#set document(title: "4.7 Graphs of 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")) == 4.7#h(0.6em)Graphs of Linear Inequalities #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Solve: #math.equation(block: false, alt: "4 x plus 3 greater than 23")[$4 x + 3 > 23$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "x greater than 5")[$x > 5$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate from algebra to English: #math.equation(block: false, alt: "x less than 5")[$x < 5$]. #linebreak() If you missed this problem, review . #solutionbox[ #emph[x] is less than 5. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "3 x minus 2 y")[$3 x − 2 y$] when #math.equation(block: false, alt: "x equals 1")[$x = 1$], #math.equation(block: false, alt: "y equals −2")[$y = −2$]. #linebreak() If you missed this problem, review . #solutionbox[ 7 ] ] === Verify Solutions to an Inequality in Two Variables We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Linear Inequality] A #strong[linear inequality] is an inequality that can be written in one of the following forms: #math.equation(block: true, alt: "A x plus B y greater than C, A x plus B y greater than or equal to C, A x plus B y less than C, A x plus B y less than or equal to C")[$A x + B y > C & & & A x + B y ≥ C & & & A x + B y < C & & & A x + B y ≤ C$]where #math.equation(block: false, alt: "A and B")[$A #h(0.2em) "and" #h(0.2em) B$] are not both zero. ] Do you remember that an inequality with one variable had many solutions? The solution to the inequality #math.equation(block: false, alt: "x greater than 3")[$x > 3$] is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. #figure(figph[The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity.], alt: "The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity.", caption: none) Similarly, inequalities in two variables have many solutions. Any ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] that makes the inequality true when we substitute in the values is a solution of the inequality. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solution of a Linear Inequality] An ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] is a #strong[solution of a linear inequality] if the inequality is true when we substitute the values of #emph[x] and #emph[y]. ] #examplebox("Example 1")[][ Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]: ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$] ⓔ #math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$] #solutionbox[ + ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]], [#figure(figph[The image shows the mathematical inequality y \> x + 4.], alt: "The image shows the mathematical inequality y > x + 4.", caption: none)]), [#figure(figph[Substitute 0 for x and 0 for y.], alt: "Substitute 0 for x and 0 for y.", caption: none)], [#figure(figph[A mathematical expression 0 \> 0 + 4 with a question mark above the greater than sign, asking if 0 is greater than 4.], alt: "A mathematical expression 0 > 0 + 4 with a question mark above the greater than sign, asking if 0 is greater than 4.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression reads '0 is not greater than or equal to 4', indicating that zero is strictly less than four.], alt: "A mathematical expression reads '0 is not greater than or equal to 4', indicating that zero is strictly less than four.", caption: none) #linebreak() So, #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is not a solution to #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$].], )) + ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$]], [#figure(figph[A mathematical inequality is shown, displaying 'y \> x + 4' in black text against a plain white background. The inequality indicates that the variable 'y' is greater than the sum of 'x' and 4.], alt: "A mathematical inequality is shown, displaying 'y > x + 4' in black text against a plain white background. The inequality indicates that the variable 'y' is greater than the sum of 'x' and 4.", caption: none)]), [#figure(figph[Substitute 1 for x and 6 for y.], alt: "Substitute 1 for x and 6 for y.", caption: none)], [#figure(figph[A math problem asking if 6 is greater than 1 + 4. The question mark above the greater than symbol indicates uncertainty, prompting evaluation of the inequality.], alt: "A math problem asking if 6 is greater than 1 + 4. The question mark above the greater than symbol indicates uncertainty, prompting evaluation of the inequality.", caption: none)], [Simplify.], [#figure(figph[The simple mathematical expression '6 \> 5' is displayed in gray text on a white background, indicating that six is greater than five.], alt: "The simple mathematical expression '6 > 5' is displayed in gray text on a white background, indicating that six is greater than five.", caption: none) #linebreak() So, #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] is a solution to #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$].], )) + ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$]], [#figure(figph[The image shows a mathematical inequality: y \> x + 4.], alt: "The image shows a mathematical inequality: y > x + 4.", caption: none)]), [#figure(figph[The image shows the text 'Substitute 2 for x and 6 for y.' The number 2 is highlighted in red, and the number 6 is highlighted in light blue.], alt: "The image shows the text 'Substitute 2 for x and 6 for y.' The number 2 is highlighted in red, and the number 6 is highlighted in light blue.", caption: none)], [#figure(figph[A mathematical equation displays '6 \>? 2 + 4', featuring a blue '6' and a red '2'. The question mark above the '\>' symbol suggests determining if 6 is strictly greater than the sum of 2 and 4.], alt: "A mathematical equation displays '6 >? 2 + 4', featuring a blue '6' and a red '2'. The question mark above the '>' symbol suggests determining if 6 is strictly greater than the sum of 2 and 4.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression '6≯6' is displayed against a white background, which means '6 is not greater than 6'.], alt: "A mathematical expression '6≯6' is displayed against a white background, which means '6 is not greater than 6'.", caption: none) #linebreak() So, #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$] is not a solution to #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$].], )) + ⓓ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$]], [#figure(figph[The mathematical inequality y \> x + 4 is displayed on a white background.], alt: "The mathematical inequality y > x + 4 is displayed on a white background.", caption: none)]), [#figure(figph[The image shows the text 'Substitute -5 for x and -15 for y.' The numbers -5 and -15 are highlighted in red and blue respectively.], alt: "The image shows the text 'Substitute -5 for x and -15 for y.' The numbers -5 and -15 are highlighted in red and blue respectively.", caption: none)], [#figure(figph[The image shows a mathematical inequality: -15 \> -5 + 4. A question mark above the greater-than symbol implies a query on its truth. The right side simplifies to -1, making the full inequality -15 \> -1, which is false.], alt: "The image shows a mathematical inequality: -15 > -5 + 4. A question mark above the greater-than symbol implies a query on its truth. The right side simplifies to -1, making the full inequality -15 > -1, which is false.", caption: none)], [Simplify.], [#figure(figph[A mathematical inequality expression showing '-15 is not greater than or equal to -1'. The symbol ≱ signifies 'not greater than or equal to'.], alt: "A mathematical inequality expression showing '-15 is not greater than or equal to -1'. The symbol ≱ signifies 'not greater than or equal to'.", caption: none) #linebreak() So, #math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$] is not a solution to #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$].], )) + ⓔ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$]], [#figure(figph[A mathematical inequality is displayed on a white background, showing 'y \> x + 4' in black text.], alt: "A mathematical inequality is displayed on a white background, showing 'y > x + 4' in black text.", caption: none)]), [#figure(figph[The image shows the text 'Substitute -8 for x and 12 for y.', with -8 highlighted in red and 12 highlighted in light blue.], alt: "The image shows the text 'Substitute -8 for x and 12 for y.', with -8 highlighted in red and 12 highlighted in light blue.", caption: none)], [#figure(figph[A mathematical inequality expression showing 12 followed by a greater than symbol, with a question mark above it, and then -8 + 4. This asks if 12 is greater than -8 + 4.], alt: "A mathematical inequality expression showing 12 followed by a greater than symbol, with a question mark above it, and then -8 + 4. This asks if 12 is greater than -8 + 4.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression displays '12 \> -4', indicating that twelve is greater than negative four.], alt: "A mathematical expression displays '12 > -4', indicating that twelve is greater than negative four.", caption: none) #linebreak() So, #math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$] is a solution to #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$].], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y greater than x minus 3")[$y > x − 3$]: ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 , 9 close parenthesis")[$( 4 , 9 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$] ⓔ #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] #solutionbox[ ⓐ yes ⓑ yes ⓒ yes ⓓ yes ⓔ no ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y less than x plus 1")[$y < x + 1$]: ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 , 6 close parenthesis")[$( 8 , 6 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$( −2 , −1 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 3 , 4 close parenthesis")[$( 3 , 4 )$] ⓔ #math.equation(block: false, alt: "open parenthesis −1 , −4 close parenthesis")[$( −1 , −4 )$] #solutionbox[ ⓐ yes ⓑ yes ⓒ no ⓓ no ⓔ yes ] ] === Recognize the Relation Between the Solutions of an Inequality and its Graph Now, we will look at how the solutions of an inequality relate to its graph. Let’s think about the number line in again. The point #math.equation(block: false, alt: "x equals 3")[$x = 3$] separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. #figure(figph[The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity. An arrow above the number line extends from 3 and points to the left. It is labeled “numbers less than 3.” An arrow above the number line extends from 3 and points to the right. It is labeled “numbers greater than 3.”], alt: "The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity. An arrow above the number line extends from 3 and points to the left. It is labeled “numbers less than 3.” An arrow above the number line extends from 3 and points to the right. It is labeled “numbers greater than 3.”", caption: none) The solution to #math.equation(block: false, alt: "x greater than 3")[$x > 3$] is the shaded part of the number line to the right of #math.equation(block: false, alt: "x equals 3")[$x = 3$]. Similarly, the line #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$] separates the plane into two regions. On one side of the line are points with #math.equation(block: false, alt: "y less than x plus 4")[$y < x + 4$]. On the other side of the line are the points with #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]. We call the line #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$] a boundary line. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Boundary Line] The line with equation #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$] is the #strong[boundary line] that separates the region where #math.equation(block: false, alt: "A x plus B y greater than C")[$A x + B y > C$] from the region where #math.equation(block: false, alt: "A x plus B y less than C")[$A x + B y < C$]. ] For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not #math.equation(block: false, alt: "a")[$a$] is included in the solution: #figure(figph[The figure shows two number lines. The number line on the left is labeled x is less than a. The number line shows a parenthesis at a and an arrow that points to the left. The number line on the right is labeled x is less than or equal to a. The number line shows a bracket at a and an arrow that points to the left.], alt: "The figure shows two number lines. The number line on the left is labeled x is less than a. The number line shows a parenthesis at a and an arrow that points to the left. The number line on the right is labeled x is less than or equal to a. The number line shows a bracket at a and an arrow that points to the left.", caption: none) Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to indicate whether or not it the line is included in the solution. This is summarized in #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "A x plus B y less than C")[$A x + B y < C$]], [#math.equation(block: false, alt: "A x plus B y less than or equal to C")[$A x + B y ≤ C$]]), [#math.equation(block: false, alt: "A x plus B y greater than C")[$A x + B y > C$]], [#math.equation(block: false, alt: "A x plus B y greater than or equal to C")[$A x + B y ≥ C$]], [Boundary line is not included in solution.], [Boundary line is included in solution.], [#strong[Boundary line is dashed.]], [#strong[Boundary line is solid.]], )) Now, let’s take a look at what we found in . We’ll start by graphing the line #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$], and then we’ll plot the five points we tested. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15).], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15).", caption: none) In we found that some of the points were solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$] and some were not. Which of the points we plotted are solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]? The points #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] and #math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$] are solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]. Notice that they are both on the same side of the boundary line #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$]. The two points #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$] are on the other side of the #strong[boundary line] #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$], and they are not solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]. For those two points, #math.equation(block: false, alt: "y less than x plus 4")[$y < x + 4$]. What about the point #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$]? Because #math.equation(block: false, alt: "6 equals 2 plus 4")[$6 = 2 + 4$], the point is a solution to the equation #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$]. So the point #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$] is on the boundary line. Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]. The point #math.equation(block: false, alt: "open parenthesis 0 , 10 close parenthesis")[$( 0 , 10 )$] clearly looks to be to the left of the boundary line, doesn’t it? Is it a solution to the inequality? #math.equation(block: true, alt: "y greater than x plus 4; 10 greater than ? 0 plus 4; 10 greater than 4, So , open parenthesis 0 , 10 close parenthesis is a solution to y greater than x plus 4 .")[$y > x + 4 & & & \ 10 limits(>)^(?) 0 + 4 & & & \ 10 > 4 & & & "So" , #h(0.2em) ( 0 , 10 ) #h(0.2em) "is a solution to" #h(0.2em) y > x + 4 .$]Any point you choose on the left side of the boundary line is a solution to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]. All points on the left are solutions. Similarly, all points on the right side of the boundary line, the side with #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$], are not solutions to #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15). To the upper left of the line is the inequality y is greater than x plus 4. To the right of the line is the inequality y is less than x plus 4.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15). To the upper left of the line is the inequality y is greater than x plus 4. To the right of the line is the inequality y is less than x plus 4.", caption: none) The graph of the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$] is shown in below. The line #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$] divides the plane into two regions. The shaded side shows the solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$]. The points on the boundary line, those where #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$], are not solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$], so the line itself is not part of the solution. We show that by making the line dashed, not solid. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as a dashed arrow extending from the bottom left toward the upper right. The coordinate plane to the upper left of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as a dashed arrow extending from the bottom left toward the upper right. The coordinate plane to the upper left of the line is shaded.", caption: [The graph of the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$].]) #examplebox("Example 2")[][ The boundary line shown is #math.equation(block: false, alt: "y equals 2 x minus 1")[$y = 2 x − 1$]. Write the inequality shown by the graph. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 x minus 1 is plotted as a solid arrow extending from the bottom left toward the upper right. The coordinate plane to the left of the line is shaded], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 x minus 1 is plotted as a solid arrow extending from the bottom left toward the upper right. The coordinate plane to the left of the line is shaded", caption: none) #solutionbox[ The line #math.equation(block: false, alt: "y equals 2 x minus 1")[$y = 2 x − 1$] is the boundary line. On one side of the line are the points with #math.equation(block: false, alt: "y greater than 2 x minus 1")[$y > 2 x − 1$] and on the other side of the line are the points with #math.equation(block: false, alt: "y less than 2 x minus 1")[$y < 2 x − 1$]. Let’s test the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and see which inequality describes its side of the boundary line. At #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$], which inequality is true: #math.equation(block: true, alt: "y greater than 2 x minus 1, or, y less than 2 x minus 1 ?; y greater than 2 x minus 1, y less than 2 x minus 1; 0 greater than ? 2 times 0 minus 1, 0 less than ? 2 times 0 minus 1; 0 greater than −1 True, 0 less than −1 False")[$y > 2 x − 1 & & "or" & & y < 2 x − 1 ? \ y > 2 x − 1 & & & & y < 2 x − 1 \ 0 limits(>)^(?) 2 · 0 − 1 & & & & 0 limits(<)^(?) 2 · 0 − 1 \ 0 > −1 #h(0.2em) "True" & & & & 0 < −1 #h(0.2em) "False"$]Since, #math.equation(block: false, alt: "y greater than 2 x minus 1")[$y > 2 x − 1$] is true, the side of the line with #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$], is the solution. The shaded region shows the solution of the inequality #math.equation(block: false, alt: "y greater than 2 x minus 1")[$y > 2 x − 1$]. Since the boundary line is graphed with a solid line, the inequality includes the equal sign. The graph shows the inequality #math.equation(block: false, alt: "y greater than or equal to 2 x minus 1")[$y ≥ 2 x − 1$]. We could use any point as a test point, provided it is not on the line. Why did we choose #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that #math.equation(block: false, alt: "y less than 2 x minus 1")[$y < 2 x − 1$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals −2 x plus 3")[$y = −2 x + 3$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x plus 3 is plotted as a solid arrow extending from the top left toward the bottom right. The coordinate plane to the right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x plus 3 is plotted as a solid arrow extending from the top left toward the bottom right. The coordinate plane to the right of the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y greater than or equal to −2 x plus 3")[$y ≥ −2 x + 3$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x minus 4")[$y = frac(1, 2) x − 4$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals one half x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the bottom right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals one half x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the bottom right of the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y less than or equal to the fraction 1 over 2 x minus 4")[$y ≤ frac(1, 2) x − 4$] ] ] #examplebox("Example 3")[][ The boundary line shown is #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$]. Write the inequality shown by the graph. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus 3 y equals 6 is plotted as a dashed arrow extending from the top left toward the bottom right. The coordinate plane to the bottom of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus 3 y equals 6 is plotted as a dashed arrow extending from the top left toward the bottom right. The coordinate plane to the bottom of the line is shaded.", caption: none) #solutionbox[ The line #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$] is the boundary line. On one side of the line are the points with #math.equation(block: false, alt: "2 x plus 3 y greater than 6")[$2 x + 3 y > 6$] and on the other side of the line are the points with #math.equation(block: false, alt: "2 x plus 3 y less than 6")[$2 x + 3 y < 6$]. Let’s test the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and see which inequality describes its side of the boundary line. At #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$], which inequality is true: #math.equation(block: true, alt: "2 x plus 3 y, greater than, 6, or, 2 x plus 3 y, less than, 6 ?; 2 x plus 3 y, greater than, 6, 2 x plus 3 y, less than, 6; 2 open parenthesis 0 close parenthesis plus 3 open parenthesis 0 close parenthesis, greater than ?, 6, 2 open parenthesis 0 close parenthesis plus 3 open parenthesis 0 close parenthesis, less than ?, 6; 0, greater than, 6 False, 0, less than, 6 True")[$2 x + 3 y & > & 6 & & & "or" & & & 2 x + 3 y & < & 6 ? \ 2 x + 3 y & > & 6 & & & & & & 2 x + 3 y & < & 6 \ 2 ( 0 ) + 3 ( 0 ) & limits(>)^(?) & 6 & & & & & & 2 ( 0 ) + 3 ( 0 ) & limits(<)^(?) & 6 \ 0 & > & 6 #h(0.2em) "False" & & & & & & 0 & < & 6 #h(0.2em) "True"$]So the side with #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is the side where #math.equation(block: false, alt: "2 x plus 3 y less than 6")[$2 x + 3 y < 6$]. (You may want to pick a point on the other side of the boundary line and check that #math.equation(block: false, alt: "2 x plus 3 y greater than 6")[$2 x + 3 y > 6$].) Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign. The graph shows the solution to the inequality #math.equation(block: false, alt: "2 x plus 3 y less than 6")[$2 x + 3 y < 6$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x minus 4 y equals 8")[$x − 4 y = 8$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 4 y equals 8 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the top of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 4 y equals 8 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the top of the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "x minus 4 y less than or equal to 8")[$x − 4 y ≤ 8$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "3 x minus y equals 6")[$3 x − y = 6$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the right of the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "3 x minus y greater than or equal to 6")[$3 x − y ≥ 6$] ] ] === Graph Linear Inequalities Now, we’re ready to put all this together to graph linear inequalities. #examplebox("Example 4")[How to Graph Linear Inequalities][ Graph the #strong[linear inequality] #math.equation(block: false, alt: "y greater than or equal to the fraction 3 over 4 x minus 2")[$y ≥ frac(3, 4) x − 2$]. #solutionbox[ #figure(figph[This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Identify and graph the boundary line. If the inequality is less than or equal to or greater than or equal to, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. The text in the second cell reads: “Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y equals three-fourths x minus 2. The inequality sign is greater than or equal to, so we draw a solid line. The third cell contains the graph of the line three-fourths x minus 2 on a coordinate plane.], alt: "This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Identify and graph the boundary line. If the inequality is less than or equal to or greater than or equal to, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. The text in the second cell reads: “Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y equals three-fourths x minus 2. The inequality sign is greater than or equal to, so we draw a solid line. The third cell contains the graph of the line three-fourths x minus 2 on a coordinate plane.", caption: none) #figure(figph[In the second row of the table, the first cell says: “Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality? In the second cell, the instructions say: “We’ll test (0, 0). Is it a solution of the inequality?” The third cell asks: At (0, 0), is y greater than or equal to three-fourths x minus 2? Below that is the inequality 0 is greater than or equal to three-fourths 0 minus 2, with a question mark above the inequality symbol. Below that is the inequality 0 is greater than or equal to negative 2. Below that is: “So (0, 0) is a solution.], alt: "In the second row of the table, the first cell says: “Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality? In the second cell, the instructions say: “We’ll test (0, 0). Is it a solution of the inequality?” The third cell asks: At (0, 0), is y greater than or equal to three-fourths x minus 2? Below that is the inequality 0 is greater than or equal to three-fourths 0 minus 2, with a question mark above the inequality symbol. Below that is the inequality 0 is greater than or equal to negative 2. Below that is: “So (0, 0) is a solution.", caption: none) #figure(figph[In the third row of the table, the first cell says: “Step 3. Shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. In the second cell, the instructions say: The test point (0, 0) is a solution to y is greater than or equal to three-fourths x minus 2. So we shade in that side.” In the third cell is the graph of the line three-fourths x minus 2 on a coordinate plane with the region above the line shaded.], alt: "In the third row of the table, the first cell says: “Step 3. Shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. In the second cell, the instructions say: The test point (0, 0) is a solution to y is greater than or equal to three-fourths x minus 2. So we shade in that side.” In the third cell is the graph of the line three-fourths x minus 2 on a coordinate plane with the region above the line shaded.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the linear inequality #math.equation(block: false, alt: "y greater than the fraction 5 over 2 x minus 4")[$y > frac(5, 2) x − 4$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-halves x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-halves x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the linear inequality #math.equation(block: false, alt: "y less than the fraction 2 over 3 x minus 5")[$y < frac(2, 3) x − 5$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals two-thirds x minus 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals two-thirds x minus 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) ] ] The steps we take to graph a linear inequality are summarized here. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph a linear inequality.] + Identify and graph the boundary line. - If the inequality is #math.equation(block: false, alt: "less than or equal to or greater than or equal to")[$≤ "or" ≥$], the boundary line is solid. - If the inequality is \< or \>, the boundary line is dashed. + Test a point that is not on the boundary line. Is it a solution of the inequality? + Shade in one side of the boundary line. - If the test point is a solution, shade in the side that includes the point. - If the test point is not a solution, shade in the opposite side. ] #examplebox("Example 5")[][ Graph the linear inequality #math.equation(block: false, alt: "x minus 2 y less than 5")[$x − 2 y < 5$]. #solutionbox[ First we graph the boundary line #math.equation(block: false, alt: "x minus 2 y equals 5")[$x − 2 y = 5$]. The inequality is #math.equation(block: false, alt: "less than")[$<$] so we draw a dashed line. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right.", caption: none) Then we test a point. We’ll use #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] again because it is easy to evaluate and it is not on the boundary line. Is #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] a solution of #math.equation(block: false, alt: "x minus 2 y less than 5")[$x − 2 y < 5$]? #linebreak() #figure(figph[The figure shows the inequality 0 minus 2 times 0 in parentheses is less than 5, with a question mark above the inequality symbol. The next line shows 0 minus 0 is less than 5, with a question mark above the inequality symbol. The third line shows 0 is less than 5.], alt: "The figure shows the inequality 0 minus 2 times 0 in parentheses is less than 5, with a question mark above the inequality symbol. The next line shows 0 minus 0 is less than 5, with a question mark above the inequality symbol. The third line shows 0 is less than 5.", caption: none) The point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is a solution of #math.equation(block: false, alt: "x minus 2 y less than 5")[$x − 2 y < 5$], so we shade in that side of the boundary line. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The point (0, 0) is plotted, but not labeled. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The point (0, 0) is plotted, but not labeled. The region above the line is shaded.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the linear inequality #math.equation(block: false, alt: "2 x minus 3 y less than or equal to 6")[$2 x − 3 y ≤ 6$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 3 y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 3 y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the linear inequality #math.equation(block: false, alt: "2 x minus y greater than 3")[$2 x − y > 3$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 3 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 3 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) ] ] What if the boundary line goes through the origin? Then we won’t be able to use #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] as a test point. No problem—we’ll just choose some other point that is not on the boundary line. #examplebox("Example 6")[][ Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to −4 x")[$y ≤ −4 x$]. #solutionbox[ First we graph the boundary line #math.equation(block: false, alt: "y equals −4 x")[$y = −4 x$]. It is in slope–intercept form, with #math.equation(block: false, alt: "m equals −4 and b equals 0")[$m = −4 #h(0.2em) "and" #h(0.2em) b = 0$]. The inequality is #math.equation(block: false, alt: "less than or equal to")[$≤$] so we draw a solid line. #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line s y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line s y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right.", caption: none) Now, we need a test point. We can see that the point #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$] is not on the boundary line. Is #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$] a solution of #math.equation(block: false, alt: "y less than or equal to −4 x")[$y ≤ −4 x$]? #linebreak() #figure(figph[The figure shows 0 is less than or equal to negative 4 times 1 in parentheses, with a question mark above the inequality symbol. The next line shows 0 is not less than or equal to negative 4.], alt: "The figure shows 0 is less than or equal to negative 4 times 1 in parentheses, with a question mark above the inequality symbol. The next line shows 0 is not less than or equal to negative 4.", caption: none) The point #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$] is not a solution to #math.equation(block: false, alt: "y less than or equal to −4 x")[$y ≤ −4 x$], so we shade in the opposite side of the boundary line. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right. The point (1, 0) is plotted, but not labeled. The region to the left of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right. The point (1, 0) is plotted, but not labeled. The region to the left of the line is shaded.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the linear inequality #math.equation(block: false, alt: "y greater than −3 x")[$y > −3 x$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a dashed arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a dashed arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the linear inequality #math.equation(block: false, alt: "y greater than or equal to −2 x")[$y ≥ −2 x$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x is plotted as a solid arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x is plotted as a solid arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.", caption: none) ] ] Some linear inequalities have only one variable. They may have an #emph[x] but no #emph[y], or a #emph[y] but no #emph[x]. In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember? #math.equation(block: true, alt: "x equals a, vertical line; y equals b, horizontal line")[$x = a & & & "vertical line" \ y = b & & & "horizontal line"$]#examplebox("Example 7")[][ Graph the linear inequality #math.equation(block: false, alt: "y greater than 3")[$y > 3$]. #solutionbox[ First we graph the boundary line #math.equation(block: false, alt: "y equals 3")[$y = 3$]. It is a horizontal line. The inequality is \> so we draw a dashed line. We test the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]. #math.equation(block: true, alt: "y greater than 3; 0 greater than 3")[$y > 3 \ \ 0 cancel(>) 3$]#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is not a solution to #math.equation(block: false, alt: "y greater than 3")[$y > 3$]. So we shade the side that does not include (0, 0). #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the linear inequality #math.equation(block: false, alt: "y less than 5")[$y < 5$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 5 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 5 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to −1")[$y ≤ −1$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 1 is plotted as a dashed arrow horizontally across the plane. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 1 is plotted as a dashed arrow horizontally across the plane. The region below the line is shaded.", caption: none) ] ] === Key Concepts - #strong[To Graph a Linear Inequality] + Identify and graph the boundary line. #linebreak() If the inequality is #math.equation(block: false, alt: "less than or equal to or greater than or equal to")[$≤ "or" ≥$], the boundary line is solid. #linebreak() If the inequality is \< or \>, the boundary line is dashed. + Test a point that is not on the boundary line. Is it a solution of the inequality? + Shade in one side of the boundary line. #linebreak() If the test point is a solution, shade in the side that includes the point. #linebreak() If the test point is not a solution, shade in the opposite side. === Section Exercises ==== Practice Makes Perfect #strong[Verify Solutions to an Inequality in Two Variables] In the following exercises, determine whether each ordered pair is a solution to the given inequality. Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y greater than x minus 1")[$y > x − 1$]: + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −4 , −1 close parenthesis")[$( −4 , −1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis")[$( 4 , 2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$] Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y greater than x minus 3")[$y > x − 3$]: + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −6 , −3 close parenthesis")[$( −6 , −3 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$] #solutionbox[ ⓐ yes ⓑ yes ⓒ no ⓓ yes ⓔ yes ] Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y less than x plus 2")[$y < x + 2$]: + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −3 , −2 close parenthesis")[$( −3 , −2 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis −1 , 4 close parenthesis")[$( −1 , 4 )$] Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y less than x plus 5")[$y < x + 5$]: + ⓐ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −6 , −2 close parenthesis")[$( −6 , −2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 5 , −4 close parenthesis")[$( 5 , −4 )$] #solutionbox[ ⓐ yes ⓑ no ⓒ yes ⓓ yes ⓔ yes ] Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "x plus y greater than 4")[$x + y > 4$]: + ⓐ #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 10 , −5 close parenthesis")[$( 10 , −5 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "x plus y greater than 2")[$x + y > 2$]: + ⓐ #math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$( 1 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] #solutionbox[ ⓐ no ⓑ no ⓒ no ⓓ yes ⓔ yes ] #strong[Recognize the Relation Between the Solutions of an Inequality and its Graph] In the following exercises, write the inequality shown by the shaded region. Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals 3 x minus 4 .")[$y = 3 x − 4 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3x minus 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3x minus 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the right of the line is shaded.", caption: none) Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals 2 x minus 4 .")[$y = 2 x − 4 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2x minus 4 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2x minus 4 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y less than 2 x minus 4")[$y < 2 x − 4$] ] Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 1 .")[$y = frac(1, 2) x + 1 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-half x plus 1 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-half x plus 1 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals minus the fraction 1 over 3 x minus 2 .")[$y = − frac(1, 3) x − 2 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y less than or equal to minus the fraction 1 over 3 x minus 2")[$y ≤ − frac(1, 3) x − 2$] ] Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x plus y equals 5 .")[$x + y = 5 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 5 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 5 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.", caption: none) Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x plus y equals 3 .")[$x + y = 3 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 3 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 3 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "x plus y greater than or equal to 3")[$x + y ≥ 3$] ] Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "2 x plus y equals −4 .")[$2 x + y = −4 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus y equals negative 4 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus y equals negative 4 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.", caption: none) Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x plus 2 y equals −2 .")[$x + 2 y = −2 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 2 y equals negative 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 2 y equals negative 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "x plus 2 y less than or equal to −2")[$x + 2 y ≤ −2$] ] Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "3 x minus y equals 6 .")[$3 x − y = 6 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.", caption: none) Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "2 x minus y equals 4 .")[$2 x − y = 4 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "2 x minus y less than 4")[$2 x − y < 4$] ] Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "2 x minus 5 y equals 10 .")[$2 x − 5 y = 10 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 5 y equals 10 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 5 y equals 10 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "4 x minus 3 y equals 12 .")[$4 x − 3 y = 12 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x minus 3 y equals 12 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x minus 3 y equals 12 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "4 x minus 3 y greater than 12")[$4 x − 3 y > 12$] ] #strong[Graph Linear Inequalities] In the following exercises, graph each linear inequality. Graph the linear inequality #math.equation(block: false, alt: "y greater than the fraction 2 over 3 x minus 1")[$y > frac(2, 3) x − 1$]. Graph the linear inequality #math.equation(block: false, alt: "y less than the fraction 3 over 5 x plus 2")[$y < frac(3, 5) x + 2$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals three-fifths x plus 2 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals three-fifths x plus 2 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to minus the fraction 1 over 2 x plus 4")[$y ≤ − frac(1, 2) x + 4$]. Graph the linear inequality #math.equation(block: false, alt: "y greater than or equal to minus the fraction 1 over 3 x minus 2")[$y ≥ − frac(1, 3) x − 2$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "x minus y less than or equal to 3")[$x − y ≤ 3$]. Graph the linear inequality #math.equation(block: false, alt: "x minus y greater than or equal to −2")[$x − y ≥ −2$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 2 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 2 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "4 x plus y greater than −4")[$4 x + y > −4$]. Graph the linear inequality #math.equation(block: false, alt: "x plus 5 y less than −5")[$x + 5 y < −5$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 5 y equals negative 5 is plotted as a dashed line extending from the top left toward the bottom right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 5 y equals negative 5 is plotted as a dashed line extending from the top left toward the bottom right. The region below the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "3 x plus 2 y greater than or equal to −6")[$3 x + 2 y ≥ −6$]. Graph the linear inequality #math.equation(block: false, alt: "4 x plus 2 y greater than or equal to −8")[$4 x + 2 y ≥ −8$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x plus 2 y equals negative 8 is plotted as a solid line extending from the top left toward the bottom right. The region to the right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x plus 2 y equals negative 8 is plotted as a solid line extending from the top left toward the bottom right. The region to the right of the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y greater than 4 x")[$y > 4 x$]. Graph the linear inequality #math.equation(block: false, alt: "y greater than x")[$y > x$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x is plotted as a solid line extending from the bottom left toward the top right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x is plotted as a solid line extending from the bottom left toward the top right. The region above the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to − x")[$y ≤ "−" x$]. Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to −3 x")[$y ≤ −3 x$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a solid line extending from the top left toward the bottom right. The region to the left of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a solid line extending from the top left toward the bottom right. The region to the left of the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y greater than or equal to −2")[$y ≥ −2$]. Graph the linear inequality #math.equation(block: false, alt: "y less than −1")[$y < −1$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 1 is plotted as a dashed horizontal line. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 1 is plotted as a dashed horizontal line. The region below the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y less than 4")[$y < 4$]. Graph the linear inequality #math.equation(block: false, alt: "y greater than or equal to 2")[$y ≥ 2$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 is plotted as a solid horizontal line. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 is plotted as a solid horizontal line. The region above the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "x less than or equal to 5")[$x ≤ 5$]. Graph the linear inequality #math.equation(block: false, alt: "x greater than −2")[$x > −2$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 2 is plotted as a dashed vertical line. The region to the right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 2 is plotted as a dashed vertical line. The region to the right of the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "x greater than −3")[$x > −3$]. Graph the linear inequality #math.equation(block: false, alt: "x less than or equal to 4")[$x ≤ 4$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals 4 is plotted as a solid vertical line. The region to the left of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals 4 is plotted as a solid vertical line. The region to the left of the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "x minus y less than 4")[$x − y < 4$]. Graph the linear inequality #math.equation(block: false, alt: "x minus y less than −3")[$x − y < −3$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y greater than or equal to the fraction 3 over 2 x")[$y ≥ frac(3, 2) x$]. Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to the fraction 5 over 4 x")[$y ≤ frac(5, 4) x$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-fourths x is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-fourths x is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y greater than −2 x plus 1")[$y > −2 x + 1$]. Graph the linear inequality #math.equation(block: false, alt: "y less than −3 x minus 4")[$y < −3 x − 4$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x minus 4 is plotted as a dashed line extending from the top left toward the bottom right. The region to the left of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x minus 4 is plotted as a dashed line extending from the top left toward the bottom right. The region to the left of the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "x less than or equal to −1")[$x ≤ −1$]. Graph the linear inequality #math.equation(block: false, alt: "x greater than or equal to 0")[$x ≥ 0$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 0 is plotted as a solid vertical line along the y-axis. The region to the right of the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 0 is plotted as a solid vertical line along the y-axis. The region to the right of the line is shaded.", caption: none) ] ==== Everyday Math #strong[Money.] Gerry wants to have a maximum of \$100 cash at the ticket booth when his church carnival opens. He will have \$1 bills and \$5 bills. If #emph[x] is the number of \$1 bills and #emph[y] is the number of \$5 bills, the inequality #math.equation(block: false, alt: "x plus 5 y less than or equal to 100")[$x + 5 y ≤ 100$] models the situation. + ⓐ Graph the inequality. + ⓑ List three solutions to the inequality #math.equation(block: false, alt: "x plus 5 y less than or equal to 100")[$x + 5 y ≤ 100$] where both #emph[x] and #emph[y] are integers. #strong[Shopping.] Tula has \$20 to spend at the used book sale. Hardcover books cost \$2 each and paperback books cost \$0.50 each. If #emph[x] is the number of hardcover books Tula can buy and #emph[y] is the number of paperback books she can buy, the inequality #math.equation(block: false, alt: "2 x plus the fraction 1 over 2 y less than or equal to 20")[$2 x + frac(1, 2) y ≤ 20$] models the situation. + ⓐ Graph the inequality. + ⓑ List three solutions to the inequality #math.equation(block: false, alt: "2 x plus the fraction 1 over 2 y less than or equal to 20")[$2 x + frac(1, 2) y ≤ 20$] where both #emph[x] and #emph[y] are whole numbers. #solutionbox[ + ⓐ #linebreak() #figure(figph[The graph shows the x y-coordinate plane. The x- axis runs from 0 to 20 and the y-axis runs from 0 to 30. The line 2 x plus one-half y equals 20 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- axis runs from 0 to 20 and the y-axis runs from 0 to 30. The line 2 x plus one-half y equals 20 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.", caption: none) + ⓑ Answers will vary. ] ==== Writing Exercises Lester thinks that the solution of any inequality with a \> sign is the region above the line and the solution of any inequality with a \< sign is the region below the line. Is Lester correct? Explain why or why not. Explain why in some graphs of linear inequalities the boundary line is solid but in other graphs it is dashed. ==== 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 four 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 “verify solutions to an inequality in two variables,”, “recognize the relation between the solutions of an inequality and its graph,” and “graph linear inequalities.” The rest of the cells are blank.], alt: "This is a table that has four 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 “verify solutions to an inequality in two variables,”, “recognize the relation between the solutions of an inequality and its graph,” and “graph linear inequalities.” 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 4 Review Exercises ==== Rectangular Coordinate System #strong[Plot Points in a Rectangular Coordinate System] In the following exercises, plot each point in a rectangular coordinate system. + ⓐ #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis")[$( −3 , 4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 1 , the fraction 5 over 2 close parenthesis")[$( 1 , frac(5, 2) )$] + ⓐ #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$( −4 , 3 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis −4 , −3 close parenthesis")[$( −4 , −3 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (4, 3) is plotted and labeled "a". The point (negative 4, 3) is plotted and labeled "b". The point (negative 4, negative 3) is plotted and labeled "c". The point (4, negative 3) is plotted and labeled “d”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (4, 3) is plotted and labeled \"a\". The point (negative 4, 3) is plotted and labeled \"b\". The point (negative 4, negative 3) is plotted and labeled \"c\". The point (4, negative 3) is plotted and labeled “d”.", caption: none) ] + ⓐ #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] + ⓐ #math.equation(block: false, alt: "open parenthesis 2 , the fraction 3 over 2 close parenthesis")[$( 2 , frac(3, 2) )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 3 , the fraction 4 over 3 close parenthesis")[$( 3 , frac(4, 3) )$] + ⓒ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 3 , −4 close parenthesis")[$( frac(1, 3) , −4 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 2 , −5 close parenthesis")[$( frac(1, 2) , −5 )$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (2, three halves) is plotted and labeled "a". The point (3, four thirds) is plotted and labeled "b". The point (one third, negative 4) is plotted and labeled "c". The point (one-half, negative 5) is plotted and labeled “d”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (2, three halves) is plotted and labeled \"a\". The point (3, four thirds) is plotted and labeled \"b\". The point (one third, negative 4) is plotted and labeled \"c\". The point (one-half, negative 5) is plotted and labeled “d”.", caption: none) ] #strong[Identify Points on a Graph] In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (5, 3) is plotted and labeled "a". The point (2, negative 1) is plotted and labeled "b". The point (negative 3, negative 2) is plotted and labeled "c". The point (negative 1, 4) is plotted and labeled “d”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (5, 3) is plotted and labeled \"a\". The point (2, negative 1) is plotted and labeled \"b\". The point (negative 3, negative 2) is plotted and labeled \"c\". The point (negative 1, 4) is plotted and labeled “d”.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (2, 0) is plotted and labeled "a". The point (0, negative 5) is plotted and labeled "b". The point (negative 4, 0) is plotted and labeled "c". The point (0, 3) is plotted and labeled “d”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (2, 0) is plotted and labeled \"a\". The point (0, negative 5) is plotted and labeled \"b\". The point (negative 4, 0) is plotted and labeled \"c\". The point (0, 3) is plotted and labeled “d”.", caption: none) #solutionbox[ ⓐ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −4.0 close parenthesis")[$( −4.0 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$] ] #strong[Verify Solutions to an Equation in Two Variables] In the following exercises, which ordered pairs are solutions to the given equations? #math.equation(block: true, alt: "5 x plus y equals 10")[$5 x + y = 10$] + ⓐ #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 4 , −10 close parenthesis")[$( 4 , −10 )$] #math.equation(block: true, alt: "y equals 6 x minus 2")[$y = 6 x − 2$] + ⓐ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 3 , 0 close parenthesis")[$( frac(1, 3) , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 6 , −2 close parenthesis")[$( 6 , −2 )$] #solutionbox[ a, b ] #strong[Complete a Table of Solutions to a Linear Equation in Two Variables] In the following exercises, complete the table to find solutions to each linear equation. #math.equation(block: true, alt: "y equals 4 x minus 1")[$y = 4 x − 1$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]), [0], [], [], [1], [], [], [#math.equation(block: false, alt: "−2")[$−2$]], [], [], )) #math.equation(block: true, alt: "y equals minus the fraction 1 over 2 x plus 3")[$y = − frac(1, 2) x + 3$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]), [0], [], [], [4], [], [], [#math.equation(block: false, alt: "−2")[$−2$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]), [0], [3], [#math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]], [4], [1], [(4, 1)], [#math.equation(block: false, alt: "−2")[$−2$]], [4], [#math.equation(block: false, alt: "open parenthesis −2 , 4 close parenthesis")[$( −2 , 4 )$]], )) ] #math.equation(block: true, alt: "x plus 2 y equals 5")[$x + 2 y = 5$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]), [], [0], [], [1], [], [], [#math.equation(block: false, alt: "−1")[$−1$]], [], [], )) #math.equation(block: true, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$] #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [0], [], [], [], [0], [], [#math.equation(block: false, alt: "−2")[$−2$]], [], [], )) #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]), [0], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]], [2], [0], [#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$]], )) ] #strong[Find Solutions to a Linear Equation in Two Variables] In the following exercises, find three solutions to each linear equation. #math.equation(block: true, alt: "x plus y equals 3")[$x + y = 3$] #math.equation(block: true, alt: "x plus y equals −4")[$x + y = −4$] #solutionbox[ Answers will vary. ] #math.equation(block: true, alt: "y equals 3 x plus 1")[$y = 3 x + 1$] #math.equation(block: true, alt: "y equals − x minus 1")[$y = "−" x − 1$] #solutionbox[ Answers will vary. ] ==== Graphing Linear Equations #strong[Recognize the Relation Between the Solutions of an Equation and its Graph] In the following exercises, for each ordered pair, decide: + ⓐ Is the ordered pair a solution to the equation? + ⓑ Is the point on the line? #math.equation(block: true, alt: "y equals − x plus 4")[$y = "−" x + 4$] ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] ⓑ #math.equation(block: false, alt: "open parenthesis −1 , 3 close parenthesis")[$( −1 , 3 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$( 2 , 2 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 4 is plotted as an arrow extending from the top left toward the bottom right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 4 is plotted as an arrow extending from the top left toward the bottom right.", caption: none) #math.equation(block: true, alt: "y equals the fraction 2 over 3 x minus 1")[$y = frac(2, 3) x − 1$] ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis")[$( 3 , 1 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −3 , −3 close parenthesis")[$( −3 , −3 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 6 , 4 close parenthesis")[$( 6 , 4 )$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 1 is plotted as an arrow extending from the bottom left toward the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 1 is plotted as an arrow extending from the bottom left toward the top right.", caption: none) #solutionbox[ ⓐ yes; yes ⓑ yes; yes ⓒ yes; yes ⓓ no; no ] #strong[Graph a Linear Equation by Plotting Points] In the following exercises, graph by plotting points. #math.equation(block: true, alt: "y equals 4 x minus 3")[$y = 4 x − 3$] #math.equation(block: true, alt: "y equals −3 x")[$y = −3 x$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 3 x is plotted as an arrow extending from the top left toward the bottom right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 3 x is plotted as an arrow extending from the top left toward the bottom right.", caption: none) ] #math.equation(block: true, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$] #math.equation(block: true, alt: "x minus y equals 6")[$x − y = 6$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus y equals 6 is plotted as an arrow extending from the bottom left toward the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus y equals 6 is plotted as an arrow extending from the bottom left toward the top right.", caption: none) ] #math.equation(block: true, alt: "2 x plus y equals 7")[$2 x + y = 7$] #math.equation(block: true, alt: "3 x minus 2 y equals 6")[$3 x − 2 y = 6$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 3 x minus 2 y equals 6 is plotted as an arrow extending from the bottom left toward the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 3 x minus 2 y equals 6 is plotted as an arrow extending from the bottom left toward the top right.", caption: none) ] #strong[Graph Vertical and Horizontal lines] In the following exercises, graph each equation. #math.equation(block: true, alt: "y equals −2")[$y = −2$] #math.equation(block: true, alt: "x equals 3")[$x = 3$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x equals 3 is plotted as a vertical line.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x equals 3 is plotted as a vertical line.", caption: none) ] In the following exercises, graph each pair of equations in the same rectangular coordinate system. #math.equation(block: false, alt: "y equals −2 x")[$y = −2 x$] and #math.equation(block: false, alt: "y equals −2")[$y = −2$] #math.equation(block: false, alt: "y equals the fraction 4 over 3 x")[$y = frac(4, 3) x$] and #math.equation(block: false, alt: "y equals the fraction 4 over 3")[$y = frac(4, 3)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals four-thirds x is plotted as an arrow extending from the bottom left toward the top right. The line y equals four-thirds is plotted as a horizontal line.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals four-thirds x is plotted as an arrow extending from the bottom left toward the top right. The line y equals four-thirds is plotted as a horizontal line.", caption: none) ] ==== Graphing with Intercepts #strong[Identify the #emph[x]- and #emph[y]-Intercepts on a Graph] In the following exercises, find the #emph[x]- and #emph[y]-intercepts. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (3, 0) and (0, 3) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (3, 0) and (0, 3) is plotted.", caption: none) #solutionbox[ #math.equation(block: true, alt: "open parenthesis 3 , 0 close parenthesis , open parenthesis 0 , 3 close parenthesis")[$( 3 , 0 ) , ( 0 , 3 )$] ] #strong[Find the #emph[x]- and #emph[y]-Intercepts from an Equation of a Line] In the following exercises, find the intercepts of each equation. #math.equation(block: true, alt: "x plus y equals 5")[$x + y = 5$] #math.equation(block: true, alt: "x minus y equals −1")[$x − y = −1$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis −1 , 0 close parenthesis , open parenthesis 0 , 1 close parenthesis")[$( −1 , 0 ) , ( 0 , 1 )$] ] #math.equation(block: true, alt: "x plus 2 y equals 6")[$x + 2 y = 6$] #math.equation(block: true, alt: "2 x plus 3 y equals 12")[$2 x + 3 y = 12$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis 6 , 0 close parenthesis , open parenthesis 0 , 4 close parenthesis")[$( 6 , 0 ) , ( 0 , 4 )$] ] #math.equation(block: true, alt: "y equals the fraction 3 over 4 x minus 12")[$y = frac(3, 4) x − 12$] #math.equation(block: true, alt: "y equals 3 x")[$y = 3 x$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ] #strong[Graph a Line Using the Intercepts] In the following exercises, graph using the intercepts. #math.equation(block: true, alt: "− x plus 3 y equals 3")[$"−" x + 3 y = 3$] #math.equation(block: true, alt: "x plus y equals −2")[$x + y = −2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 2 is plotted as an arrow extending from the top left toward the bottom right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 2 is plotted as an arrow extending from the top left toward the bottom right.", caption: none) ] #math.equation(block: true, alt: "x minus y equals 4")[$x − y = 4$] #math.equation(block: true, alt: "2 x minus y equals 5")[$2 x − y = 5$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 2 x minus y equals 5 is plotted as an arrow extending from the bottom left toward the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 2 x minus y equals 5 is plotted as an arrow extending from the bottom left toward the top right.", caption: none) ] #math.equation(block: true, alt: "2 x minus 4 y equals 8")[$2 x − 4 y = 8$] #math.equation(block: true, alt: "y equals 2 x")[$y = 2 x$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 x is plotted as an arrow extending from the bottom left toward the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 x is plotted as an arrow extending from the bottom left toward the top right.", caption: none) ] ==== Slope of a Line #strong[Use Geoboards to Model Slope] In the following exercises, find the slope modeled on each geoboard. #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.", caption: none) #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$] ] #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.", caption: none) #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] ] In the following exercises, model each slope. Draw a picture to show your results. #math.equation(block: true, alt: "the fraction 1 over 3")[$frac(1, 3)$] #math.equation(block: true, alt: "the fraction 3 over 2")[$frac(3, 2)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 3 row 2.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 3 row 2.", caption: none) ] #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 2 and the point in column 3 row 3.], alt: "The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 2 and the point in column 3 row 3.", caption: none) ] #strong[Use #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to find the Slope of a Line from its Graph] In the following exercises, find the slope of each line shown. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 1, 3), (0, 0), and (1, negative 3) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 1, 3), (0, 0), and (1, negative 3) is plotted.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.", caption: none) #solutionbox[ 1 ] #figure(figph[A graph on a Cartesian coordinate plane displays a straight line with a positive slope, passing through the x-axis at (5, 0) and the y-axis at (0, -2.5), extending infinitely.], alt: "A graph on a Cartesian coordinate plane displays a straight line with a positive slope, passing through the x-axis at (5, 0) and the y-axis at (0, -2.5), extending infinitely.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 6) and (5, 2) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 6) and (5, 2) is plotted.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] ] #strong[Find the Slope of Horizontal and Vertical Lines] In the following exercises, find the slope of each line. #math.equation(block: true, alt: "y equals 2")[$y = 2$] #math.equation(block: true, alt: "x equals 5")[$x = 5$] #solutionbox[ undefined ] #math.equation(block: true, alt: "x equals −3")[$x = −3$] #math.equation(block: true, alt: "y equals −1")[$y = −1$] #solutionbox[ 0 ] #strong[Use the Slope Formula to find the Slope of a Line between Two Points] In the following exercises, use the slope formula to find the slope of the line between each pair of points. #math.equation(block: true, alt: "open parenthesis −1 , −1 close parenthesis , open parenthesis 0 , 5 close parenthesis")[$( −1 , −1 ) , ( 0 , 5 )$] #math.equation(block: true, alt: "open parenthesis 3 , 5 close parenthesis , open parenthesis 4 , −1 close parenthesis")[$( 3 , 5 ) , ( 4 , −1 )$] #solutionbox[ #math.equation(block: true, alt: "−6")[$−6$] ] #math.equation(block: true, alt: "open parenthesis −5 , −2 close parenthesis , open parenthesis 3 , 2 close parenthesis")[$( −5 , −2 ) , ( 3 , 2 )$] #math.equation(block: true, alt: "open parenthesis 2 , 1 close parenthesis , open parenthesis 4 , 6 close parenthesis")[$( 2 , 1 ) , ( 4 , 6 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 2")[$frac(5, 2)$] ] #strong[Graph a Line Given a Point and the Slope] #emph[In the following exercises, graph each line with the given point and slope.] #math.equation(block: true, alt: "open parenthesis 2 , −2 close parenthesis")[$( 2 , −2 )$]; #math.equation(block: true, alt: "m equals the fraction 5 over 2")[$m = frac(5, 2)$] #math.equation(block: true, alt: "open parenthesis −3 , 4 close parenthesis")[$( −3 , 4 )$]; #math.equation(block: true, alt: "m equals minus the fraction 1 over 3")[$m = − frac(1, 3)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 4) and (0, 3) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 4) and (0, 3) is plotted.", caption: none) ] #emph[x]-intercept #math.equation(block: false, alt: "−4")[$−4$]; #math.equation(block: false, alt: "m equals 3")[$m = 3$] #emph[y]-intercept 1; #math.equation(block: false, alt: "m equals minus the fraction 3 over 4")[$m = − frac(3, 4)$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (0, 1) and (4, negative 2) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (0, 1) and (4, negative 2) is plotted.", caption: none) ] #strong[Solve Slope Applications] In the following exercises, solve these slope applications. The roof pictured below has a rise of 10 feet and a run of 15 feet. What is its slope? #figure(figph[The figure shows a person on a ladder using a hammer on the roof of a building.], alt: "The figure shows a person on a ladder using a hammer on the roof of a building.", caption: none) A mountain road rises 50 feet for a 500-foot run. What is its slope? #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 10")[$frac(1, 10)$] ] ==== Intercept Form of an Equation of a Line #strong[Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line] In the following exercises, use the graph to find the slope and #emph[y]-intercept of each line. Compare the values to the equation #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 4 x minus 1 is plotted from the lower left to the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 4 x minus 1 is plotted from the lower left to the top right.", caption: none) #math.equation(block: true, alt: "y equals 4 x minus 1")[$y = 4 x − 1$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x plus 4 is plotted from the top left to the bottom right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x plus 4 is plotted from the top left to the bottom right.", caption: none) #math.equation(block: true, alt: "y equals minus the fraction 2 over 3 x plus 4")[$y = − frac(2, 3) x + 4$] #solutionbox[ slope #math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] ] #strong[Identify the Slope and y-Intercept from an Equation of a Line] In the following exercises, identify the slope and #emph[y]-intercept of each line. #math.equation(block: true, alt: "y equals −4 x plus 9")[$y = −4 x + 9$] #math.equation(block: true, alt: "y equals the fraction 5 over 3 x minus 6")[$y = frac(5, 3) x − 6$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 3 ; open parenthesis 0 , −6 close parenthesis")[$frac(5, 3) ; ( 0 , −6 )$] ] #math.equation(block: true, alt: "5 x plus y equals 10")[$5 x + y = 10$] #math.equation(block: true, alt: "4 x minus 5 y equals 8")[$4 x − 5 y = 8$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 5 ; open parenthesis 0 , minus the fraction 8 over 5 close parenthesis")[$frac(4, 5) ; ( 0 , − frac(8, 5) )$] ] #strong[Graph a Line Using Its Slope and Intercept] In the following exercises, graph the line of each equation using its slope and #emph[y]-intercept. #math.equation(block: true, alt: "y equals 2 x plus 3")[$y = 2 x + 3$] #math.equation(block: true, alt: "y equals − x minus 1")[$y = "−" x − 1$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x minus 1 is plotted from the top left to the bottom right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x minus 1 is plotted from the top left to the bottom right.", caption: none) ] #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x plus 3")[$y = − frac(2, 5) x + 3$] #math.equation(block: true, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 4 x minus 3 y equals 12 is plotted from the bottom left to the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 4 x minus 3 y equals 12 is plotted from the bottom left to the top right.", caption: none) ] In the following exercises, determine the most convenient method to graph each line. #math.equation(block: true, alt: "x equals 5")[$x = 5$] #math.equation(block: true, alt: "y equals −3")[$y = −3$] #solutionbox[ horizontal line ] #math.equation(block: true, alt: "2 x plus y equals 5")[$2 x + y = 5$] #math.equation(block: true, alt: "x minus y equals 2")[$x − y = 2$] #solutionbox[ intercepts ] #math.equation(block: true, alt: "y equals x plus 2")[$y = x + 2$] #math.equation(block: true, alt: "y equals the fraction 3 over 4 x minus 1")[$y = frac(3, 4) x − 1$] #solutionbox[ plotting points ] #strong[Graph and Interpret Applications of Slope–Intercept] Katherine is a private chef. The equation #math.equation(block: false, alt: "C equals 6.5 m plus 42")[$C = 6.5 m + 42$] models the relation between her weekly cost, #emph[C], in dollars and the number of meals, #emph[m], that she serves. + ⓐ Find Katherine’s cost for a week when she serves no meals. + ⓑ Find the cost for a week when she serves 14 meals. + ⓒ Interpret the slope and #emph[C]-intercept of the equation. + ⓓ Graph the equation. Marjorie teaches piano. The equation #math.equation(block: false, alt: "P equals 35 s minus 250")[$P = 35 s − 250$] models the relation between her weekly profit, #emph[P], in dollars and the number of student lessons, #emph[s], that she teaches. + ⓐ Find Marjorie’s profit for a week when she teaches no student lessons. + ⓑ Find the profit for a week when she teaches 20 student lessons. + ⓒ Interpret the slope and #emph[P]–intercept of the equation. + ⓓ Graph the equation. #solutionbox[ ⓐ −\$250 ⓑ \$450 ⓒ The slope, 35, means that Marjorie’s weekly profit, #emph[P], increases by \$35 for each additional student lesson she teaches. The #emph[P]–intercept means that when the number of lessons is 0, Marjorie loses \$250. ⓓ #linebreak() #figure(figph[The graph shows the x y-coordinate plane where h is plotted along the x-axis and P is potted along the y-axis. The x-axis runs from 0 to 24. The y-axis runs from negative 300 to 500. The line P equals 35 h minus 250 is plotted from the bottom left to the top right.], alt: "The graph shows the x y-coordinate plane where h is plotted along the x-axis and P is potted along the y-axis. The x-axis runs from 0 to 24. The y-axis runs from negative 300 to 500. The line P equals 35 h minus 250 is plotted from the bottom left to the top right.", caption: none) ] #strong[Use Slopes to Identify Parallel Lines] In the following exercises, use slopes and y-intercepts to determine if the lines are parallel. #math.equation(block: true, alt: "4 x minus 3 y equals −1 ; y equals the fraction 4 over 3 x minus 3")[$4 x − 3 y = −1 ; #h(0.2em) y = frac(4, 3) x − 3$] #math.equation(block: true, alt: "2 x minus y equals 8 ; x minus 2 y equals 4")[$2 x − y = 8 ; #h(0.2em) x − 2 y = 4$] #solutionbox[ not parallel ] #strong[Use Slopes to Identify Perpendicular Lines] In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular. #math.equation(block: true, alt: "y equals 5 x minus 1 ; 10 x plus 2 y equals 0")[$y = 5 x − 1 ; 10 x + 2 y = 0$] #math.equation(block: true, alt: "3 x minus 2 y equals 5 ; 2 x plus 3 y equals 6")[$3 x − 2 y = 5 ; 2 x + 3 y = 6$] #solutionbox[ perpendicular ] ==== Find the Equation of a Line #strong[Find an Equation of the Line Given the Slope and #emph[y]-Intercept] In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form. slope #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] and #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] #math.equation(block: false, alt: "open parenthesis 0 , −6 close parenthesis")[$( 0 , −6 )$] slope #math.equation(block: false, alt: "−5")[$−5$] and #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −5 x minus 3")[$y = −5 x − 3$] ] slope #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] slope #math.equation(block: false, alt: "−2")[$−2$] and #math.equation(block: false, alt: "y -intercept")[$y "-intercept"$] #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −2 x")[$y = −2 x$] ] In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 x plus 1 is plotted from the bottom left to the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 x plus 1 is plotted from the bottom left to the top right.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 3 x plus 5 is plotted from the top left to the bottom right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 3 x plus 5 is plotted from the top left to the bottom right.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals −3 x plus 5")[$y = −3 x + 5$] ] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals three-fourths x minus 2 is plotted from the bottom left to the top right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals three-fourths x minus 2 is plotted from the bottom left to the top right.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 4 is plotted as a horizontal line.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 4 is plotted as a horizontal line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals −4")[$y = −4$] ] #strong[Find an Equation of the Line Given the Slope and a Point] In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form. #math.equation(block: false, alt: "m equals minus the fraction 1 over 4")[$m = − frac(1, 4)$], point #math.equation(block: false, alt: "open parenthesis −8 , 3 close parenthesis")[$( −8 , 3 )$] #math.equation(block: false, alt: "m equals the fraction 3 over 5")[$m = frac(3, 5)$], point #math.equation(block: false, alt: "open parenthesis 10 , 6 close parenthesis")[$( 10 , 6 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 3 over 5 x")[$y = frac(3, 5) x$] ] Horizontal line containing #math.equation(block: false, alt: "open parenthesis −2 , 7 close parenthesis")[$( −2 , 7 )$] #math.equation(block: false, alt: "m equals −2")[$m = −2$], point #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −2 x minus 5")[$y = −2 x − 5$] ] #strong[Find an Equation of the Line Given Two Points] In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form. #math.equation(block: false, alt: "open parenthesis 2 , 10 close parenthesis")[$( 2 , 10 )$] and #math.equation(block: false, alt: "open parenthesis −2 , −2 close parenthesis")[$( −2 , −2 )$] #math.equation(block: false, alt: "open parenthesis 7 , 1 close parenthesis")[$( 7 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 over 2 x minus the fraction 5 over 2")[$y = frac(1, 2) x − frac(5, 2)$] ] #math.equation(block: false, alt: "open parenthesis 3 , 8 close parenthesis")[$( 3 , 8 )$] and #math.equation(block: false, alt: "open parenthesis 3 , −4 close parenthesis")[$( 3 , −4 )$]. #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis")[$( 5 , 2 )$] and #math.equation(block: false, alt: "open parenthesis −1 , 2 close parenthesis")[$( −1 , 2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 2")[$y = 2$] ] #strong[Find an Equation of a Line Parallel to a Given Line] In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form. line #math.equation(block: false, alt: "y equals −3 x plus 6")[$y = −3 x + 6$], point #math.equation(block: false, alt: "open parenthesis 1 , −5 close parenthesis")[$( 1 , −5 )$] line #math.equation(block: false, alt: "2 x plus 5 y equals −10")[$2 x + 5 y = −10$], point #math.equation(block: false, alt: "open parenthesis 10 , 4 close parenthesis")[$( 10 , 4 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x plus 8")[$y = − frac(2, 5) x + 8$] ] line #math.equation(block: false, alt: "x equals 4")[$x = 4$], point #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$( −2 , −1 )$] line #math.equation(block: false, alt: "y equals −5")[$y = −5$], point #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$( −4 , 3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 3")[$y = 3$] ] #strong[Find an Equation of a Line Perpendicular to a Given Line] In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form. line #math.equation(block: false, alt: "y equals minus the fraction 4 over 5 x plus 2")[$y = − frac(4, 5) x + 2$], point #math.equation(block: false, alt: "open parenthesis 8 , 9 close parenthesis")[$( 8 , 9 )$] line #math.equation(block: false, alt: "2 x minus 3 y equals 9")[$2 x − 3 y = 9$], point #math.equation(block: false, alt: "open parenthesis −4 , 0 close parenthesis")[$( −4 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x minus 6")[$y = − frac(3, 2) x − 6$] ] line #math.equation(block: false, alt: "y equals 3")[$y = 3$], point #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] line #math.equation(block: false, alt: "x equals −5")[$x = −5$] point #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 1")[$y = 1$] ] ==== Graph Linear Inequalities #strong[Verify Solutions to an Inequality in Two Variables] In the following exercises, determine whether each ordered pair is a solution to the given inequality. Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y less than x minus 3")[$y < x − 3$]: + ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis")[$( 5 , 2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis 3 , −1 close parenthesis")[$( 3 , −1 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$] Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "x plus y greater than 4")[$x + y > 4$]: + ⓐ #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$( 6 , 1 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 6 close parenthesis")[$( −3 , 6 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −5 , 10 close parenthesis")[$( −5 , 10 )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #solutionbox[ ⓐ yes ⓑ no ⓒ yes ⓓ yes ⓔ no ] #strong[Recognize the Relation Between the Solutions of an Inequality and its Graph] In the following exercises, write the inequality shown by the shaded region. Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals − x plus 2")[$y = "−" x + 2$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.", caption: none) Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals the fraction 2 over 3 x minus 3")[$y = frac(2, 3) x − 3$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y greater than or equal to the fraction 2 over 3 x minus 3")[$y ≥ frac(2, 3) x − 3$] ] Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x plus y equals −4")[$x + y = −4$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 4 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 4 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.", caption: none) Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x minus 2 y equals 6 .")[$x − 2 y = 6 .$] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus 2 y equals 6 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus 2 y equals 6 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "x minus 2 y greater than or equal to 6")[$x − 2 y ≥ 6$] ] #strong[Graph Linear Inequalities] In the following exercises, graph each linear inequality. Graph the linear inequality #math.equation(block: false, alt: "y greater than the fraction 2 over 5 x minus 4")[$y > frac(2, 5) x − 4$]. Graph the linear inequality #math.equation(block: false, alt: "y less than minus the fraction 1 over 4 x plus 3")[$y < − frac(1, 4) x + 3$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative one-fourth x plus 3 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative one-fourth x plus 3 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "x minus y less than or equal to 5")[$x − y ≤ 5$]. Graph the linear inequality #math.equation(block: false, alt: "3 x plus 2 y greater than 10")[$3 x + 2 y > 10$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 3 x plus 2 y equals 10 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 3 x plus 2 y equals 10 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to −3 x")[$y ≤ −3 x$]. Graph the linear inequality #math.equation(block: false, alt: "y less than 6")[$y < 6$]. #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 6 is plotted as a dashed, horizontal line. The region below the line is shaded.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 6 is plotted as a dashed, horizontal line. The region below the line is shaded.", caption: none) ] === Practice Test Plot each point in a rectangular coordinate system. + ⓐ #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] + ⓓ #math.equation(block: false, alt: "open parenthesis −4 , the fraction 3 over 2 close parenthesis")[$( −4 , frac(3, 2) )$] + ⓔ #math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$] Which of the given ordered pairs are solutions to the equation #math.equation(block: false, alt: "3 x minus y equals 6")[$3 x − y = 6$]? + ⓐ #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$] + ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$] + ⓒ #math.equation(block: false, alt: "open parenthesis 4 , −6 close parenthesis")[$( 4 , −6 )$] #solutionbox[ ⓐ yes ⓑ yes ⓒ no ] Find three solutions to the linear equation #math.equation(block: false, alt: "y equals −2 x minus 4")[$y = −2 x − 4$]. Find the #emph[x]- and #emph[y]-intercepts of the equation #math.equation(block: false, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$]. #solutionbox[ #math.equation(block: true, alt: "open parenthesis 3 , 0 close parenthesis , open parenthesis 0 , −4 close parenthesis")[$( 3 , 0 ) , ( 0 , −4 )$] ] Find the slope of each line shown. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 5, 2) and (0, negative 1) is plotted from the top left toward the bottom right.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 5, 2) and (0, negative 1) is plotted from the top left toward the bottom right.", caption: none) #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A vertical line passing through the point (2, 0) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A vertical line passing through the point (2, 0) is plotted.", caption: none) #solutionbox[ undefined ] #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A horizontal line passing through the point (0, 5) is plotted.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A horizontal line passing through the point (0, 5) is plotted.", caption: none) Find the slope of the line between the points #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis")[$( 5 , 2 )$] and #math.equation(block: false, alt: "open parenthesis −1 , −4 close parenthesis")[$( −1 , −4 )$]. #solutionbox[ #math.equation(block: true, alt: "1")[$1$] ] Graph the line with slope #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] containing the point #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$]. Graph the line for each of the following equations. #math.equation(block: true, alt: "y equals the fraction 5 over 3 x minus 1")[$y = frac(5, 3) x − 1$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals five-thirds x minus 1 is plotted. The line passes through the points (0, negative 1) and (three-fifths, 0).], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals five-thirds x minus 1 is plotted. The line passes through the points (0, negative 1) and (three-fifths, 0).", caption: none) ] #math.equation(block: true, alt: "y equals − x")[$y = "−" x$] #math.equation(block: true, alt: "x minus y equals 2")[$x − y = 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus y equals 2 is plotted. The line passes through the points (0, negative 2) and (2, 0).], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus y equals 2 is plotted. The line passes through the points (0, negative 2) and (2, 0).", caption: none) ] #math.equation(block: true, alt: "4 x plus 2 y equals −8")[$4 x + 2 y = −8$] #math.equation(block: true, alt: "y equals 2")[$y = 2$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 is plotted as a horizontal line passing through the point (0, 2).], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 is plotted as a horizontal line passing through the point (0, 2).", caption: none) ] #math.equation(block: true, alt: "x equals −3")[$x = −3$] Find the equation of each line. Write the equation in slope–intercept form. slope #math.equation(block: false, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 4 x minus 2")[$y = − frac(3, 4) x − 2$] ] #math.equation(block: false, alt: "m equals 2")[$m = 2$], point #math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$( −3 , −1 )$] containing #math.equation(block: false, alt: "open parenthesis 10 , 1 close parenthesis")[$( 10 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 6 , −1 close parenthesis")[$( 6 , −1 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 over 2 x minus 4")[$y = frac(1, 2) x − 4$] ] parallel to the line #math.equation(block: false, alt: "y equals minus the fraction 2 over 3 x minus 1")[$y = − frac(2, 3) x − 1$], containing the point #math.equation(block: false, alt: "open parenthesis −3 , 8 close parenthesis")[$( −3 , 8 )$] perpendicular to the line #math.equation(block: false, alt: "y equals the fraction 5 over 4 x plus 2")[$y = frac(5, 4) x + 2$], containing the point #math.equation(block: false, alt: "open parenthesis −10 , 3 close parenthesis")[$( −10 , 3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 4 over 5 x minus 5")[$y = − frac(4, 5) x − 5$] ] Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals − x minus 3")[$y = "−" x − 3$]. #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x minus 3 is plotted. The solid line passes through the points (negative 3, 0) and (0, negative 3).], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x minus 3 is plotted. The solid line passes through the points (negative 3, 0) and (0, negative 3).", caption: none) Graph each linear inequality. #math.equation(block: true, alt: "y greater than the fraction 3 over 2 x plus 5")[$y > frac(3, 2) x + 5$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals three-halves x plus 5 is plotted. The dashed line passes through the points (0, 5) and (2, 8).], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals three-halves x plus 5 is plotted. The dashed line passes through the points (0, 5) and (2, 8).", caption: none) ] #math.equation(block: true, alt: "x minus y greater than or equal to −4")[$x − y ≥ −4$] #math.equation(block: true, alt: "y less than or equal to −5 x")[$y ≤ −5 x$] #solutionbox[ #figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 5 x is plotted. The solid line passes through the points (0, 0) and (1, negative 5).], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 5 x is plotted. The solid line passes through the points (0, 0) and (1, negative 5).", caption: none) ] #math.equation(block: true, alt: "y less than 3")[$y < 3$]