figure is the the inequality 90c is less than 450. Below this is the solution to the inequality: c is less than 5. Below the solution is the solution written in interval notation: parenthesis, negative infinity comma 5, parenthesis. Below the interval notation is a number line ranging from 3 to 7 with tick marks for each integer. The inequality c is less than 5 is graphed on the number line, with an open parenthesis at c equals 5, and a dark line extending to the left of the parenthesis.], alt: "At the top of this figure is the the inequality 90c is less than 450. Below this is the solution to the inequality: c is less than 5. Below the solution is the solution written in interval notation: parenthesis, negative infinity comma 5, parenthesis. Below the interval notation is a number line ranging from 3 to 7 with tick marks for each integer. The inequality c is less than 5 is graphed on the number line, with an open parenthesis at c equals 5, and a dark line extending to the left of the parenthesis.", caption: none) ] Eight times #emph[z] is smaller than #math.equation(block: false, alt: "−40")[$−40$]. Ten times #emph[y] is at most #math.equation(block: false, alt: "−110")[$−110$]. #solutionbox[ #figure(figph[At the top of this figure is the the inequality 10y is less than or equal to negative 110. Below this is the solution to the inequality: y is less than or equal to negative 11. Below the solution is the solution written in interval notation: parenthesis, negative infinity comma negative 11, bracket. Below the interval notation is a number line ranging from negative 13 to negative 9 with tick marks for each integer. The inequality y is less than or equal to negative 11 is graphed on the number line, with an open bracket at y equals negative 11, and a dark line extending to the left of the bracket.], alt: "At the top of this figure is the the inequality 10y is less than or equal to negative 110. Below this is the solution to the inequality: y is less than or equal to negative 11. Below the solution is the solution written in interval notation: parenthesis, negative infinity comma negative 11, bracket. Below the interval notation is a number line ranging from negative 13 to negative 9 with tick marks for each integer. The inequality y is less than or equal to negative 11 is graphed on the number line, with an open bracket at y equals negative 11, and a dark line extending to the left of the bracket.", caption: none) ] Three more than #emph[h] is no less than 25. Six more than #emph[k] exceeds 25. #solutionbox[ #figure(figph[At the top of this figure is the the inequality k plus 6 is greater than 25. Below this is the solution to the inequality: k is greater than 19. Below the the solution written in interval notation: parenthesis, 19 comma infinity, parenthesis. Below the interval notation is a number line ranging from 17 to 21 with tick marks for each integer. The inequality k is greater than 19 is graphed on the number line, with an open parenthesis at k equals 19, and a dark line extending to the right of the parenthesis.], alt: "At the top of this figure is the the inequality k plus 6 is greater than 25. Below this is the solution to the inequality: k is greater than 19. Below the the solution written in interval notation: parenthesis, 19 comma infinity, parenthesis. Below the interval notation is a number line ranging from 17 to 21 with tick marks for each integer. The inequality k is greater than 19 is graphed on the number line, with an open parenthesis at k equals 19, and a dark line extending to the right of the parenthesis.", caption: none) ] Ten less than #emph[w] is at least 39. Twelve less than #emph[x] is no less than 21. #solutionbox[ #figure(figph[At the top of this figure is the the inequality x minus 12 is greater than or equal to 21. Below this is the solution to the inequality: x is greater than or equal to 33. Below the solution is the solution written in interval notation: bracket, 33 comma infinity, parenthesis. Below the interval notation is a number line ranging from 32 to 36 with tick marks for each integer. The inequality x is greater than or equal to 33 is graphed on the number line, with an open bracket at x equals 33, and a dark line extending to the right of the bracket.], alt: "At the top of this figure is the the inequality x minus 12 is greater than or equal to 21. Below this is the solution to the inequality: x is greater than or equal to 33. Below the solution is the solution written in interval notation: bracket, 33 comma infinity, parenthesis. Below the interval notation is a number line ranging from 32 to 36 with tick marks for each integer. The inequality x is greater than or equal to 33 is graphed on the number line, with an open bracket at x equals 33, and a dark line extending to the right of the bracket.", caption: none) ] Negative five times #emph[r] is no more than 95. Negative two times #emph[s] is lower than 56. #solutionbox[ #figure(figph[At the top of this figure is the the inequality negative 2s is less than 56. Below this is the solution to the inequality: s is greater than negative 28. Below the solution is the solution written in interval notation: parenthesis, negative 28 comma infinity, parenthesis. Below the interval notation is a number line ranging from negative 30 to negative 26 with tick marks for each integer. The inequality s is greater than negative 28 is graphed on the number line, with an open parenthesis at s equals negative 28, and a dark line extending to the right of the parenthesis.], alt: "At the top of this figure is the the inequality negative 2s is less than 56. Below this is the solution to the inequality: s is greater than negative 28. Below the solution is the solution written in interval notation: parenthesis, negative 28 comma infinity, parenthesis. Below the interval notation is a number line ranging from negative 30 to negative 26 with tick marks for each integer. The inequality s is greater than negative 28 is graphed on the number line, with an open parenthesis at s equals negative 28, and a dark line extending to the right of the parenthesis.", caption: none) ] Nineteen less than #emph[b] is at most #math.equation(block: false, alt: "−22")[$−22$]. Fifteen less than #emph[a] is at least #math.equation(block: false, alt: "−7")[$−7$]. #solutionbox[ #figure(figph[At the top of this figure is the the inequality a minus 15 is greater than or equal to negative 7. Below this is the solution to the inequality: a is greater than or equal to 8. Below the solution is the solution written in interval notation: bracket, 8 comma infinity, parenthesis. Below the interval notation is a number line ranging from 0 to 10 with tick marks for each integer. The inequality a is greater than or equal to 8 is graphed on the number line, with an open bracket at a equals 8, and a dark line extending to the right of the bracket.], alt: "At the top of this figure is the the inequality a minus 15 is greater than or equal to negative 7. Below this is the solution to the inequality: a is greater than or equal to 8. Below the solution is the solution written in interval notation: bracket, 8 comma infinity, parenthesis. Below the interval notation is a number line ranging from 0 to 10 with tick marks for each integer. The inequality a is greater than or equal to 8 is graphed on the number line, with an open bracket at a equals 8, and a dark line extending to the right of the bracket.", caption: none) ] ==== Everyday Math #strong[Safety] A child’s height, #emph[h], must be at least 57 inches for the child to safely ride in the front seat of a car. Write this as an inequality. #strong[Fighter pilots] The maximum height, #emph[h], of a fighter pilot is 77 inches. Write this as an inequality. #solutionbox[ #math.equation(block: true, alt: "h less than or equal to 77")[$h ≤ 77$] ] #strong[Elevators] The total weight, #emph[w], of an elevator’s passengers can be no more than 1,200 pounds. Write this as an inequality. #strong[Shopping] The number of items, #emph[n], a shopper can have in the express check-out lane is at most 8. Write this as an inequality. #solutionbox[ #math.equation(block: true, alt: "n less than or equal to 8")[$n ≤ 8$] ] ==== Writing Exercises Give an example from your life using the phrase ‘at least’. Give an example from your life using the phrase ‘at most’. #solutionbox[ Answers will vary. ] Explain why it is necessary to reverse the inequality when solving #math.equation(block: false, alt: "−5 x greater than 10")[$−5 x > 10$]. Explain why it is necessary to reverse the inequality when solving #math.equation(block: false, alt: "the fraction n over −3 less than 12")[$frac(n, −3) < 12$]. #solutionbox[ Answers will vary. ] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “graph inequalities on the number line,” “solve inequalitites using the Subtraction and Addition Properties of Inequality,” “solve inequalitites using the Division and Multiplication Properties of Inequality,” “solve inequalities that require simplification,” and “translate to an inequality and solve.” The rest of the cells are blank.], alt: "This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “graph inequalities on the number line,” “solve inequalitites using the Subtraction and Addition Properties of Inequality,” “solve inequalitites using the Division and Multiplication Properties of Inequality,” “solve inequalities that require simplification,” and “translate to an inequality and solve.” The rest of the cells are blank.", caption: none) ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? === Chapter 2 Review Exercises ==== Solve Equations using the Subtraction and Addition Properties of Equality #strong[Verify a Solution of an Equation] In the following exercises, determine whether each number is a solution to the equation. #math.equation(block: true, alt: "10 x minus 1 equals 5 x ; x equals the fraction 1 over 5")[$10 x − 1 = 5 x ; x = frac(1, 5)$] #math.equation(block: true, alt: "w plus 2 equals the fraction 5 over 8 ; w equals the fraction 3 over 8")[$w + 2 = frac(5, 8) ; w = frac(3, 8)$] #solutionbox[ no ] #math.equation(block: true, alt: "−12 n plus 5 equals 8 n ; n equals minus the fraction 5 over 4")[$−12 n + 5 = 8 n ; n = − frac(5, 4)$] #math.equation(block: true, alt: "6 a minus 3 equals −7 a , a equals the fraction 3 over 13")[$6 a − 3 = −7 a , a = frac(3, 13)$] #solutionbox[ yes ] #strong[Solve Equations using the Subtraction and Addition Properties of Equality] In the following exercises, solve each equation using the Subtraction Property of Equality. #math.equation(block: true, alt: "x plus 7 equals 19")[$x + 7 = 19$] #math.equation(block: true, alt: "y plus 2 equals −6")[$y + 2 = −6$] #solutionbox[ #math.equation(block: true, alt: "y equals −8")[$y = −8$] ] #math.equation(block: true, alt: "a plus the fraction 1 over 3 equals the fraction 5 over 3")[$a + frac(1, 3) = frac(5, 3)$] #math.equation(block: true, alt: "n plus 3.6 equals 5.1")[$n + 3.6 = 5.1$] #solutionbox[ #math.equation(block: true, alt: "n equals 1.5")[$n = 1.5$] ] In the following exercises, solve each equation using the Addition Property of Equality. #math.equation(block: true, alt: "u minus 7 equals 10")[$u − 7 = 10$] #math.equation(block: true, alt: "x minus 9 equals −4")[$x − 9 = −4$] #solutionbox[ #math.equation(block: true, alt: "x equals 5")[$x = 5$] ] #math.equation(block: true, alt: "c minus the fraction 3 over 11 equals the fraction 9 over 11")[$c − frac(3, 11) = frac(9, 11)$] #math.equation(block: true, alt: "p minus 4.8 equals 14")[$p − 4.8 = 14$] #solutionbox[ #math.equation(block: true, alt: "p equals 18.8")[$p = 18.8$] ] In the following exercises, solve each equation. #math.equation(block: true, alt: "n minus 12 equals 32")[$n − 12 = 32$] #math.equation(block: true, alt: "y plus 16 equals −9")[$y + 16 = −9$] #solutionbox[ #math.equation(block: true, alt: "y equals −25")[$y = −25$] ] #math.equation(block: true, alt: "f plus the fraction 2 over 3 equals 4")[$f + frac(2, 3) = 4$] #math.equation(block: true, alt: "d minus 3.9 equals 8.2")[$d − 3.9 = 8.2$] #solutionbox[ #math.equation(block: true, alt: "d equals 12.1")[$d = 12.1$] ] #strong[Solve Equations That Require Simplification] In the following exercises, solve each equation. #math.equation(block: true, alt: "y plus 8 minus 15 equals −3")[$y + 8 − 15 = −3$] #math.equation(block: true, alt: "7 x plus 10 minus 6 x plus 3 equals 5")[$7 x + 10 − 6 x + 3 = 5$] #solutionbox[ #math.equation(block: true, alt: "x equals −8")[$x = −8$] ] #math.equation(block: true, alt: "6 open parenthesis n minus 1 close parenthesis minus 5 n equals −14")[$6 ( n − 1 ) − 5 n = −14$] #math.equation(block: true, alt: "8 open parenthesis 3 p plus 5 close parenthesis minus 23 open parenthesis p minus 1 close parenthesis equals 35")[$8 ( 3 p + 5 ) − 23 ( p − 1 ) = 35$] #solutionbox[ #math.equation(block: true, alt: "p equals −28")[$p = −28$] ] #strong[Translate to an Equation and Solve] In the following exercises, translate each English sentence into an algebraic equation and then solve it. The sum of #math.equation(block: false, alt: "−6")[$−6$] and #math.equation(block: false, alt: "m")[$m$] is 25. Four less than #math.equation(block: false, alt: "n")[$n$] is 13. #solutionbox[ #math.equation(block: true, alt: "n minus 4 equals 13 ; n equals 17")[$n − 4 = 13 ; n = 17$] ] #strong[Translate and Solve Applications] In the following exercises, translate into an algebraic equation and solve. Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son? Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh? #solutionbox[ 161 pounds ] Peter paid \$9.75 to go to the movies, which was \$46.25 less than he paid to go to a concert. How much did he pay for the concert? Elissa earned \$152.84 this week, which was \$21.65 more than she earned last week. How much did she earn last week? #solutionbox[ \$131.19 ] ==== Solve Equations using the Division and Multiplication Properties of Equality #strong[Solve Equations Using the Division and Multiplication Properties of Equality] In the following exercises, solve each equation using the division and multiplication properties of equality and check the solution. #math.equation(block: true, alt: "8 x equals 72")[$8 x = 72$] #math.equation(block: true, alt: "13 a equals −65")[$13 a = −65$] #solutionbox[ #math.equation(block: true, alt: "a equals −5")[$a = −5$] ] #math.equation(block: true, alt: "0.25 p equals 5.25")[$0.25 p = 5.25$] #math.equation(block: true, alt: "− y equals 4")[$"−" y = 4$] #solutionbox[ #math.equation(block: true, alt: "y equals −4")[$y = −4$] ] #math.equation(block: true, alt: "the fraction n over 6 equals 18")[$frac(n, 6) = 18$] #math.equation(block: true, alt: "the fraction y over −10 equals 30")[$frac(y, −10) = 30$] #solutionbox[ #math.equation(block: true, alt: "y equals −300")[$y = −300$] ] #math.equation(block: true, alt: "36 equals the fraction 3 over 4 x")[$36 = frac(3, 4) x$] #math.equation(block: true, alt: "the fraction 5 over 8 u equals the fraction 15 over 16")[$frac(5, 8) u = frac(15, 16)$] #solutionbox[ #math.equation(block: true, alt: "u equals the fraction 3 over 2")[$u = frac(3, 2)$] ] #math.equation(block: true, alt: "−18 m equals −72")[$−18 m = −72$] #math.equation(block: true, alt: "the fraction c over 9 equals 36")[$frac(c, 9) = 36$] #solutionbox[ #math.equation(block: true, alt: "c equals 324")[$c = 324$] ] #math.equation(block: true, alt: "0.45 x equals 6.75")[$0.45 x = 6.75$] #math.equation(block: true, alt: "the fraction 11 over 12 equals the fraction 2 over 3 y")[$frac(11, 12) = frac(2, 3) y$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 11 over 8")[$y = frac(11, 8)$] ] #strong[Solve Equations That Require Simplification] In the following exercises, solve each equation requiring simplification. #math.equation(block: true, alt: "5 r minus 3 r plus 9 r equals 35 minus 2")[$5 r − 3 r + 9 r = 35 − 2$] #math.equation(block: true, alt: "24 x plus 8 x minus 11 x equals −7 minus 14")[$24 x + 8 x − 11 x = −7 − 14$] #solutionbox[ #math.equation(block: true, alt: "x equals −1")[$x = −1$] ] #math.equation(block: true, alt: "the fraction 11 over 12 n minus the fraction 5 over 6 n equals 9 minus 5")[$frac(11, 12) n − frac(5, 6) n = 9 − 5$] #math.equation(block: true, alt: "−9 open parenthesis d minus 2 close parenthesis minus 15 equals −24")[$−9 ( d − 2 ) − 15 = −24$] #solutionbox[ #math.equation(block: true, alt: "d equals 3")[$d = 3$] ] #strong[Translate to an Equation and Solve] In the following exercises, translate to an equation and then solve. 143 is the product of #math.equation(block: false, alt: "−11")[$−11$] and #emph[y]. The quotient of #emph[b] and 9 is #math.equation(block: false, alt: "−27")[$−27$]. #solutionbox[ #math.equation(block: true, alt: "the fraction b over 9 equals −27 ; b equals −243")[$frac(b, 9) = −27 ; b = −243$] ] The sum of #emph[q] and one-fourth is one. The difference of #emph[s] and one-twelfth is one fourth. #solutionbox[ #math.equation(block: true, alt: "s minus the fraction 1 over 12 equals the fraction 1 over 4 ; s equals the fraction 1 over 3")[$s − frac(1, 12) = frac(1, 4) ; s = frac(1, 3)$] ] #strong[Translate and Solve Applications] In the following exercises, translate into an equation and solve. Ray paid \$21 for 12 tickets at the county fair. What was the price of each ticket? Janet gets paid \$24 per hour. She heard that this is #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] of what Adam is paid. How much is Adam paid per hour? #solutionbox[ \$32 ] ==== Solve Equations with Variables and Constants on Both Sides #strong[Solve an Equation with Constants on Both Sides] In the following exercises, solve the following equations with constants on both sides. #math.equation(block: true, alt: "8 p plus 7 equals 47")[$8 p + 7 = 47$] #math.equation(block: true, alt: "10 w minus 5 equals 65")[$10 w − 5 = 65$] #solutionbox[ #math.equation(block: true, alt: "w equals 7")[$w = 7$] ] #math.equation(block: true, alt: "3 x plus 19 equals −47")[$3 x + 19 = −47$] #math.equation(block: true, alt: "32 equals −4 minus 9 n")[$32 = −4 − 9 n$] #solutionbox[ #math.equation(block: true, alt: "n equals −4")[$n = −4$] ] #strong[Solve an Equation with Variables on Both Sides] In the following exercises, solve the following equations with variables on both sides. #math.equation(block: true, alt: "7 y equals 6 y minus 13")[$7 y = 6 y − 13$] #math.equation(block: true, alt: "5 a plus 21 equals 2 a")[$5 a + 21 = 2 a$] #solutionbox[ #math.equation(block: true, alt: "a equals −7")[$a = −7$] ] #math.equation(block: true, alt: "k equals −6 k minus 35")[$k = −6 k − 35$] #math.equation(block: true, alt: "4 x minus the fraction 3 over 8 equals 3 x")[$4 x − frac(3, 8) = 3 x$] #solutionbox[ #math.equation(block: true, alt: "x equals the fraction 3 over 8")[$x = frac(3, 8)$] ] #strong[Solve an Equation with Variables and Constants on Both Sides] In the following exercises, solve the following equations with variables and constants on both sides. #math.equation(block: true, alt: "12 x minus 9 equals 3 x plus 45")[$12 x − 9 = 3 x + 45$] #math.equation(block: true, alt: "5 n minus 20 equals −7 n minus 80")[$5 n − 20 = −7 n − 80$] #solutionbox[ #math.equation(block: true, alt: "n equals −5")[$n = −5$] ] #math.equation(block: true, alt: "4 u plus 16 equals −19 minus u")[$4 u + 16 = −19 − u$] #math.equation(block: true, alt: "the fraction 5 over 8 c minus 4 equals the fraction 3 over 8 c plus 4")[$frac(5, 8) c − 4 = frac(3, 8) c + 4$] #solutionbox[ #math.equation(block: true, alt: "c equals 32")[$c = 32$] ] ==== Use a General Strategy for Solving Linear Equations #strong[Solve Equations Using the General Strategy for Solving Linear Equations] In the following exercises, solve each linear equation. #math.equation(block: true, alt: "6 open parenthesis x plus 6 close parenthesis equals 24")[$6 ( x + 6 ) = 24$] #math.equation(block: true, alt: "9 open parenthesis 2 p minus 5 close parenthesis equals 72")[$9 ( 2 p − 5 ) = 72$] #solutionbox[ #math.equation(block: true, alt: "p equals the fraction 13 over 2")[$p = frac(13, 2)$] ] #math.equation(block: true, alt: "− open parenthesis s plus 4 close parenthesis equals 18")[$"−" ( s + 4 ) = 18$] #math.equation(block: true, alt: "8 plus 3 open parenthesis n minus 9 close parenthesis equals 17")[$8 + 3 ( n − 9 ) = 17$] #solutionbox[ #math.equation(block: true, alt: "n equals 12")[$n = 12$] ] #math.equation(block: true, alt: "23 minus 3 open parenthesis y minus 7 close parenthesis equals 8")[$23 − 3 ( y − 7 ) = 8$] #math.equation(block: true, alt: "the fraction 1 over 3 open parenthesis 6 m plus 21 close parenthesis equals m minus 7")[$frac(1, 3) ( 6 m + 21 ) = m − 7$] #solutionbox[ #math.equation(block: true, alt: "m equals −14")[$m = −14$] ] #math.equation(block: true, alt: "4 open parenthesis 3.5 y plus 0.25 close parenthesis equals 365")[$4 ( 3.5 y + 0.25 ) = 365$] #math.equation(block: true, alt: "0.25 open parenthesis q minus 8 close parenthesis equals 0.1 open parenthesis q plus 7 close parenthesis")[$0.25 ( q − 8 ) = 0.1 ( q + 7 )$] #solutionbox[ #math.equation(block: true, alt: "q equals 18")[$q = 18$] ] #math.equation(block: true, alt: "8 open parenthesis r minus 2 close parenthesis equals 6 open parenthesis r plus 10 close parenthesis")[$8 ( r − 2 ) = 6 ( r + 10 )$] #math.equation(block: true, alt: "5 plus 7 open parenthesis 2 minus 5 x close parenthesis equals 2 open parenthesis 9 x plus 1 close parenthesis")[$5 + 7 ( 2 − 5 x ) = 2 ( 9 x + 1 )$] #linebreak() #math.equation(block: true, alt: "minus open parenthesis 13 x minus 57 close parenthesis")[$− ( 13 x − 57 )$] #solutionbox[ #math.equation(block: true, alt: "x equals −1")[$x = −1$] ] #math.equation(block: true, alt: "open parenthesis 9 n plus 5 close parenthesis minus open parenthesis 3 n minus 7 close parenthesis")[$( 9 n + 5 ) − ( 3 n − 7 )$] #linebreak() #math.equation(block: true, alt: "equals 20 minus open parenthesis 4 n minus 2 close parenthesis")[$= 20 − ( 4 n − 2 )$] #math.equation(block: true, alt: "2 [ −16 plus 5 open parenthesis 8 k minus 6 close parenthesis ]")[$2 [ −16 + 5 ( 8 k − 6 ) ]$] #linebreak() #math.equation(block: true, alt: "equals 8 open parenthesis 3 minus 4 k close parenthesis minus 32")[$= 8 ( 3 − 4 k ) − 32$] #solutionbox[ #math.equation(block: true, alt: "k equals the fraction 3 over 4")[$k = frac(3, 4)$] ] #strong[Classify Equations] In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution. #math.equation(block: true, alt: "17 y minus 3 open parenthesis 4 minus 2 y close parenthesis equals 11 open parenthesis y minus 1 close parenthesis")[$17 y − 3 ( 4 − 2 y ) = 11 ( y − 1 )$] #linebreak() #math.equation(block: true, alt: "plus 12 y minus 1")[$+ 12 y − 1$] #math.equation(block: true, alt: "9 u plus 32 equals 15 open parenthesis u minus 4 close parenthesis")[$9 u + 32 = 15 ( u − 4 )$] #linebreak() #math.equation(block: true, alt: "minus 3 open parenthesis 2 u plus 21 close parenthesis")[$− 3 ( 2 u + 21 )$] #solutionbox[ contradiction; no solution ] #math.equation(block: true, alt: "−8 open parenthesis 7 m plus 4 close parenthesis equals −6 open parenthesis 8 m plus 9 close parenthesis")[$−8 ( 7 m + 4 ) = −6 ( 8 m + 9 )$] #math.equation(block: true, alt: "21 open parenthesis c minus 1 close parenthesis minus 19 open parenthesis c plus 1 close parenthesis")[$21 ( c − 1 ) − 19 ( c + 1 )$] #linebreak() #math.equation(block: true, alt: "equals 2 open parenthesis c minus 20 close parenthesis")[$= 2 ( c − 20 )$] #solutionbox[ identity; all real numbers ] ==== Solve Equations with Fractions and Decimals #strong[Solve Equations with Fraction Coefficients] In the following exercises, solve each equation with fraction coefficients. #math.equation(block: true, alt: "the fraction 2 over 5 n minus the fraction 1 over 10 equals the fraction 7 over 10")[$frac(2, 5) n − frac(1, 10) = frac(7, 10)$] #math.equation(block: true, alt: "the fraction 1 over 3 x plus the fraction 1 over 5 x equals 8")[$frac(1, 3) x + frac(1, 5) x = 8$] #solutionbox[ #math.equation(block: true, alt: "x equals 15")[$x = 15$] ] #math.equation(block: true, alt: "the fraction 3 over 4 a minus the fraction 1 over 3 equals the fraction 1 over 2 a minus the fraction 5 over 6")[$frac(3, 4) a − frac(1, 3) = frac(1, 2) a − frac(5, 6)$] #math.equation(block: true, alt: "the fraction 1 over 2 open parenthesis k minus 3 close parenthesis equals the fraction 1 over 3 open parenthesis k plus 16 close parenthesis")[$frac(1, 2) ( k − 3 ) = frac(1, 3) ( k + 16 )$] #solutionbox[ #math.equation(block: true, alt: "k equals 41")[$k = 41$] ] #math.equation(block: true, alt: "the fraction 3 x minus 2 over 5 equals the fraction 3 x plus 4 over 8")[$frac(3 x − 2, 5) = frac(3 x + 4, 8)$] #math.equation(block: true, alt: "the fraction 5 y minus 1 over 3 plus 4 equals the fraction −8 y plus 4 over 6")[$frac(5 y − 1, 3) + 4 = frac(−8 y + 4, 6)$] #solutionbox[ #math.equation(block: true, alt: "y equals −1")[$y = −1$] ] #strong[Solve Equations with Decimal Coefficients] In the following exercises, solve each equation with decimal coefficients. #math.equation(block: true, alt: "0.8 x minus 0.3 equals 0.7 x plus 0.2")[$0.8 x − 0.3 = 0.7 x + 0.2$] #math.equation(block: true, alt: "0.36 u plus 2.55 equals 0.41 u plus 6.8")[$0.36 u + 2.55 = 0.41 u + 6.8$] #solutionbox[ #math.equation(block: true, alt: "u equals −85")[$u = −85$] ] #math.equation(block: true, alt: "0.6 p minus 1.9 equals 0.78 p plus 1.7")[$0.6 p − 1.9 = 0.78 p + 1.7$] #math.equation(block: true, alt: "0.7 y plus 2.5 equals 0.95 y minus 9.25")[$0.7 y + 2.5 = 0.95 y − 9.25$] #solutionbox[ #math.equation(block: true, alt: "y equals 47")[$y = 47$] ] ==== Solve a Formula for a Specific Variable #strong[Use the Distance, Rate, and Time Formula] In the following exercises, solve. Natalie drove for #math.equation(block: false, alt: "7 the fraction 1 over 2")[$7 frac(1, 2)$] hours at 60 miles per hour. How much distance did she travel? Mallory is taking the bus from St. Louis to Chicago. The distance is 300 miles and the bus travels at a steady rate of 60 miles per hour. How long will the bus ride be? #solutionbox[ 5 hours ] Aaron’s friend drove him from Buffalo to Cleveland. The distance is 187 miles and the trip took 2.75 hours. How fast was Aaron’s friend driving? Link rode his bike at a steady rate of 15 miles per hour for #math.equation(block: false, alt: "2 the fraction 1 over 2")[$2 frac(1, 2)$] hours. How much distance did he travel? #solutionbox[ 37.5 miles ] #strong[Solve a Formula for a Specific Variable] In the following exercises, solve. Use the formula. #math.equation(block: false, alt: "d equals r t")[$d = r t$] to solve for #emph[t] #linebreak() ⓐ when #math.equation(block: false, alt: "d equals 510")[$d = 510$] and #math.equation(block: false, alt: "r equals 60")[$r = 60$] #linebreak() ⓑ in general Use the formula. #math.equation(block: false, alt: "d equals r t")[$d = r t$] to solve for #math.equation(block: false, alt: "r")[$r$] #linebreak() ⓐ when when #math.equation(block: false, alt: "d equals 451")[$d = 451$] and #math.equation(block: false, alt: "t equals 5.5")[$t = 5.5$] #linebreak() ⓑ in general #solutionbox[ ⓐ #math.equation(block: false, alt: "r equals 82 mph")[$r = 82 #h(0.2em) "mph"$]; ⓑ #math.equation(block: false, alt: "r equals the fraction D over t")[$r = frac(D, t)$] ] Use the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] to solve for #math.equation(block: false, alt: "b")[$b$] #linebreak() ⓐ when #math.equation(block: false, alt: "A equals 390")[$A = 390$] and #math.equation(block: false, alt: "h equals 26")[$h = 26$] #linebreak() ⓑ in general Use the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] to solve for #math.equation(block: false, alt: "h")[$h$] #linebreak() ⓐ when #math.equation(block: false, alt: "A equals 153")[$A = 153$] and #math.equation(block: false, alt: "b equals 18")[$b = 18$] #linebreak() ⓑ in general #solutionbox[ ⓐ #math.equation(block: false, alt: "h equals 17")[$h = 17$] ⓑ #math.equation(block: false, alt: "h equals the fraction 2 A over b")[$h = frac(2 A, b)$] ] Use the formula #math.equation(block: false, alt: "I equals P r t")[$I = P r t$] to solve for the principal, #emph[P] for #linebreak() ⓐ #math.equation(block: false, alt: "I equals $ 2 , 501 , r equals 4.1 % ,")[$I = "$" 2 , 501 , r = 4.1 % ,$] #linebreak() #math.equation(block: false, alt: "t equals 5 years")[$t = 5 #h(0.2em) "years"$] #linebreak() ⓑ in general Solve the formula #math.equation(block: false, alt: "4 x plus 3 y equals 6")[$4 x + 3 y = 6$] for #emph[y] #linebreak() ⓐ when #math.equation(block: false, alt: "x equals −2")[$x = −2$] #linebreak() ⓑ in general #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals the fraction 14 over 3")[$y = frac(14, 3)$] ⓑ #math.equation(block: false, alt: "y equals the fraction 6 minus 4 x over 3")[$y = frac(6 − 4 x, 3)$] ] Solve #math.equation(block: false, alt: "180 equals a plus b plus c")[$180 = a + b + c$] for #math.equation(block: false, alt: "c")[$c$]. Solve the formula #math.equation(block: false, alt: "V equals L W H")[$V = L W H$] for #math.equation(block: false, alt: "H")[$H$]. #solutionbox[ #math.equation(block: true, alt: "H equals the fraction V over L W")[$H = frac(V, L W)$] ] ==== Solve Linear Inequalities #strong[Graph Inequalities on the Number Line] In the following exercises, graph each inequality on the number line. ⓐ #math.equation(block: false, alt: "x less than or equal to 4")[$x ≤ 4$] #linebreak() ⓑ #math.equation(block: false, alt: "x greater than minus 2")[$x > − 2$] #linebreak() ⓒ #math.equation(block: false, alt: "x less than 1")[$x < 1$] ⓐ #math.equation(block: false, alt: "x greater than 0")[$x > 0$] #linebreak() ⓑ #math.equation(block: false, alt: "x less than minus 3")[$x < − 3$] #linebreak() ⓒ #math.equation(block: false, alt: "x greater than or equal to −1")[$x ≥ −1$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 0 is graphed on the number line, with an open parenthesis at x equals 0, and a dark line extending to the right of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 0 is graphed on the number line, with an open parenthesis at x equals 0, and a dark line extending to the right of the parenthesis.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a dark line extending to the right of the bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a dark line extending to the right of the bracket.", caption: none) ] In the following exercises, graph each inequality on the number line and write in interval notation. ⓐ #math.equation(block: false, alt: "x less than minus 1")[$x < − 1$] #linebreak() ⓑ #math.equation(block: false, alt: "x greater than or equal to −2.5")[$x ≥ −2.5$] #linebreak() ⓒ #math.equation(block: false, alt: "x less than or equal to the fraction 5 over 4")[$x ≤ frac(5, 4)$] ⓐ #math.equation(block: false, alt: "x greater than 2")[$x > 2$] #linebreak() ⓑ #math.equation(block: false, alt: "x less than or equal to minus 1.5")[$x ≤ − 1.5$] #linebreak() ⓒ #math.equation(block: false, alt: "x greater than or equal to the fraction 5 over 3")[$x ≥ frac(5, 3)$] #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, 2 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, 2 comma infinity, parenthesis.", caption: none) + ⓑ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 1.5, bracket.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 1.5, bracket.", caption: none) + ⓒ #linebreak() #figure(figph[This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 5/3 is graphed on the number line, with an open bracket at x equals 5/3, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 5/3 comma infinity, parenthesis.], alt: "This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 5/3 is graphed on the number line, with an open bracket at x equals 5/3, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 5/3 comma infinity, parenthesis.", caption: none) ] #strong[Solve Inequalities using the Subtraction and Addition Properties of Inequality] In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "n minus 12 less than or equal to 23")[$n − 12 ≤ 23$] #math.equation(block: true, alt: "m plus 14 less than or equal to 56")[$m + 14 ≤ 56$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: m is less than or equal to 42. Below this is a number line ranging from 40 to 44 with tick marks for each integer. The inequality m is less than or equal to 42 is graphed on the number line, with an open bracket at m equals 42, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 42, bracket], alt: "At the top of this figure is the solution to the inequality: m is less than or equal to 42. Below this is a number line ranging from 40 to 44 with tick marks for each integer. The inequality m is less than or equal to 42 is graphed on the number line, with an open bracket at m equals 42, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 42, bracket", caption: none) ] #math.equation(block: true, alt: "a plus the fraction 2 over 3 greater than or equal to the fraction 7 over 12")[$a + frac(2, 3) ≥ frac(7, 12)$] #math.equation(block: true, alt: "b minus the fraction 7 over 8 greater than or equal to minus the fraction 1 over 2")[$b − frac(7, 8) ≥ − frac(1, 2)$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: b is greater than or equal to 3/8. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The inequality b is greater than or equal to 3/8 is graphed on the number line, with an open bracket at b equals 3/8 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 3/8 comma infinity, bracket], alt: "At the top of this figure is the solution to the inequality: b is greater than or equal to 3/8. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The inequality b is greater than or equal to 3/8 is graphed on the number line, with an open bracket at b equals 3/8 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 3/8 comma infinity, bracket", caption: none) ] #strong[Solve Inequalities using the Division and Multiplication Properties of Inequality] In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "9 x greater than 54")[$9 x > 54$] #math.equation(block: true, alt: "−12 d less than or equal to 108")[$−12 d ≤ 108$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: d is greater than or equal to negative 9. Below this is a number line ranging from negative 11 to negative 7 with tick marks for each integer. The inequality d is greater than or equal to negative 9 is graphed on the number line, with an open bracket at d equals negative 9, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 9 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: d is greater than or equal to negative 9. Below this is a number line ranging from negative 11 to negative 7 with tick marks for each integer. The inequality d is greater than or equal to negative 9 is graphed on the number line, with an open bracket at d equals negative 9, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 9 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "the fraction 5 over 6 j less than minus 60")[$frac(5, 6) j < − 60$] #math.equation(block: true, alt: "the fraction q over minus 2 greater than or equal to minus 24")[$frac(q, − 2) ≥ − 24$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: q is less than or equal to 48. Below this is a number line ranging from 46 to 50 with tick marks for each integer. The inequality q is less than or equal to 48 is graphed on the number line, with an open bracket at q equals 48, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 48, bracket.], alt: "At the top of this figure is the solution to the inequality: q is less than or equal to 48. Below this is a number line ranging from 46 to 50 with tick marks for each integer. The inequality q is less than or equal to 48 is graphed on the number line, with an open bracket at q equals 48, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 48, bracket.", caption: none) ] #strong[Solve Inequalities That Require Simplification] In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "6 p greater than 15 p minus 30")[$6 p > 15 p − 30$] #math.equation(block: true, alt: "9 h minus 7 open parenthesis h minus 1 close parenthesis less than or equal to 4 h minus 23")[$9 h − 7 ( h − 1 ) ≤ 4 h − 23$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: h is greater than or equal to 15. Below this is a number line ranging from 13 to 17 with tick marks for each integer. The inequality h is greater than or equal to 15 is graphed on the number line, with an open bracket at h equals 15, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 15 comma infinity, parenthesis.], alt: "At the top of this figure is the solution to the inequality: h is greater than or equal to 15. Below this is a number line ranging from 13 to 17 with tick marks for each integer. The inequality h is greater than or equal to 15 is graphed on the number line, with an open bracket at h equals 15, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 15 comma infinity, parenthesis.", caption: none) ] #math.equation(block: true, alt: "5 n minus 15 open parenthesis 4 minus n close parenthesis less than 10 open parenthesis n minus 6 close parenthesis plus 10 n")[$5 n − 15 ( 4 − n ) < 10 ( n − 6 ) + 10 n$] #math.equation(block: true, alt: "the fraction 3 over 8 a minus the fraction 1 over 12 a greater than the fraction 5 over 12 a plus the fraction 3 over 4")[$frac(3, 8) a − frac(1, 12) a > frac(5, 12) a + frac(3, 4)$] #solutionbox[ #figure(figph[At the top of this figure is the solution to the inequality: a is less than negative 6. Below this is a number line ranging from negative 8 to negative 4 with tick marks for each integer. The inequality a is less than negative 6 is graphed on the number line, with an open parenthesis at a equals negative 6, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 6, parenthesis.], alt: "At the top of this figure is the solution to the inequality: a is less than negative 6. Below this is a number line ranging from negative 8 to negative 4 with tick marks for each integer. The inequality a is less than negative 6 is graphed on the number line, with an open parenthesis at a equals negative 6, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 6, parenthesis.", caption: none) ] #strong[Translate to an Inequality and Solve] In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line. Five more than #emph[z] is at most 19. Three less than #emph[c] is at least 360. #solutionbox[ #figure(figph[At the top of this figure is the inequality c minus 3 is greater than or equal to 360. To the right of this is the solution to the inequality: c is greater than or equal to 363. To the right of the solution is the solution written in interval notation: bracket, 363 comma infinity, parenthesis. Below all of this is a number line ranging from 361 to 365 with tick marks for each integer. The inequality c is greater than or equal to 363 is graphed on the number line, with an open bracket at c equals 363, and a dark line extending to the right of the bracket.], alt: "At the top of this figure is the inequality c minus 3 is greater than or equal to 360. To the right of this is the solution to the inequality: c is greater than or equal to 363. To the right of the solution is the solution written in interval notation: bracket, 363 comma infinity, parenthesis. Below all of this is a number line ranging from 361 to 365 with tick marks for each integer. The inequality c is greater than or equal to 363 is graphed on the number line, with an open bracket at c equals 363, and a dark line extending to the right of the bracket.", caption: none) ] Nine times #emph[n] exceeds 42. Negative two times #emph[a] is no more than 8. #solutionbox[ #figure(figph[At the top of this figure is the inequality negative 2a is less than or equal to 8. To the right of this is the solution to the inequality: a is greater than or equal to negative 4. To the right of the solution is the solution written in interval notation: bracket, negative 4 comma infinity, parenthesis. Below all of this is a number line ranging from negative 6 to negative 2 with tick marks for each integer. The inequality a is greater than or equal to negative 4 is graphed on the number line, with an open bracket at a equals negative 4, and a dark line extending to the right of the bracket.], alt: "At the top of this figure is the inequality negative 2a is less than or equal to 8. To the right of this is the solution to the inequality: a is greater than or equal to negative 4. To the right of the solution is the solution written in interval notation: bracket, negative 4 comma infinity, parenthesis. Below all of this is a number line ranging from negative 6 to negative 2 with tick marks for each integer. The inequality a is greater than or equal to negative 4 is graphed on the number line, with an open bracket at a equals negative 4, and a dark line extending to the right of the bracket.", caption: none) ] ==== Everyday Math Describe how you have used two topics from this chapter in your life outside of your math class during the past month. === Chapter 2 Practice Test Determine whether each number is a solution to the equation #math.equation(block: false, alt: "6 x minus 3 equals x plus 20")[$6 x − 3 = x + 20$]. ⓐ 5ⓑ #math.equation(block: false, alt: "the fraction 23 over 5")[$frac(23, 5)$] #solutionbox[ ⓐ no ⓑ yes ] In the following exercises, solve each equation. #math.equation(block: true, alt: "n minus the fraction 2 over 3 equals the fraction 1 over 4")[$n − frac(2, 3) = frac(1, 4)$] #math.equation(block: true, alt: "the fraction 9 over 2 c equals 144")[$frac(9, 2) c = 144$] #solutionbox[ #math.equation(block: true, alt: "c equals 32")[$c = 32$] ] #math.equation(block: true, alt: "4 y minus 8 equals 16")[$4 y − 8 = 16$] #math.equation(block: true, alt: "−8 x minus 15 plus 9 x minus 1 equals −21")[$−8 x − 15 + 9 x − 1 = −21$] #solutionbox[ #math.equation(block: true, alt: "x equals −5")[$x = −5$] ] #math.equation(block: true, alt: "−15 a equals 120")[$−15 a = 120$] #math.equation(block: true, alt: "the fraction 2 over 3 x equals 6")[$frac(2, 3) x = 6$] #solutionbox[ #math.equation(block: true, alt: "x equals 9")[$x = 9$] ] #math.equation(block: true, alt: "x minus 3.8 equals 8.2")[$x − 3.8 = 8.2$] #math.equation(block: true, alt: "10 y equals −5 y minus 60")[$10 y = −5 y − 60$] #solutionbox[ #math.equation(block: true, alt: "y equals −4")[$y = −4$] ] #math.equation(block: true, alt: "8 n minus 2 equals 6 n minus 12")[$8 n − 2 = 6 n − 12$] #math.equation(block: true, alt: "9 m minus 2 minus 4 m minus m equals 42 minus 8")[$9 m − 2 − 4 m − m = 42 − 8$] #solutionbox[ #math.equation(block: true, alt: "m equals 9")[$m = 9$] ] #math.equation(block: true, alt: "−5 open parenthesis 2 x minus 1 close parenthesis equals 45")[$−5 ( 2 x − 1 ) = 45$] #math.equation(block: true, alt: "− open parenthesis d minus 9 close parenthesis equals 23")[$"−" ( d − 9 ) = 23$] #solutionbox[ #math.equation(block: true, alt: "d equals −14")[$d = −14$] ] #math.equation(block: true, alt: "the fraction 1 over 4 open parenthesis 12 m minus 28 close parenthesis equals 6 minus 2 open parenthesis 3 m minus 1 close parenthesis")[$frac(1, 4) ( 12 m − 28 ) = 6 − 2 ( 3 m − 1 )$] #math.equation(block: true, alt: "2 open parenthesis 6 x minus 5 close parenthesis minus 8 equals −22")[$2 ( 6 x − 5 ) − 8 = −22$] #solutionbox[ #math.equation(block: true, alt: "x equals minus the fraction 1 over 3")[$x = − frac(1, 3)$] ] #math.equation(block: true, alt: "8 open parenthesis 3 a minus 5 close parenthesis minus 7 open parenthesis 4 a minus 3 close parenthesis equals 20 minus 3 a")[$8 ( 3 a − 5 ) − 7 ( 4 a − 3 ) = 20 − 3 a$] #math.equation(block: true, alt: "the fraction 1 over 4 p minus the fraction 1 over 3 equals the fraction 1 over 2")[$frac(1, 4) p − frac(1, 3) = frac(1, 2)$] #solutionbox[ #math.equation(block: true, alt: "p equals the fraction 10 over 3")[$p = frac(10, 3)$] ] #math.equation(block: true, alt: "0.1 d plus 0.25 open parenthesis d plus 8 close parenthesis equals 4.1")[$0.1 d + 0.25 ( d + 8 ) = 4.1$] #math.equation(block: true, alt: "14 n minus 3 open parenthesis 4 n plus 5 close parenthesis equals −9 plus 2 open parenthesis n minus 8 close parenthesis")[$14 n − 3 ( 4 n + 5 ) = −9 + 2 ( n − 8 )$] #solutionbox[ contradiction; no solution ] #math.equation(block: true, alt: "9 open parenthesis 3 u minus 2 close parenthesis minus 4 [ 6 minus 8 open parenthesis u minus 1 close parenthesis ]")[$9 ( 3 u − 2 ) − 4 [ 6 − 8 ( u − 1 ) ]$] #math.equation(block: true, alt: "equals 3 open parenthesis u minus 2 close parenthesis")[$= 3 ( u − 2 )$] Solve the formula #math.equation(block: false, alt: "x minus 2 y equals 5")[$x − 2 y = 5$] for #emph[y] #linebreak() ⓐ when #math.equation(block: false, alt: "x equals −3")[$x = −3$] #linebreak() ⓑ in general #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals 4")[$y = 4$] ⓑ #math.equation(block: false, alt: "y equals the fraction 5 minus x over 2")[$y = frac(5 − x, 2)$] ] In the following exercises, graph on the number line and write in interval notation. #math.equation(block: true, alt: "x greater than or equal to −3.5")[$x ≥ −3.5$] #math.equation(block: true, alt: "x less than the fraction 11 over 4")[$x < frac(11, 4)$] #solutionbox[ #figure(figph[This figure is a number line ranging from 1 to 5 with tick marks for each integer. The inequality x is less than 11/4 is graphed on the number line, with an open parenthesis at x equals 11/4, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/4, parenthesis.], alt: "This figure is a number line ranging from 1 to 5 with tick marks for each integer. The inequality x is less than 11/4 is graphed on the number line, with an open parenthesis at x equals 11/4, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/4, parenthesis.", caption: none) ] In the following exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval notation. #math.equation(block: true, alt: "8 k greater than or equal to 5 k minus 120")[$8 k ≥ 5 k − 120$] #math.equation(block: true, alt: "3 c minus 10 open parenthesis c minus 2 close parenthesis less than 5 c plus 16")[$3 c − 10 ( c − 2 ) < 5 c + 16$] #solutionbox[ #figure(figph[This figure is a number line ranging from negative 2 to 3 with tick marks for each integer. The inequality c is greater than 1/3 is graphed on the number line, with an open parenthesis at c equals 1/3, and a dark line extending to the right of the parenthesis. Below the number line is the solution: c is greater than 1/3. To the right of the solution is the solution written in interval notation: parenthesis, 1/3 comma infinity, parenthesis], alt: "This figure is a number line ranging from negative 2 to 3 with tick marks for each integer. The inequality c is greater than 1/3 is graphed on the number line, with an open parenthesis at c equals 1/3, and a dark line extending to the right of the parenthesis. Below the number line is the solution: c is greater than 1/3. To the right of the solution is the solution written in interval notation: parenthesis, 1/3 comma infinity, parenthesis", caption: none) ] In the following exercises, translate to an equation or inequality and solve. 4 less than twice #emph[x] is 16. Fifteen more than #emph[n] is at least 48. #solutionbox[ #math.equation(block: true, alt: "n plus 15 greater than or equal to 48 ; n greater than or equal to 33")[$n + 15 ≥ 48 ; n ≥ 33$] ] Samuel paid \$25.82 for gas this week, which was \$3.47 less than he paid last week. How much had he paid last week? Jenna bought a coat on sale for \$120, which was #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] of the original price. What was the original price of the coat? #solutionbox[ #math.equation(block: false, alt: "120 equals the fraction 2 over 3 p")[$120 = frac(2, 3) p$]; The original price was \$180. ] Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took #math.equation(block: false, alt: "7 the fraction 2 over 3")[$7 frac(2, 3)$] hours, what was the speed of the bus?