#set document(title: "4.7 Solve Equations with Fractions", 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)Solve Equations with Fractions #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material. Evaluate #math.equation(block: false, alt: "x plus 4")[$x + 4$] when #math.equation(block: false, alt: "x equals −3")[$x = −3$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "1")[$1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "2 y minus 3 equals 9 .")[$2 y − 3 = 9 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "y equals 6")[$y = 6$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "y minus 3 equals −9")[$y − 3 = −9$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus 6")[$− 6$] ] ] === Determine Whether a Fraction is a Solution of an Equation As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Determine whether a number is a solution to an equation.] + Substitute the number for the variable in the equation. + Simplify the expressions on both sides of the equation. + Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution. ] #examplebox("Example 1")[][ Determine whether each of the following is a solution of #math.equation(block: false, alt: "x minus the fraction 3 over 10 equals the fraction 1 over 2 .")[$x − frac(3, 10) = frac(1, 2) .$] + ⓐ #math.equation(block: false, alt: "x equals 1")[$x = 1$] + ⓑ #math.equation(block: false, alt: "x equals the fraction 4 over 5")[$x = frac(4, 5)$] + ⓒ #math.equation(block: false, alt: "x equals minus the fraction 4 over 5")[$x = − frac(4, 5)$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [], [#figure(figph[A mathematical equation shows 'x minus three tenths equals one half' on a white background. The equation is represented as x - 3/10 = 1/2.], alt: "A mathematical equation shows 'x minus three tenths equals one half' on a white background. The equation is represented as x - 3/10 = 1/2.", caption: none)], [#figure(figph[The image displays the mathematical instruction 'Substitute 1 for x.' in a bold, teal-colored font, with the number '1' highlighted in red.], alt: "The image displays the mathematical instruction 'Substitute 1 for x.' in a bold, teal-colored font, with the number '1' highlighted in red.", caption: none)], [#figure(figph[Mathematical expression 1 - 3/10 ?= 1/2, questioning the equality. The number '1' is colored red.], alt: "Mathematical expression 1 - 3/10 ?= 1/2, questioning the equality. The number '1' is colored red.", caption: none)], [Change to fractions with a LCD of 10.], [#figure(figph[A mathematical equation shows '10/10 - 3/10 ? 5/10', where the question mark represents an unknown operator or symbol between the two sides of the equation.], alt: "A mathematical equation shows '10/10 - 3/10 ? 5/10', where the question mark represents an unknown operator or symbol between the two sides of the equation.", caption: none)], [Subtract.], [#figure(figph[A mathematical expression states that 7/10 is not equal to 5/10. The numbers are written in black against a white background.], alt: "A mathematical expression states that 7/10 is not equal to 5/10. The numbers are written in black against a white background.", caption: none)], )) Since #math.equation(block: false, alt: "x equals 1")[$x = 1$] does not result in a true equation, #math.equation(block: false, alt: "1")[$1$] is not a solution to the equation. #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [], [#figure(figph[The image displays the algebraic equation x - 3/10 = 1/2, where 'x' is an unknown variable and the fractions 3/10 and 1/2 are constants. This equation requires solving for the value of x.], alt: "The image displays the algebraic equation x - 3/10 = 1/2, where 'x' is an unknown variable and the fractions 3/10 and 1/2 are constants. This equation requires solving for the value of x.", caption: none)], [#figure(figph[The image shows the text 'Substitute 4/5 for x.' in a mathematical context, likely instructing to replace the variable 'x' with the fraction '4/5'.], alt: "The image shows the text 'Substitute 4/5 for x.' in a mathematical context, likely instructing to replace the variable 'x' with the fraction '4/5'.", caption: none)], [#figure(figph[A mathematical equation asking whether 4/5 minus 3/10 equals 1/2. The fraction 4/5 is colored red.], alt: "A mathematical equation asking whether 4/5 minus 3/10 equals 1/2. The fraction 4/5 is colored red.", caption: none)], [], [#figure(figph[The image displays a fractional subtraction problem: 8/10 - 3/10. A question mark over the equals sign asks if the result is 5/10, evaluating the truth of the statement.], alt: "The image displays a fractional subtraction problem: 8/10 - 3/10. A question mark over the equals sign asks if the result is 5/10, evaluating the truth of the statement.", caption: none)], [Subtract.], [#figure(figph[The image displays a mathematical equation: 5/10 = 5/10, followed by a checkmark, indicating that the equality is correct.], alt: "The image displays a mathematical equation: 5/10 = 5/10, followed by a checkmark, indicating that the equality is correct.", caption: none)], )) Since #math.equation(block: false, alt: "x equals the fraction 4 over 5")[$x = frac(4, 5)$] results in a true equation, #math.equation(block: false, alt: "the fraction 4 over 5")[$frac(4, 5)$] is a solution to the equation #math.equation(block: false, alt: "x minus the fraction 3 over 10 equals the fraction 1 over 2 .")[$x − frac(3, 10) = frac(1, 2) .$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓒ], []), [], [#figure(figph[A mathematical equation shows 'x - 3/10 = 1/2' on a white background, representing an algebraic problem to solve for the variable x.], alt: "A mathematical equation shows 'x - 3/10 = 1/2' on a white background, representing an algebraic problem to solve for the variable x.", caption: none)], [#figure(figph[The image shows the instruction 'Substitute -4/5 for x.' with the fraction -4/5 written in red.], alt: "The image shows the instruction 'Substitute -4/5 for x.' with the fraction -4/5 written in red.", caption: none)], [#figure(figph[A mathematical expression shows the equation: negative four-fifths minus three-tenths equals, with a question mark over the equal sign, one-half. The fraction negative four-fifths is in red.], alt: "A mathematical expression shows the equation: negative four-fifths minus three-tenths equals, with a question mark over the equal sign, one-half. The fraction negative four-fifths is in red.", caption: none)], [], [#figure(figph[A mathematical equation shows '-8/10 - 3/10' on the left, an equals sign with a question mark above it in the middle, and '5/10' on the right. The first fraction's numerator and denominator are red.], alt: "A mathematical equation shows '-8/10 - 3/10' on the left, an equals sign with a question mark above it in the middle, and '5/10' on the right. The first fraction's numerator and denominator are red.", caption: none)], [Subtract.], [#figure(figph[A mathematical expression showing that -11/10 is not equal to 5/10.], alt: "A mathematical expression showing that -11/10 is not equal to 5/10.", caption: none)], )) Since #math.equation(block: false, alt: "x equals minus the fraction 4 over 5")[$x = − frac(4, 5)$] does not result in a true equation, #math.equation(block: false, alt: "minus the fraction 4 over 5")[$− frac(4, 5)$] is not a solution to the equation. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether each number is a solution of the given equation. #math.equation(block: true, alt: "x minus the fraction 2 over 3 equals the fraction 1 over 6")[$x − frac(2, 3) = frac(1, 6)$]: + ⓐ #math.equation(block: false, alt: "x equals 1")[$x = 1$] + ⓑ #math.equation(block: false, alt: "x equals the fraction 5 over 6")[$x = frac(5, 6)$] + ⓒ #math.equation(block: false, alt: "x equals minus the fraction 5 over 6")[$x = − frac(5, 6)$] #solutionbox[ + ⓐ no + ⓑ yes + ⓒ no ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether each number is a solution of the given equation. #math.equation(block: true, alt: "y minus the fraction 1 over 4 equals the fraction 3 over 8")[$y − frac(1, 4) = frac(3, 8)$]: + ⓐ #math.equation(block: false, alt: "y equals 1")[$y = 1$] + ⓑ #math.equation(block: false, alt: "y equals minus the fraction 5 over 8")[$y = − frac(5, 8)$] + ⓒ #math.equation(block: false, alt: "y equals the fraction 5 over 8")[$y = frac(5, 8)$] #solutionbox[ + ⓐ no + ⓑ no + ⓒ yes ] ] === Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Addition, Subtraction, and Division Properties of Equality] For any numbers #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c ,")[$c ,$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a plus c equals b plus c .")[$a + c = b + c .$]], [Addition Property of Equality]), [$$ if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a minus c equals b minus c .")[$a − c = b − c .$]], [Subtraction Property of Equality], [if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "the fraction a over c equals the fraction b over c , c not equal to 0 .")[$frac(a, c) = frac(b, c) , c ≠ 0 .$]], [Division Property of Equality], )) In other words, when you add or subtract the same quantity from both sides of an equation, or divide both sides by the same quantity, you still have equality. ] #examplebox("Example 2")[][ Solve: #math.equation(block: false, alt: "y plus the fraction 9 over 16 equals the fraction 5 over 16 .")[$y + frac(9, 16) = frac(5, 16) .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed, reading 'y + 9/16 = 5/16'.], alt: "A mathematical equation is displayed, reading 'y + 9/16 = 5/16'.", caption: none)], [], [Subtract #math.equation(block: false, alt: "the fraction 9 over 16")[$frac(9, 16)$] from each side to undo the addition.], [#figure(figph[An equation showing y + 9/16 - 9/16 = 5/16 - 9/16. The subtracted 9/16 terms are highlighted in red on both sides of the equality, suggesting they might be canceled out.], alt: "An equation showing y + 9/16 - 9/16 = 5/16 - 9/16. The subtracted 9/16 terms are highlighted in red on both sides of the equality, suggesting they might be canceled out.", caption: none)], [], [Simplify on each side of the equation.], [#figure(figph[A mathematical equation shows 'y + 0 = -4/16' centered on a white background, representing a simplified algebraic expression where y is equal to -1/4.], alt: "A mathematical equation shows 'y + 0 = -4/16' centered on a white background, representing a simplified algebraic expression where y is equal to -1/4.", caption: none)], [], [Simplify the fraction.], [#figure(figph[The mathematical equation y = -1/4 is displayed on a white background, representing a horizontal line at y equals negative one-fourth.], alt: "The mathematical equation y = -1/4 is displayed on a white background, representing a horizontal line at y equals negative one-fourth.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation displays 'y + 9/16 = 5/16'.], alt: "A mathematical equation displays 'y + 9/16 = 5/16'.", caption: none)], [], [Substitute #math.equation(block: false, alt: "y equals minus the fraction 1 over 4")[$y = − frac(1, 4)$].], [#figure(figph[A mathematical equation shows -1/4 + 9/16 =? 5/16, representing the sum of two fractions with a question mark indicating a verification or unknown result.], alt: "A mathematical equation shows -1/4 + 9/16 =? 5/16, representing the sum of two fractions with a question mark indicating a verification or unknown result.", caption: none)], [], [Rewrite as fractions with the LCD.], [#figure(figph[The mathematical expression -4/16 + 9/16 ?= 5/16, asking to verify the equality of the fractional sum.], alt: "The mathematical expression -4/16 + 9/16 ?= 5/16, asking to verify the equality of the fractional sum.", caption: none)], [], [Add.], [#figure(figph[The image shows the fraction 5/16 equals 5/16, followed by a checkmark, indicating correctness or verification of the equality.], alt: "The image shows the fraction 5/16 equals 5/16, followed by a checkmark, indicating correctness or verification of the equality.", caption: none)], [], )) Since #math.equation(block: false, alt: "y equals minus the fraction 1 over 4")[$y = − frac(1, 4)$] makes #math.equation(block: false, alt: "y plus the fraction 9 over 16 equals the fraction 5 over 16")[$y + frac(9, 16) = frac(5, 16)$] a true statement, we know we have found the solution to this equation. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "y plus the fraction 11 over 12 equals the fraction 5 over 12 .")[$y + frac(11, 12) = frac(5, 12) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "y plus the fraction 8 over 15 equals the fraction 4 over 15 .")[$y + frac(8, 15) = frac(4, 15) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 over 15")[$− frac(4, 15)$] ] ] We used the Subtraction Property of Equality in . Now we’ll use the Addition Property of Equality. #examplebox("Example 3")[][ Solve: #math.equation(block: false, alt: "a minus the fraction 5 over 9 equals minus the fraction 8 over 9 .")[$a − frac(5, 9) = − frac(8, 9) .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed on a white background. The equation reads as 'a - 5/9 = -8/9', where 'a' is a variable, and '5/9' and '-8/9' are fractions.], alt: "A mathematical equation is displayed on a white background. The equation reads as 'a - 5/9 = -8/9', where 'a' is a variable, and '5/9' and '-8/9' are fractions.", caption: none)], [], [Add #math.equation(block: false, alt: "the fraction 5 over 9")[$frac(5, 9)$] from each side to undo the subtraction.], [#figure(figph[The image shows a mathematical equation: a - 5/9 + 5/9 = -8/9 + 5/9. The fractions 5/9 on both sides of the equation are highlighted in red.], alt: "The image shows a mathematical equation: a - 5/9 + 5/9 = -8/9 + 5/9. The fractions 5/9 on both sides of the equation are highlighted in red.", caption: none)], [], [Simplify on each side of the equation.], [#figure(figph[A mathematical equation shows 'a + 0 = -3/9' written in black text on a white background.], alt: "A mathematical equation shows 'a + 0 = -3/9' written in black text on a white background.", caption: none)], [], [Simplify the fraction.], [#figure(figph[The image displays a mathematical equation: a = -1/3. The variable 'a' is shown to be equal to negative one-third, presented in a clear, standard mathematical notation against a white background.], alt: "The image displays a mathematical equation: a = -1/3. The variable 'a' is shown to be equal to negative one-third, presented in a clear, standard mathematical notation against a white background.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation shows 'a minus five ninths equals negative eight ninths'.], alt: "A mathematical equation shows 'a minus five ninths equals negative eight ninths'.", caption: none)], [], [Substitute #math.equation(block: false, alt: "a equals minus the fraction 1 over 3")[$a = − frac(1, 3)$].], [#figure(figph[A mathematical equation with fractions: -1/3 (in red) - 5/9 with a question mark above the equals sign, followed by -8/9. It asks to verify the equality of the expression.], alt: "A mathematical equation with fractions: -1/3 (in red) - 5/9 with a question mark above the equals sign, followed by -8/9. It asks to verify the equality of the expression.", caption: none)], [], [Change to common denominator.], [#figure(figph[The equation -3/9 - 5/9 ?= -8/9, where a question mark above the equals sign prompts verification of the fractional subtraction.], alt: "The equation -3/9 - 5/9 ?= -8/9, where a question mark above the equals sign prompts verification of the fractional subtraction.", caption: none)], [], [Subtract.], [#figure(figph[A mathematical equation displays -8/9 = -8/9, accompanied by a checkmark, confirming its correctness.], alt: "A mathematical equation displays -8/9 = -8/9, accompanied by a checkmark, confirming its correctness.", caption: none)], [], )) Since #math.equation(block: false, alt: "a equals minus the fraction 1 over 3")[$a = − frac(1, 3)$] makes the equation true, we know that #math.equation(block: false, alt: "a equals minus the fraction 1 over 3")[$a = − frac(1, 3)$] is the solution to the equation. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "a minus the fraction 3 over 5 equals minus the fraction 8 over 5 .")[$a − frac(3, 5) = − frac(8, 5) .$] #solutionbox[ −1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "n minus the fraction 3 over 7 equals minus the fraction 9 over 7 .")[$n − frac(3, 7) = − frac(9, 7) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 6 over 7")[$− frac(6, 7)$] ] ] The next example may not seem to have a fraction, but let’s see what happens when we solve it. #examplebox("Example 4")[][ Solve: #math.equation(block: false, alt: "10 q equals 44 .")[$10 q = 44 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#math.equation(block: false, alt: "10 q equals 44")[$10 q = 44$]], [], [Divide both sides by 10 to undo the multiplication.], [#math.equation(block: false, alt: "the fraction 10 q over 10 equals the fraction 44 over 10")[$frac(10 q, 10) = frac(44, 10)$]], [], [Simplify.], [#math.equation(block: false, alt: "q equals the fraction 22 over 5")[$q = frac(22, 5)$]], [], [Check:], [], [], [Substitute #math.equation(block: false, alt: "q equals the fraction 22 over 5")[$q = frac(22, 5)$] into the original equation.], [#math.equation(block: false, alt: "10 open parenthesis the fraction 22 over 5 close parenthesis equals ? 44")[$10 ( frac(22, 5) ) limits(=)^(?) 44$]], [], [Simplify.], [#math.equation(block: false, alt: "10 2 open parenthesis the fraction 22 over 5 close parenthesis equals ? 44")[$limits(cancel(10))^(2) ( frac(22, cancel(5)) ) limits(=)^(?) 44$]], [], [Multiply.], [#math.equation(block: false, alt: "44 equals 44 ✓")[$44 = 44 #h(0.2em) ✓$]], [], )) The solution to the equation was the fraction #math.equation(block: false, alt: "the fraction 22 over 5 .")[$frac(22, 5) .$] We leave it as an improper fraction. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "12 u equals −76 .")[$12 u = −76 .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 19 over 3")[$− frac(19, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "8 m equals 92 .")[$8 m = 92 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 23 over 2")[$frac(23, 2)$] ] ] === Solve Equations with Fractions Using the Multiplication Property of Equality Consider the equation #math.equation(block: false, alt: "the fraction x over 4 equals 3 .")[$frac(x, 4) = 3 .$] We want to know what number divided by #math.equation(block: false, alt: "4")[$4$] gives #math.equation(block: false, alt: "3 .")[$3 .$] So to “undo” the division, we will need to multiply by #math.equation(block: false, alt: "4 .")[$4 .$] The #emph[Multiplication Property of Equality] will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[The Multiplication Property of Equality] For any numbers #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c ,")[$c ,$] #math.equation(block: true, alt: "if a equals b , then a c equals b c .")[$"if" #h(0.2em) a = b , "then" #h(0.2em) a c = b c .$]If you multiply both sides of an equation by the same quantity, you still have equality. ] Let’s use the Multiplication Property of Equality to solve the equation #math.equation(block: false, alt: "the fraction x over 7 equals −9 .")[$frac(x, 7) = −9 .$] #examplebox("Example 5")[][ Solve: #math.equation(block: false, alt: "the fraction x over 7 equals −9 .")[$frac(x, 7) = −9 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed on a white background, reading 'x/7 = -9'.], alt: "A mathematical equation is displayed on a white background, reading 'x/7 = -9'.", caption: none)], [], [Use the Multiplication Property of Equality to multiply both sides by #math.equation(block: false, alt: "7")[$7$]. This will isolate the variable.], [#figure(figph[An algebraic step where both sides of the equation are multiplied by 7 to isolate 'x', shown as 7 \* (x/7) = 7(-9).], alt: "An algebraic step where both sides of the equation are multiplied by 7 to isolate 'x', shown as 7 * (x/7) = 7(-9).", caption: none)], [], [Multiply.], [#figure(figph[A mathematical equation is displayed on a white background: 7x/7 = -63.], alt: "A mathematical equation is displayed on a white background: 7x/7 = -63.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation displays 'x = -63' in black text against a white background.], alt: "A mathematical equation displays 'x = -63' in black text against a white background.", caption: none)], [], [#figure(figph[The text reads 'Check.Substitute -63 for x for in the original equation.' with '-63' highlighted in red, indicating a step in solving an algebraic problem.], alt: "The text reads 'Check.Substitute -63 for x for in the original equation.' with '-63' highlighted in red, indicating a step in solving an algebraic problem.", caption: none)], [#figure(figph[A mathematical equation shows the fraction negative sixty-three over seven, followed by a question mark above an equals sign, and then negative nine. This setup asks if negative sixty-three divided by seven is equal to negative nine.], alt: "A mathematical equation shows the fraction negative sixty-three over seven, followed by a question mark above an equals sign, and then negative nine. This setup asks if negative sixty-three divided by seven is equal to negative nine.", caption: none)], [], [The equation is true.], [#figure(figph[The mathematical equation -9 = -9 is shown, followed by a checkmark, indicating that the equality is correct.], alt: "The mathematical equation -9 = -9 is shown, followed by a checkmark, indicating that the equality is correct.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction f over 5 equals −25 .")[$frac(f, 5) = −25 .$] #solutionbox[ −125 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction h over 9 equals −27 .")[$frac(h, 9) = −27 .$] #solutionbox[ −243 ] ] #examplebox("Example 6")[][ Solve: #math.equation(block: false, alt: "the fraction p over −8 equals −40 .")[$frac(p, −8) = −40 .$] #solutionbox[ Here, #math.equation(block: false, alt: "p")[$p$] is divided by #math.equation(block: false, alt: "−8 .")[$−8 .$] We must multiply by #math.equation(block: false, alt: "−8")[$−8$] to isolate #math.equation(block: false, alt: "p .")[$p .$] #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[The mathematical equation p over -8 equals -40 is displayed on a white background.], alt: "The mathematical equation p over -8 equals -40 is displayed on a white background.", caption: none)], [], [Multiply both sides by #math.equation(block: false, alt: "−8")[$−8$]], [#figure(figph[A mathematical equation shows '-8' multiplied by 'P divided by -8' on the left side, which equals '-8' multiplied by '-40' on the right side. The -8 coefficients are highlighted in red.], alt: "A mathematical equation shows '-8' multiplied by 'P divided by -8' on the left side, which equals '-8' multiplied by '-40' on the right side. The -8 coefficients are highlighted in red.", caption: none)], [], [Multiply.], [#figure(figph[A mathematical equation shows a fraction with '-8p' in the numerator and '-8' in the denominator, set equal to '320'.], alt: "A mathematical equation shows a fraction with '-8p' in the numerator and '-8' in the denominator, set equal to '320'.", caption: none)], [], [Simplify.], [#figure(figph[The image displays the mathematical equation p = 320 in black text against a white background.], alt: "The image displays the mathematical equation p = 320 in black text against a white background.", caption: none)], [], [Check:], [], [], [Substitute #math.equation(block: false, alt: "p equals 320")[$p = 320$].], [#figure(figph[A mathematical equation shows '320' in red, divided by '-8', with a question mark over an equals sign, followed by '-40'. It asks whether 320 divided by -8 is equal to -40.], alt: "A mathematical equation shows '320' in red, divided by '-8', with a question mark over an equals sign, followed by '-40'. It asks whether 320 divided by -8 is equal to -40.", caption: none)], [], [The equation is true.], [#figure(figph[A mathematical equation shows '-40 = -40' with a black checkmark symbol to its right, indicating that the equality is correct. The numbers and symbols are in a dark gray font against a white background.], alt: "A mathematical equation shows '-40 = -40' with a black checkmark symbol to its right, indicating that the equality is correct. The numbers and symbols are in a dark gray font against a white background.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction c over −7 equals −35 .")[$frac(c, −7) = −35 .$] #solutionbox[ 245 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction x over −11 equals −12 .")[$frac(x, −11) = −12 .$] #solutionbox[ 132 ] ] ==== Solve Equations with a Coefficient of #math.equation(block: false, alt: "−1")[$−1$] Look at the equation #math.equation(block: false, alt: "minus y equals 15 .")[$− y = 15 .$] Does it look as if #math.equation(block: false, alt: "y")[$y$] is already isolated? But there is a negative sign in front of #math.equation(block: false, alt: "y ,")[$y ,$] so it is not isolated. There are three different ways to isolate the variable in this type of equation. We will show all three ways in . #examplebox("Example 7")[][ Solve: #math.equation(block: false, alt: "minus y equals 15 .")[$− y = 15 .$] #solutionbox[ One way to solve the equation is to rewrite #math.equation(block: false, alt: "minus y")[$− y$] as #math.equation(block: false, alt: "−1 y ,")[$−1 y ,$] and then use the Division Property of Equality to isolate #math.equation(block: false, alt: "y .")[$y .$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the equation -y = 15, representing a simple linear algebraic expression.], alt: "The image displays the equation -y = 15, representing a simple linear algebraic expression.", caption: none)]), [Rewrite #math.equation(block: false, alt: "minus y")[$− y$] as #math.equation(block: false, alt: "−1 y")[$−1 y$].], [#figure(figph[A mathematical equation is displayed, showing -1y = 15.], alt: "A mathematical equation is displayed, showing -1y = 15.", caption: none)], [Divide both sides by −1.], [#figure(figph[The equation shows -1y divided by -1 equals 15 divided by -1, demonstrating a step in solving for y by dividing both sides by -1.], alt: "The equation shows -1y divided by -1 equals 15 divided by -1, demonstrating a step in solving for y by dividing both sides by -1.", caption: none)], [Simplify each side.], [#figure(figph[The image displays the equation y = -15, rendered in a clear, standard mathematical font on a white background.], alt: "The image displays the equation y = -15, rendered in a clear, standard mathematical font on a white background.", caption: none)], )) Another way to solve this equation is to multiply both sides of the equation by #math.equation(block: false, alt: "−1 .")[$−1 .$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image shows a mathematical equation with a variable 'y'. The equation reads '-y = 15'.], alt: "The image shows a mathematical equation with a variable 'y'. The equation reads '-y = 15'.", caption: none)]), [Multiply both sides by −1.], [#figure(figph[A mathematical equation shows '-1(-y) = -1(15)', where both sides are multiplied by -1. The -1 is highlighted in red on both sides.], alt: "A mathematical equation shows '-1(-y) = -1(15)', where both sides are multiplied by -1. The -1 is highlighted in red on both sides.", caption: none)], [Simplify each side.], [#figure(figph[The image displays a mathematical equation in black text on a white background, stating 'y = -15'.], alt: "The image displays a mathematical equation in black text on a white background, stating 'y = -15'.", caption: none)], )) The third way to solve the equation is to read #math.equation(block: false, alt: "minus y")[$− y$] as “the opposite of #math.equation(block: false, alt: "y")[$y$].” What number has #math.equation(block: false, alt: "15")[$15$] as its opposite? The opposite of #math.equation(block: false, alt: "15")[$15$] is #math.equation(block: false, alt: "−15 .")[$−15 .$] So #math.equation(block: false, alt: "y equals −15 .")[$y = −15 .$] For all three methods, we isolated #math.equation(block: false, alt: "y")[$y$] is isolated and solved the equation. Check: #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image shows a mathematical equation with a variable 'y'. The equation reads '-y = 15'.], alt: "The image shows a mathematical equation with a variable 'y'. The equation reads '-y = 15'.", caption: none)]), [Substitute #math.equation(block: false, alt: "y equals −15")[$y = −15$].], [#figure(figph[A mathematical equation shows -(-15) ?= (15), asking if the negative of negative 15 is equal to 15, with the -15 in red text.], alt: "A mathematical equation shows -(-15) ?= (15), asking if the negative of negative 15 is equal to 15, with the -15 in red text.", caption: none)], [Simplify. The equation is true.], [#figure(figph[The number 15 is equal to 15, confirmed by a checkmark indicating correctness or verification.], alt: "The number 15 is equal to 15, confirmed by a checkmark indicating correctness or verification.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "minus y equals 48 .")[$− y = 48 .$] #solutionbox[ −48 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "minus c equals −23 .")[$− c = −23 .$] #solutionbox[ 23 ] ] ==== Solve Equations with a Fraction Coefficient When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to #math.equation(block: false, alt: "1 .")[$1 .$] For example, in the equation: #math.equation(block: true, alt: "the fraction 3 over 4 x equals 24")[$frac(3, 4) x = 24$]The coefficient of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$] To solve for #math.equation(block: false, alt: "x ,")[$x ,$] we need its coefficient to be #math.equation(block: false, alt: "1 .")[$1 .$] Since the product of a number and its reciprocal is #math.equation(block: false, alt: "1 ,")[$1 ,$] our strategy here will be to isolate #math.equation(block: false, alt: "x")[$x$] by multiplying by the reciprocal of #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$] We will do this in . #examplebox("Example 8")[][ Solve: #math.equation(block: false, alt: "the fraction 3 over 4 x equals 24 .")[$frac(3, 4) x = 24 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed on a white background: 3/4x = 24. The equation represents a linear algebraic problem involving a fraction and a variable.], alt: "A mathematical equation is displayed on a white background: 3/4x = 24. The equation represents a linear algebraic problem involving a fraction and a variable.", caption: none)], [], [Multiply both sides by the reciprocal of the coefficient.], [#figure(figph[An algebraic equation is shown: (4/3) multiplied by (3/4)x equals (4/3) multiplied by 24. The fraction 4/3 appears in red on both sides of the equation.], alt: "An algebraic equation is shown: (4/3) multiplied by (3/4)x equals (4/3) multiplied by 24. The fraction 4/3 appears in red on both sides of the equation.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation showing 1x equals the product of 4/3 and 24/1, which simplifies to 1x = 32. It demonstrates a step in solving for 'x' by multiplying fractions.], alt: "A mathematical equation showing 1x equals the product of 4/3 and 24/1, which simplifies to 1x = 32. It demonstrates a step in solving for 'x' by multiplying fractions.", caption: none)], [], [Multiply.], [#figure(figph[The mathematical expression 'x = 32' is displayed on a white background.], alt: "The mathematical expression 'x = 32' is displayed on a white background.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed on a white background, which reads '3/4x = 24'.], alt: "A mathematical equation is displayed on a white background, which reads '3/4x = 24'.", caption: none)], [], [Substitute #math.equation(block: false, alt: "x equals 32")[$x = 32$].], [#figure(figph[A mathematical equation asks if three-fourths multiplied by thirty-two equals twenty-four, with a question mark over the equals sign to indicate inquiry.], alt: "A mathematical equation asks if three-fourths multiplied by thirty-two equals twenty-four, with a question mark over the equals sign to indicate inquiry.", caption: none)], [], [Rewrite #math.equation(block: false, alt: "32")[$32$] as a fraction.], [#figure(figph[A mathematical equation shows (3/4) multiplied by (32/1) with a question mark over the equality sign, followed by the number 24, suggesting a check to see if the product equals 24.], alt: "A mathematical equation shows (3/4) multiplied by (32/1) with a question mark over the equality sign, followed by the number 24, suggesting a check to see if the product equals 24.", caption: none)], [], [Multiply. The equation is true.], [#figure(figph[A mathematical statement showing 24 = 24, followed by a checkmark to indicate correctness or verification.], alt: "A mathematical statement showing 24 = 24, followed by a checkmark to indicate correctness or verification.", caption: none)], [], )) Notice that in the equation #math.equation(block: false, alt: "the fraction 3 over 4 x equals 24 ,")[$frac(3, 4) x = 24 ,$] we could have divided both sides by #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] to get #math.equation(block: false, alt: "x")[$x$] by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 2 over 5 n equals 14 .")[$frac(2, 5) n = 14 .$] #solutionbox[ 35 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 5 over 6 y equals 15 .")[$frac(5, 6) y = 15 .$] #solutionbox[ 18 ] ] #examplebox("Example 9")[][ Solve: #math.equation(block: false, alt: "minus the fraction 3 over 8 w equals 72 .")[$− frac(3, 8) w = 72 .$] #solutionbox[ The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative. #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation showing negative three-eighths multiplied by 'w' equals 72.], alt: "A mathematical equation showing negative three-eighths multiplied by 'w' equals 72.", caption: none)], [], [Multiply both sides by the reciprocal of #math.equation(block: false, alt: "− the fraction 3 over 8")[$− frac(3, 8)$].], [#figure(figph[The equation -8/3(-3/8w) = (-8/3)72 is displayed, illustrating a linear equation involving fractions and the variable 'w'.], alt: "The equation -8/3(-3/8w) = (-8/3)72 is displayed, illustrating a linear equation involving fractions and the variable 'w'.", caption: none)], [], [Simplify; reciprocals multiply to one.], [#figure(figph[The equation shows '1w = -8/3 ×72/1', representing a multiplication of two fractions, one negative and one positive, to solve for 'w'.], alt: "The equation shows '1w = -8/3 ×72/1', representing a multiplication of two fractions, one negative and one positive, to solve for 'w'.", caption: none)], [], [Multiply.], [#figure(figph[A mathematical equation is displayed, showing the variable 'w' equal to the negative integer -192, presented in a clear, dark font against a stark white background.], alt: "A mathematical equation is displayed, showing the variable 'w' equal to the negative integer -192, presented in a clear, dark font against a stark white background.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation displays '-3/8 w = 72' in black text against a white background.], alt: "A mathematical equation displays '-3/8 w = 72' in black text against a white background.", caption: none)], [], [Let #math.equation(block: false, alt: "w equals −192")[$w = −192$].], [#figure(figph[A mathematical expression showing negative three-eighths multiplied by negative one hundred ninety-two, followed by an equals sign with a question mark above it, and then the number seventy-two.], alt: "A mathematical expression showing negative three-eighths multiplied by negative one hundred ninety-two, followed by an equals sign with a question mark above it, and then the number seventy-two.", caption: none)], [], [Multiply. It checks.], [#figure(figph[The equation 72=72 is marked as correct with a checkmark.], alt: "The equation 72=72 is marked as correct with a checkmark.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "minus the fraction 4 over 7 a equals 52 .")[$− frac(4, 7) a = 52 .$] #solutionbox[ −91 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "minus the fraction 7 over 9 w equals 84 .")[$− frac(7, 9) w = 84 .$] #solutionbox[ −108 ] ] === Translate Sentences to Equations and Solve Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference. #figure(table( columns: 2, align: left, inset: 6pt, table.header([#strong[Subtraction Property of Equality:] #linebreak() For any real numbers #math.equation(block: false, alt: "a, b,")[$"a, b,"$] and #math.equation(block: false, alt: "c,")[$"c,"$] #linebreak() #linebreak() if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a minus c equals b minus c .")[$a − c = b − c .$]], [#strong[Addition Property of Equality:] #linebreak() For any real numbers #math.equation(block: false, alt: "a, b,")[$"a, b,"$] and #math.equation(block: false, alt: "c,")[$"c,"$] #linebreak() #linebreak() if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a plus c equals b plus c .")[$a + c = b + c .$]]), [#strong[Division Property of Equality:] #linebreak() For any numbers #math.equation(block: false, alt: "a, b,")[$"a, b,"$] and #math.equation(block: false, alt: "c,")[$"c,"$] where #math.equation(block: false, alt: "c not equal to 0")[$"c" ≠ 0$] #linebreak() #linebreak() if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "the fraction a over c equals the fraction b over c")[$frac(a, c) = frac(b, c)$]], [#strong[Multiplication Property of Equality:] #linebreak() For any real numbers #math.equation(block: false, alt: "a, b,")[$"a, b,"$] and #math.equation(block: false, alt: "c")[$"c"$] #linebreak() #linebreak() if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a c equals b c")[$a c = b c$]], )) When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality. In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions. #examplebox("Example 10")[][ Translate and solve: #math.equation(block: false, alt: "n")[$n$] divided by #math.equation(block: false, alt: "6")[$6$] is #math.equation(block: false, alt: "−24 .")[$−24 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [Translate.], [#figure(figph[The image shows the phrase 'n divided by 6 is 24' with brackets indicating its parts. Below, the equation 'n/6 = -24' is displayed, translating the phrase but with a changed sign for the number 24.], alt: "The image shows the phrase 'n divided by 6 is 24' with brackets indicating its parts. Below, the equation 'n/6 = -24' is displayed, translating the phrase but with a changed sign for the number 24.", caption: none)], [], [Multiply both sides by #math.equation(block: false, alt: "6")[$6$].], [#figure(figph[The equation 6 times n/6 equals 6 times -24, illustrating a step in solving for 'n'.], alt: "The equation 6 times n/6 equals 6 times -24, illustrating a step in solving for 'n'.", caption: none)], [], [Simplify.], [#figure(figph[The image displays a mathematical equation in a black serif font on a white background, which states 'n = -144'.], alt: "The image displays a mathematical equation in a black serif font on a white background, which states 'n = -144'.", caption: none)], [], [Check:], [Is #math.equation(block: false, alt: "−144")[$−144$] divided by #math.equation(block: false, alt: "6")[$6$] equal to #math.equation(block: false, alt: "−24")[$−24$]?], [], [Translate.], [#figure(figph[A math problem displays the fraction -144/6, followed by a question mark above an equals sign, and then -24. It asks whether -144 divided by 6 is equal to -24.], alt: "A math problem displays the fraction -144/6, followed by a question mark above an equals sign, and then -24. It asks whether -144 divided by 6 is equal to -24.", caption: none)], [], [Simplify. It checks.], [#figure(figph[A mathematical equation displays '-24 = -24' followed by a checkmark, confirming the equality.], alt: "A mathematical equation displays '-24 = -24' followed by a checkmark, confirming the equality.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve: #math.equation(block: false, alt: "n")[$n$] divided by #math.equation(block: false, alt: "7")[$7$] is equal to #math.equation(block: false, alt: "−21 .")[$−21 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction n over 7 equals −21 ; n equals −147")[$frac(n, 7) = −21 ; n = −147$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve: #math.equation(block: false, alt: "n")[$n$] divided by #math.equation(block: false, alt: "8")[$8$] is equal to #math.equation(block: false, alt: "−56 .")[$−56 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction n over 8 equals −56 ; n equals −448")[$frac(n, 8) = −56 ; n = −448$] ] ] #examplebox("Example 11")[][ Translate and solve: The quotient of #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "−5")[$−5$] is #math.equation(block: false, alt: "70 .")[$70 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [Translate.], [#figure(figph[The image shows the verbal expression 'The quotient of q and -5 is 70' translated into the mathematical equation q/-5 = 70. Brackets link corresponding parts of the phrase and the equation.], alt: "The image shows the verbal expression 'The quotient of q and -5 is 70' translated into the mathematical equation q/-5 = 70. Brackets link corresponding parts of the phrase and the equation.", caption: none)], [], [Multiply both sides by #math.equation(block: false, alt: "−5")[$−5$].], [#figure(figph[A mathematical equation shows -5 multiplied by q over -5, which equals -5 multiplied by 70.], alt: "A mathematical equation shows -5 multiplied by q over -5, which equals -5 multiplied by 70.", caption: none)], [], [Simplify.], [#figure(figph[The mathematical equation 'q = -350' is displayed on a white background, representing a variable 'q' equal to the negative integer three hundred fifty.], alt: "The mathematical equation 'q = -350' is displayed on a white background, representing a variable 'q' equal to the negative integer three hundred fifty.", caption: none)], [], [Check:], [Is the quotient of #math.equation(block: false, alt: "−350")[$−350$] and #math.equation(block: false, alt: "−5")[$−5$] equal to #math.equation(block: false, alt: "70")[$70$]?], [], [Translate.], [#figure(figph[A mathematical problem demonstrating the division of negative numbers, asking if -350 divided by -5 equals 70, symbolized by a question mark over an equals sign between the two values.], alt: "A mathematical problem demonstrating the division of negative numbers, asking if -350 divided by -5 equals 70, symbolized by a question mark over an equals sign between the two values.", caption: none)], [], [Simplify. It checks.], [#figure(figph[The equation 70 = 70 is displayed, followed by a black checkmark, indicating correctness or verification.], alt: "The equation 70 = 70 is displayed, followed by a black checkmark, indicating correctness or verification.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve: The quotient of #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "−8")[$−8$] is #math.equation(block: false, alt: "72 .")[$72 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction q over −8 equals 72 ; q equals −576")[$frac(q, −8) = 72 ; q = −576$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve: The quotient of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "−9")[$−9$] is #math.equation(block: false, alt: "81 .")[$81 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction p over −9 equals 81 ; p equals −729")[$frac(p, −9) = 81 ; p = −729$] ] ] #examplebox("Example 12")[][ Translate and solve: Two-thirds of #math.equation(block: false, alt: "f")[$f$] is #math.equation(block: false, alt: "18 .")[$18 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [Translate.], [#figure(figph[A mathematical expression shows how the phrase 'Two-thirds of f is 18' translates into the algebraic equation '(2/3)f = 18,' with curly brackets indicating the corresponding parts.], alt: "A mathematical expression shows how the phrase 'Two-thirds of f is 18' translates into the algebraic equation '(2/3)f = 18,' with curly brackets indicating the corresponding parts.", caption: none)], [], [Multiply both sides by #math.equation(block: false, alt: "the fraction 3 over 2")[$frac(3, 2)$].], [#figure(figph[A math equation shows (3/2) \* (2/3)f = (3/2) \* 18, with the red fraction 3/2 applied to both sides, likely to solve for 'f' after simplification.], alt: "A math equation shows (3/2) * (2/3)f = (3/2) * 18, with the red fraction 3/2 applied to both sides, likely to solve for 'f' after simplification.", caption: none)], [], [Simplify.], [#figure(figph[The image displays the equation 'f = 27' in black text on a white background.], alt: "The image displays the equation 'f = 27' in black text on a white background.", caption: none)], [], [Check:], [Is two-thirds of #math.equation(block: false, alt: "27")[$27$] equal to #math.equation(block: false, alt: "18")[$18$]?], [], [Translate.], [#figure(figph[A mathematical equation shows two-thirds multiplied by twenty-seven, followed by a question mark over an equals sign, then eighteen. The equation asks if 2/3(27) is equal to 18.], alt: "A mathematical equation shows two-thirds multiplied by twenty-seven, followed by a question mark over an equals sign, then eighteen. The equation asks if 2/3(27) is equal to 18.", caption: none)], [], [Simplify. It checks.], [#figure(figph[The image displays the equation '18 = 18' followed by a checkmark, indicating that the mathematical statement is correct.], alt: "The image displays the equation '18 = 18' followed by a checkmark, indicating that the mathematical statement is correct.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve: Two-fifths of #math.equation(block: false, alt: "f")[$f$] is #math.equation(block: false, alt: "16 .")[$16 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 5 f equals 16 ; f equals 40")[$frac(2, 5) f = 16 ; f = 40$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve: Three-fourths of #math.equation(block: false, alt: "f")[$f$] is #math.equation(block: false, alt: "21 .")[$21 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4 f equals 21 ; f equals 28")[$frac(3, 4) f = 21 ; f = 28$] ] ] #examplebox("Example 13")[][ Translate and solve: The quotient of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$] is #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [The quotient of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$] is #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$].], [], [Translate.], [#math.equation(block: false, alt: "the fraction m over the fraction 5 over 6 equals the fraction 3 over 4")[$frac(#h(0.2em) m #h(0.2em), frac(5, 6)) = frac(3, 4)$]], [], [Multiply both sides by #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$] to isolate #math.equation(block: false, alt: "m")[$m$].], [#math.equation(block: false, alt: "the fraction 5 over 6 open parenthesis the fraction m over the fraction 5 over 6 close parenthesis equals the fraction 5 over 6 open parenthesis the fraction 3 over 4 close parenthesis")[$frac(5, 6) ( frac(#h(0.2em) m #h(0.2em), frac(5, 6)) ) = frac(5, 6) ( frac(3, 4) )$]], [], [Simplify.], [#math.equation(block: false, alt: "m equals the fraction 5 times 3 over 6 times 4")[$m = frac(5 · 3, 6 · 4)$]], [], [Remove common factors and multiply.], [#math.equation(block: false, alt: "m equals the fraction 5 over 8")[$m = frac(5, 8)$]], [], [Check:], [], [], [Is the quotient of #math.equation(block: false, alt: "the fraction 5 over 8")[$frac(5, 8)$] and #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$] equal to #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$]?], [#math.equation(block: false, alt: "the fraction the fraction 5 over 8 over the fraction 5 over 6 equals ? the fraction 3 over 4")[$frac(#h(0.2em) frac(5, 8) #h(0.2em), #h(0.2em) frac(5, 6) #h(0.2em)) limits(=)^(?) frac(3, 4)$]], [], [Rewrite as division.], [#math.equation(block: false, alt: "the fraction 5 over 8 divided by the fraction 5 over 6 equals ? the fraction 3 over 4")[$frac(5, 8) ÷ frac(5, 6) limits(=)^(?) frac(3, 4)$]], [], [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 5 over 8 times the fraction 6 over 5 equals ? the fraction 3 over 4")[$frac(5, 8) · frac(6, 5) limits(=)^(?) frac(3, 4)$]], [], [Simplify.], [#math.equation(block: false, alt: "the fraction 3 over 4 equals the fraction 3 over 4 ✓")[$frac(3, 4) = frac(3, 4) ✓$]], [], )) Our solution checks. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve. The quotient of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] is #math.equation(block: false, alt: "the fraction 5 over 12 .")[$frac(5, 12) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction n over the fraction 2 over 3 equals the fraction 5 over 12 ; n equals the fraction 5 over 18")[$frac(#h(0.2em) n #h(0.2em), #h(0.2em) frac(2, 3) #h(0.2em)) = frac(5, 12) ; n = frac(5, 18)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve The quotient of #math.equation(block: false, alt: "c")[$c$] and #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] is #math.equation(block: false, alt: "the fraction 4 over 9 .")[$frac(4, 9) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction c over the fraction 3 over 8 equals the fraction 4 over 9 ; c equals the fraction 1 over 6")[$frac(#h(0.2em) c #h(0.2em), #h(0.2em) frac(3, 8) #h(0.2em)) = frac(4, 9) ; c = frac(1, 6)$] ] ] #examplebox("Example 14")[][ Translate and solve: The sum of three-eighths and #math.equation(block: false, alt: "x")[$x$] is three and one-half. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Translate.], [#figure(figph[This image translates the phrase 'The sum of three-eighths and x is three and one-half' into the algebraic equation 3/8 + x = 3 1/2, illustrating how to set up an equation from a word problem.], alt: "This image translates the phrase 'The sum of three-eighths and x is three and one-half' into the algebraic equation 3/8 + x = 3 1/2, illustrating how to set up an equation from a word problem.", caption: none)]), [Use the Subtraction Property of Equality to subtract #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] from both sides.], [#figure(figph[A mathematical equation is shown with the expression '3/8 + x - 3/8 = 3 1/2 - 3/8' on a white background, requiring the solving for the variable 'x'.], alt: "A mathematical equation is shown with the expression '3/8 + x - 3/8 = 3 1/2 - 3/8' on a white background, requiring the solving for the variable 'x'.", caption: none)], [Combine like terms on the left side.], [#figure(figph[A mathematical equation shows 'x = 3 1/2 - 3/8' centered on a white background, representing a mixed number subtraction problem.], alt: "A mathematical equation shows 'x = 3 1/2 - 3/8' centered on a white background, representing a mixed number subtraction problem.", caption: none)], [Convert mixed number to improper fraction.], [#figure(figph[A mathematical equation displays 'x = 7/2 - 3/8' centered on a white background.], alt: "A mathematical equation displays 'x = 7/2 - 3/8' centered on a white background.", caption: none)], [Convert to equivalent fractions with LCD of 8.], [#figure(figph[A mathematical equation shows x equals 28 over 8 minus 3 over 8, demonstrating subtraction of fractions with a common denominator.], alt: "A mathematical equation shows x equals 28 over 8 minus 3 over 8, demonstrating subtraction of fractions with a common denominator.", caption: none)], [Subtract.], [#figure(figph[The equation x = 25/8 is displayed on a white background, representing a mathematical solution.], alt: "The equation x = 25/8 is displayed on a white background, representing a mathematical solution.", caption: none)], [Write as a mixed number.], [#figure(figph[A mathematical expression displays 'x = 3 1/8' on a white background.], alt: "A mathematical expression displays 'x = 3 1/8' on a white background.", caption: none)], )) We write the answer as a mixed number because the original problem used a mixed number. Check: Is the sum of three-eighths and #math.equation(block: false, alt: "3 the fraction 1 over 8")[$3 frac(1, 8)$] equal to three and one-half? #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 3 over 8 plus 3 the fraction 1 over 8 equals ? 3 the fraction 1 over 2")[$frac(3, 8) + 3 frac(1, 8) limits(=)^(?) 3 frac(1, 2)$]]), [Add.], [#math.equation(block: false, alt: "3 the fraction 4 over 8 equals ? 3 the fraction 1 over 2")[$3 frac(4, 8) limits(=)^(?) 3 frac(1, 2)$]], [Simplify.], [#math.equation(block: false, alt: "3 the fraction 1 over 2 equals 3 the fraction 1 over 2 ✓")[$3 frac(1, 2) = 3 frac(1, 2) #h(0.2em) ✓$]], )) The solution checks. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve: The sum of five-eighths and #math.equation(block: false, alt: "x")[$x$] is one-fourth. #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 8 plus x equals the fraction 1 over 4 ; x equals minus the fraction 3 over 8")[$frac(5, 8) + x = frac(1, 4) ; x = − frac(3, 8)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and solve: The difference of one-and-three-fourths and #math.equation(block: false, alt: "x")[$x$] is five-sixths. #solutionbox[ #math.equation(block: true, alt: "1 the fraction 3 over 4 minus x equals the fraction 5 over 6 ; x equals the fraction 11 over 12")[$1 frac(3, 4) − x = frac(5, 6) ; #h(0.2em) x = frac(11, 12)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://www.openstax.org/l/24SolveOneStep")[Solve One Step Equations With Fractions] - #link("https://www.openstax.org/l/24OneStepAdd")[Solve One Step Equations With Fractions by Adding or Subtracting] - #link("https://www.openstax.org/l/24OneStepMulti")[Solve One Step Equations With Fraction by Multiplying] ] === Key Concepts - #emph[Determine whether a number is a solution to an equation.]+ Substitute the number for the variable in the equation. + Simplify the expressions on both sides of the equation. + Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution. - #emph[Addition, Subtraction, and Division Properties of Equality]- For any numbers a, b, and c, #linebreak() if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "a plus c equals b plus c")[$a + c = b + c$]. Addition Property of Equality - if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "a minus c equals b minus c")[$a − c = b − c$]. Subtraction Property of Equality - if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "the fraction a over c equals the fraction b over c")[$frac(a, c) = frac(b, c)$], #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$]. Division Property of Equality - #emph[The Multiplication Property of Equality]- For any numbers #math.equation(block: false, alt: "a b")[$a b$] and #math.equation(block: false, alt: "c , a equals b")[$c , a = b$], then #math.equation(block: false, alt: "a c equals b c")[$a c = b c$]. - If you multiply both sides of an equation by the same quantity, you still have equality. === Section Exercises ==== Practice Makes Perfect #strong[Determine Whether a Fraction is a Solution of an Equation] In the following exercises, determine whether each number is a solution of the given equation. #math.equation(block: true, alt: "x minus the fraction 2 over 5 equals the fraction 1 over 10")[$x − frac(2, 5) = frac(1, 10)$]: + ⓐ #math.equation(block: false, alt: "x equals 1")[$x = 1$] + ⓑ #math.equation(block: false, alt: "x equals the fraction 1 over 2")[$x = frac(1, 2)$] + ⓒ #math.equation(block: false, alt: "x equals minus the fraction 1 over 2")[$x = − frac(1, 2)$] #math.equation(block: true, alt: "y minus the fraction 1 over 3 equals the fraction 5 over 12")[$y − frac(1, 3) = frac(5, 12)$]: + ⓐ #math.equation(block: false, alt: "y equals 1")[$y = 1$] + ⓑ #math.equation(block: false, alt: "y equals the fraction 3 over 4")[$y = frac(3, 4)$] + ⓒ #math.equation(block: false, alt: "y equals minus the fraction 3 over 4")[$y = − frac(3, 4)$] #solutionbox[ + ⓐ no + ⓑ yes + ⓒ no ] #math.equation(block: true, alt: "h plus the fraction 3 over 4 equals the fraction 2 over 5")[$h + frac(3, 4) = frac(2, 5)$]: + ⓐ #math.equation(block: false, alt: "h equals 1")[$h = 1$] + ⓑ #math.equation(block: false, alt: "h equals the fraction 7 over 20")[$h = frac(7, 20)$] + ⓒ #math.equation(block: false, alt: "h equals minus the fraction 7 over 20")[$h = − frac(7, 20)$] #math.equation(block: true, alt: "k plus the fraction 2 over 5 equals the fraction 5 over 6")[$k + frac(2, 5) = frac(5, 6)$]: + ⓐ #math.equation(block: false, alt: "k equals 1")[$k = 1$] + ⓑ #math.equation(block: false, alt: "k equals the fraction 13 over 30")[$k = frac(13, 30)$] + ⓒ #math.equation(block: false, alt: "k equals minus the fraction 13 over 30")[$k = − frac(13, 30)$] #solutionbox[ + ⓐ no + ⓑ yes + ⓒ no ] #strong[Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality] In the following exercises, solve. #math.equation(block: true, alt: "y plus the fraction 1 over 3 equals the fraction 4 over 3")[$y + frac(1, 3) = frac(4, 3)$] #math.equation(block: true, alt: "m plus the fraction 3 over 8 equals the fraction 7 over 8")[$m + frac(3, 8) = frac(7, 8)$] #solutionbox[ #math.equation(block: true, alt: "m equals the fraction 1 over 2")[$m = frac(1, 2)$] ] #math.equation(block: true, alt: "f plus the fraction 9 over 10 equals the fraction 2 over 5")[$f + frac(9, 10) = frac(2, 5)$] #math.equation(block: true, alt: "h plus the fraction 5 over 6 equals the fraction 1 over 6")[$h + frac(5, 6) = frac(1, 6)$] #solutionbox[ #math.equation(block: true, alt: "h equals minus the fraction 2 over 3")[$h = − frac(2, 3)$] ] #math.equation(block: true, alt: "a minus the fraction 5 over 8 equals minus the fraction 7 over 8")[$a − frac(5, 8) = − frac(7, 8)$] #math.equation(block: true, alt: "c minus the fraction 1 over 4 equals minus the fraction 5 over 4")[$c − frac(1, 4) = − frac(5, 4)$] #solutionbox[ #emph[c] = −1 ] #math.equation(block: true, alt: "x minus open parenthesis minus the fraction 3 over 20 close parenthesis equals minus the fraction 11 over 20")[$x − ( − frac(3, 20) ) = − frac(11, 20)$] #math.equation(block: true, alt: "z minus open parenthesis minus the fraction 5 over 12 close parenthesis equals minus the fraction 7 over 12")[$z − ( − frac(5, 12) ) = − frac(7, 12)$] #solutionbox[ #emph[z] = −1 ] #math.equation(block: true, alt: "n minus the fraction 1 over 6 equals the fraction 3 over 4")[$n − frac(1, 6) = frac(3, 4)$] #math.equation(block: true, alt: "p minus the fraction 3 over 10 equals the fraction 5 over 8")[$p − frac(3, 10) = frac(5, 8)$] #solutionbox[ #math.equation(block: true, alt: "p equals the fraction 37 over 40")[$p = frac(37, 40)$] ] #math.equation(block: true, alt: "s plus open parenthesis minus the fraction 1 over 2 close parenthesis equals minus the fraction 8 over 9")[$s + ( − frac(1, 2) ) = − frac(8, 9)$] #math.equation(block: true, alt: "k plus open parenthesis minus the fraction 1 over 3 close parenthesis equals minus the fraction 4 over 5")[$k + ( − frac(1, 3) ) = − frac(4, 5)$] #solutionbox[ #math.equation(block: true, alt: "k equals minus the fraction 7 over 15")[$k = − frac(7, 15)$] ] #math.equation(block: true, alt: "5 j equals 17")[$5 j = 17$] #math.equation(block: true, alt: "7 k equals 18")[$7 k = 18$] #solutionbox[ #math.equation(block: true, alt: "k equals the fraction 18 over 7")[$k = frac(18, 7)$] ] #math.equation(block: true, alt: "−4 w equals 26")[$−4 w = 26$] #math.equation(block: true, alt: "−9 v equals 33")[$−9 v = 33$] #solutionbox[ #math.equation(block: true, alt: "v equals minus the fraction 11 over 3")[$v = − frac(11, 3)$] ] #strong[Solve Equations with Fractions Using the Multiplication Property of Equality] In the following exercises, solve. #math.equation(block: true, alt: "the fraction f over 4 equals −20")[$frac(f, 4) = −20$] #math.equation(block: true, alt: "the fraction b over 3 equals −9")[$frac(b, 3) = −9$] #solutionbox[ #emph[b] = −27 ] #math.equation(block: true, alt: "the fraction y over 7 equals −21")[$frac(y, 7) = −21$] #math.equation(block: true, alt: "the fraction x over 8 equals −32")[$frac(x, 8) = −32$] #solutionbox[ #emph[x] = −256 ] #math.equation(block: true, alt: "the fraction p over −5 equals −40")[$frac(p, −5) = −40$] #math.equation(block: true, alt: "the fraction q over −4 equals −40")[$frac(q, −4) = −40$] #solutionbox[ #emph[q] = 160 ] #math.equation(block: true, alt: "the fraction r over −12 equals −6")[$frac(r, −12) = −6$] #math.equation(block: true, alt: "the fraction s over −15 equals −3")[$frac(s, −15) = −3$] #solutionbox[ #emph[s] = 45 ] #math.equation(block: true, alt: "minus x equals 23")[$− x = 23$] #math.equation(block: true, alt: "minus y equals 42")[$− y = 42$] #solutionbox[ #emph[y] = −42 ] #math.equation(block: true, alt: "minus h equals minus the fraction 5 over 12")[$− h = − frac(5, 12)$] #math.equation(block: true, alt: "minus k equals minus the fraction 17 over 20")[$− k = − frac(17, 20)$] #solutionbox[ #math.equation(block: true, alt: "k equals the fraction 17 over 20")[$k = frac(17, 20)$] ] #math.equation(block: true, alt: "the fraction 4 over 5 n equals 20")[$frac(4, 5) n = 20$] #math.equation(block: true, alt: "the fraction 3 over 10 p equals 30")[$frac(3, 10) p = 30$] #solutionbox[ #emph[p] = 100 ] #math.equation(block: true, alt: "the fraction 3 over 8 q equals −48")[$frac(3, 8) q = −48$] #math.equation(block: true, alt: "the fraction 5 over 2 m equals −40")[$frac(5, 2) m = −40$] #solutionbox[ #emph[m] = −16 ] #math.equation(block: true, alt: "minus the fraction 2 over 9 a equals 16")[$− frac(2, 9) a = 16$] #math.equation(block: true, alt: "minus the fraction 3 over 7 b equals 9")[$− frac(3, 7) b = 9$] #solutionbox[ #emph[b] = −21 ] #math.equation(block: true, alt: "minus the fraction 6 over 11 u equals −24")[$− frac(6, 11) u = −24$] #math.equation(block: true, alt: "minus the fraction 5 over 12 v equals −15")[$− frac(5, 12) v = −15$] #solutionbox[ #emph[v] = 36 ] #strong[Mixed Practice] In the following exercises, solve. #math.equation(block: true, alt: "3 x equals 0")[$3 x = 0$] #math.equation(block: true, alt: "8 y equals 0")[$8 y = 0$] #solutionbox[ #emph[y] = 0 ] #math.equation(block: true, alt: "4 f equals the fraction 4 over 5")[$4 f = frac(4, 5)$] #math.equation(block: true, alt: "7 g equals the fraction 7 over 9")[$7 g = frac(7, 9)$] #solutionbox[ #math.equation(block: true, alt: "g equals the fraction 1 over 9")[$g = frac(1, 9)$] ] #math.equation(block: true, alt: "p plus the fraction 2 over 3 equals the fraction 1 over 12")[$p + frac(2, 3) = frac(1, 12)$] #math.equation(block: true, alt: "q plus the fraction 5 over 6 equals the fraction 1 over 12")[$q + frac(5, 6) = frac(1, 12)$] #solutionbox[ #math.equation(block: true, alt: "q equals minus the fraction 3 over 4")[$q = − frac(3, 4)$] ] #math.equation(block: true, alt: "the fraction 7 over 8 m equals the fraction 1 over 10")[$frac(7, 8) m = frac(1, 10)$] #math.equation(block: true, alt: "the fraction 1 over 4 n equals the fraction 7 over 10")[$frac(1, 4) n = frac(7, 10)$] #solutionbox[ #math.equation(block: true, alt: "n equals the fraction 14 over 5")[$n = frac(14, 5)$] ] #math.equation(block: true, alt: "minus the fraction 2 over 5 equals x plus the fraction 3 over 4")[$− frac(2, 5) = x + frac(3, 4)$] #math.equation(block: true, alt: "minus the fraction 2 over 3 equals y plus the fraction 3 over 8")[$− frac(2, 3) = y + frac(3, 8)$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 25 over 24")[$y = − frac(25, 24)$] ] #math.equation(block: true, alt: "the fraction 11 over 20 equals minus f")[$frac(11, 20) = − f$] #math.equation(block: true, alt: "the fraction 8 over 15 equals minus d")[$frac(8, 15) = − "d"$] #solutionbox[ #math.equation(block: true, alt: "d equals minus the fraction 8 over 15")[$d = − frac(8, 15)$] ] #strong[Translate Sentences to Equations and Solve] In the following exercises, translate to an algebraic equation and solve. #math.equation(block: false, alt: "n")[$n$] divided by eight is #math.equation(block: false, alt: "−16 .")[$−16 .$] #math.equation(block: false, alt: "n")[$n$] divided by six is #math.equation(block: false, alt: "−24 .")[$−24 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction n over 6 equals −24 ; n equals −144")[$frac(n, 6) = −24 ; n = −144$] ] #math.equation(block: false, alt: "m")[$m$] divided by #math.equation(block: false, alt: "−9")[$−9$] is #math.equation(block: false, alt: "−7 .")[$−7 .$] #math.equation(block: false, alt: "m")[$m$] divided by #math.equation(block: false, alt: "−7")[$−7$] is #math.equation(block: false, alt: "−8 .")[$−8 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction m over −7 equals −8 ; m equals 56")[$frac(m, −7) = −8 ; m = 56$] ] The quotient of #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "−3")[$−3$] is #math.equation(block: false, alt: "−18 .")[$−18 .$] The quotient of #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "−4")[$−4$] is #math.equation(block: false, alt: "−20 .")[$−20 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction f over −4 equals −20 ; f equals 80")[$frac(f, −4) = −20 ; f = 80$] ] The quotient of #math.equation(block: false, alt: "g")[$g$] and twelve is #math.equation(block: false, alt: "8 .")[$8 .$] The quotient of #math.equation(block: false, alt: "g")[$g$] and nine is #math.equation(block: false, alt: "14 .")[$14 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction g over 9 equals 14 ; g equals 126")[$frac(g, 9) = 14 ; g = 126$] ] Three-fourths of #math.equation(block: false, alt: "q")[$q$] is #math.equation(block: false, alt: "12 .")[$12 .$] Two-fifths of #math.equation(block: false, alt: "q")[$q$] is #math.equation(block: false, alt: "20 .")[$20 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 5 q equals 20 ; q equals 50")[$frac(2, 5) q = 20 ; q = 50$] ] Seven-tenths of #math.equation(block: false, alt: "p")[$p$] is #math.equation(block: false, alt: "−63 .")[$−63 .$] Four-ninths of #math.equation(block: false, alt: "p")[$p$] is #math.equation(block: false, alt: "−28 .")[$−28 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 9 p equals −28 ; p equals −63")[$frac(4, 9) p = −28 ; p = −63$] ] #math.equation(block: false, alt: "m")[$m$] divided by #math.equation(block: false, alt: "4")[$4$] equals negative #math.equation(block: false, alt: "6 .")[$6 .$] The quotient of #math.equation(block: false, alt: "h")[$h$] and #math.equation(block: false, alt: "2")[$2$] is #math.equation(block: false, alt: "43 .")[$43 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction h over 2 equals 43 ; h equals 86")[$frac(h, 2) = 43 ; h = 86$] ] Three-fourths of #math.equation(block: false, alt: "z")[$z$] is #math.equation(block: false, alt: "15 .")[$15 .$] The quotient of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] is #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction a over the fraction 2 over 3 equals the fraction 3 over 4 ; a equals the fraction 1 over 2")[$frac(#h(0.2em) a #h(0.2em), frac(2, 3)) = frac(3, 4) ; #h(0.2em) a = frac(1, 2)$] ] The sum of five-sixths and #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "the fraction 1 over 2 .")[$frac(1, 2) .$] The sum of three-fourths and #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "the fraction 1 over 8 .")[$frac(1, 8) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4 plus x equals the fraction 1 over 8 ; x equals minus the fraction 5 over 8")[$frac(3, 4) + x = frac(1, 8) ; x = − frac(5, 8)$] ] The difference of #math.equation(block: false, alt: "y")[$y$] and one-fourth is #math.equation(block: false, alt: "minus the fraction 1 over 8 .")[$− frac(1, 8) .$] The difference of #math.equation(block: false, alt: "y")[$y$] and one-third is #math.equation(block: false, alt: "minus the fraction 1 over 6 .")[$− frac(1, 6) .$] #solutionbox[ #math.equation(block: true, alt: "y minus the fraction 1 over 3 equals minus the fraction 1 over 6 ; y equals the fraction 1 over 6")[$y − frac(1, 3) = − frac(1, 6) ; y = frac(1, 6)$] ] ==== Everyday Math #strong[Shopping] Teresa bought a pair of shoes on sale for #math.equation(block: false, alt: "$ 48")[$\$ 48$]. The sale price was #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] of the regular price. Find the regular price of the shoes by solving the equation #math.equation(block: false, alt: "the fraction 2 over 3 p equals 48")[$frac(2, 3) p = 48$] #strong[Playhouse] The table in a child’s playhouse is #math.equation(block: false, alt: "the fraction 3 over 5")[$frac(3, 5)$] of an adult-size table. The playhouse table is #math.equation(block: false, alt: "18")[$18$] inches high. Find the height of an adult-size table by solving the equation #math.equation(block: false, alt: "the fraction 3 over 5 h equals 18 .")[$frac(3, 5) h = 18 .$] #solutionbox[ 30 inches ] ==== Writing Exercises describes three methods to solve the equation #math.equation(block: false, alt: "minus y equals 15 .")[$− y = 15 .$] Which method do you prefer? Why? Richard thinks the solution to the equation #math.equation(block: false, alt: "the fraction 3 over 4 x equals 24")[$frac(3, 4) x = 24$] is #math.equation(block: false, alt: "16 .")[$16 .$] Explain why Richard is wrong. #solutionbox[ Answers will vary. ] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A self-assessment chart with 'I can...' statements related to solving equations with fractions, including determining solutions, using properties of equality, and translating sentences, with columns for confidence levels: Confidently, With some help, and No-I don't get it!], alt: "A self-assessment chart with 'I can...' statements related to solving equations with fractions, including determining solutions, using properties of equality, and translating sentences, with columns for confidence levels: Confidently, With some help, and No-I don't get it!", caption: none) ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not? === Chapter Review Exercises ==== Visualize Fractions In the following exercises, name the fraction of each figure that is shaded. #figure(figph[A circle is shown. It is divided into 8 equal pieces. 5 pieces are shaded.], alt: "A circle is shown. It is divided into 8 equal pieces. 5 pieces are shaded.", caption: none) #figure(figph[A square is shown. It is divided into 9 equal pieces. 5 pieces are shaded.], alt: "A square is shown. It is divided into 9 equal pieces. 5 pieces are shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 9")[$frac(5, 9)$] ] In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number. #figure(figph[Two squares are shown. Both are divided into four equal pieces. The square on the left has all 4 pieces shaded. The square on the right has one piece shaded.], alt: "Two squares are shown. Both are divided into four equal pieces. The square on the left has all 4 pieces shaded. The square on the right has one piece shaded.", caption: none) #figure(figph[Two circles are shown. Both are divided into two equal pieces. The circle on the left has both pieces shaded. The circle on the right has one piece shaded.], alt: "Two circles are shown. Both are divided into two equal pieces. The circle on the left has both pieces shaded. The circle on the right has one piece shaded.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 2 equals 1 the fraction 1 over 2")[$frac(3, 2) = 1 frac(1, 2)$] ] In the following exercises, convert the improper fraction to a mixed number. #math.equation(block: true, alt: "the fraction 58 over 15")[$frac(58, 15)$] #math.equation(block: true, alt: "the fraction 63 over 11")[$frac(63, 11)$] #solutionbox[ #math.equation(block: true, alt: "5 the fraction 8 over 11")[$5 frac(8, 11)$] ] In the following exercises, convert the mixed number to an improper fraction. #math.equation(block: true, alt: "12 the fraction 1 over 4")[$12 frac(1, 4)$] #math.equation(block: true, alt: "9 the fraction 4 over 5")[$9 frac(4, 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 49 over 5")[$frac(49, 5)$] ] Find three fractions equivalent to #math.equation(block: false, alt: "the fraction 2 over 5 .")[$frac(2, 5) .$] Show your work, using figures or algebra. Find three fractions equivalent to #math.equation(block: false, alt: "minus the fraction 4 over 3 .")[$− frac(4, 3) .$] Show your work, using figures or algebra. #solutionbox[ Answers may vary. ] In the following exercises, locate the numbers on a number line. #math.equation(block: true, alt: "the fraction 5 over 8 , the fraction 4 over 3 , 3 the fraction 3 over 4 , 4")[$frac(5, 8) , frac(4, 3) , 3 frac(3, 4) , 4$] #math.equation(block: true, alt: "the fraction 1 over 4 , minus the fraction 1 over 4 , 1 the fraction 1 over 3 , −1 the fraction 1 over 3 , the fraction 7 over 2 , minus the fraction 7 over 2")[$frac(1, 4) , − frac(1, 4) , 1 frac(1, 3) , −1 frac(1, 3) , frac(7, 2) , − frac(7, 2)$] #solutionbox[ #figure(figph[A number line is shown. Integers from negative 4 to 4 are labeled. Between negative 4 and negative 3, negative 7 halves is labeled and marked with a red dot. Between negative 2 and negative 1, negative 1 and 1 third is labeled and marked with a red dot. Between negative 1 and 0, negative 1 fourth is labeled and marked with a red dot. Between 0 and 1, 1 fourth is labeled and marked with a red dot. Between 1 and 2, 1 and 1 third is labeled and marked with a red dot. Between 3 and 4, 7 halves is labeled and marked with a red dot.], alt: "A number line is shown. Integers from negative 4 to 4 are labeled. Between negative 4 and negative 3, negative 7 halves is labeled and marked with a red dot. Between negative 2 and negative 1, negative 1 and 1 third is labeled and marked with a red dot. Between negative 1 and 0, negative 1 fourth is labeled and marked with a red dot. Between 0 and 1, 1 fourth is labeled and marked with a red dot. Between 1 and 2, 1 and 1 third is labeled and marked with a red dot. Between 3 and 4, 7 halves is labeled and marked with a red dot.", caption: none) ] In the following exercises, order each pair of numbers, using #math.equation(block: false, alt: "less than")[$<$] or #math.equation(block: false, alt: ">.")[$>.$] #math.equation(block: true, alt: "−1 ___ minus the fraction 2 over 5")[$−1 \_\_\_ − frac(2, 5)$] #math.equation(block: true, alt: "−2 the fraction 1 over 2 ___ −3")[$−2 frac(1, 2) \_\_\_ −3$] #solutionbox[ \> ] ==== Multiply and Divide Fractions In the following exercises, simplify. #math.equation(block: true, alt: "minus the fraction 63 over 84")[$− frac(63, 84)$] #math.equation(block: true, alt: "minus the fraction 90 over 120")[$− frac(90, 120)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] ] #math.equation(block: true, alt: "minus the fraction 14 a over 14 b")[$− frac(14 a, 14 b)$] #math.equation(block: true, alt: "minus the fraction 8 x over 8 y")[$− frac(8 x, 8 y)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction x over y")[$− frac(x, y)$] ] In the following exercises, multiply. #math.equation(block: true, alt: "the fraction 2 over 5 times the fraction 8 over 13")[$frac(2, 5) · frac(8, 13)$] #math.equation(block: true, alt: "minus the fraction 1 over 3 times the fraction 12 over 7")[$− frac(1, 3) · frac(12, 7)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 over 7")[$− frac(4, 7)$] ] #math.equation(block: true, alt: "the fraction 2 over 9 times open parenthesis minus the fraction 45 over 32 close parenthesis")[$frac(2, 9) · ( − frac(45, 32) )$] #math.equation(block: true, alt: "6 m times the fraction 4 over 11")[$6 m · frac(4, 11)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 24 m over 11")[$frac(24 m, 11)$] ] #math.equation(block: true, alt: "minus the fraction 1 over 4 open parenthesis −32 close parenthesis")[$− frac(1, 4) ( −32 )$] #math.equation(block: true, alt: "the fraction 16 over 5 times the fraction 15 over 8")[$frac(16, 5) · frac(15, 8)$] #solutionbox[ 6 ] In the following exercises, find the reciprocal. #math.equation(block: true, alt: "the fraction 2 over 9")[$frac(2, 9)$] #math.equation(block: true, alt: "the fraction 15 over 4")[$frac(15, 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 15")[$frac(4, 15)$] ] #math.equation(block: true, alt: "3")[$3$] #math.equation(block: true, alt: "minus the fraction 1 over 4")[$− frac(1, 4)$] #solutionbox[ −4 ] Fill in the chart. #figure(table( columns: 4, align: left, inset: 6pt, table.header([], [Opposite], [Absolute Value], [Reciprocal]), [#math.equation(block: false, alt: "minus the fraction 5 over 13")[$− frac(5, 13)$]], [], [], [], [#math.equation(block: false, alt: "the fraction 3 over 10")[$frac(3, 10)$]], [], [], [], [#math.equation(block: false, alt: "the fraction 9 over 4")[$frac(9, 4)$]], [], [], [], [#math.equation(block: false, alt: "−12")[$−12$]], [], [], [], )) In the following exercises, divide. #math.equation(block: true, alt: "the fraction 2 over 3 divided by the fraction 1 over 6")[$frac(2, 3) ÷ frac(1, 6)$] #solutionbox[ 4 ] #math.equation(block: true, alt: "open parenthesis minus the fraction 3 x over 5 close parenthesis divided by open parenthesis minus the fraction 2 y over 3 close parenthesis")[$( − frac(3 x, 5) ) ÷ ( − frac(2 y, 3) )$] #math.equation(block: true, alt: "the fraction 4 over 5 divided by 3")[$frac(4, 5) ÷ 3$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 15")[$frac(4, 15)$] ] #math.equation(block: true, alt: "8 divided by the fraction 8 over 3")[$8 ÷ frac(8, 3)$] #math.equation(block: true, alt: "the fraction 5 over 18 divided by open parenthesis minus the fraction b over 9 close parenthesis")[$frac(5, 18) ÷ ( − frac(b, 9) )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 2 b")[$− frac(5, 2 b)$] ] ==== Multiply and Divide Mixed Numbers and Complex Fractions In the following exercises, perform the indicated operation. #math.equation(block: true, alt: "3 the fraction 1 over 5 times 1 the fraction 7 over 8")[$3 frac(1, 5) · 1 frac(7, 8)$] #math.equation(block: true, alt: "−5 the fraction 7 over 12 times 4 the fraction 4 over 11")[$−5 frac(7, 12) · 4 frac(4, 11)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 268 over 11")[$− frac(268, 11)$] ] #math.equation(block: true, alt: "8 divided by 2 the fraction 2 over 3")[$8 ÷ 2 frac(2, 3)$] #math.equation(block: true, alt: "8 the fraction 2 over 3 divided by 1 the fraction 1 over 12")[$8 frac(2, 3) ÷ 1 frac(1, 12)$] #solutionbox[ 8 ] In the following exercises, translate the English phrase into an algebraic expression. the quotient of #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "y")[$y$] the quotient of #math.equation(block: false, alt: "V")[$V$] and the difference of #math.equation(block: false, alt: "h")[$h$] and #math.equation(block: false, alt: "6")[$6$] #solutionbox[ #math.equation(block: true, alt: "the fraction V over h minus 6")[$frac(V, h − 6)$] ] In the following exercises, simplify the complex fraction #math.equation(block: true, alt: "the fraction the fraction 5 over 8 over the fraction 4 over 5")[$frac(#h(0.2em) frac(5, 8) #h(0.2em), #h(0.2em) frac(4, 5) #h(0.2em))$] #math.equation(block: true, alt: "the fraction the fraction 8 over 9 over −4")[$frac(frac(8, 9), −4)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 9")[$− frac(2, 9)$] ] #math.equation(block: true, alt: "the fraction the fraction n over 4 over the fraction 3 over 8")[$frac(#h(0.2em) frac(n, 4) #h(0.2em), #h(0.2em) frac(3, 8) #h(0.2em))$] #math.equation(block: true, alt: "the fraction −1 the fraction 5 over 6 over minus the fraction 1 over 12")[$frac(−1 frac(5, 6), − frac(1, 12))$] #solutionbox[ 22 ] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction 5 plus 16 over 5")[$frac(5 + 16, 5)$] #math.equation(block: true, alt: "the fraction 8 times 4 minus 5 squared over 3 times 12")[$frac(8 · 4 − 5^(2), 3 · 12)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 36")[$frac(7, 36)$] ] #math.equation(block: true, alt: "the fraction 8 times 7 plus 5 open parenthesis 8 minus 10 close parenthesis over 9 times 3 minus 6 times 4")[$frac(8 · 7 + 5 ( 8 − 10 ), 9 · 3 − 6 · 4)$] ==== Add and Subtract Fractions with Common Denominators In the following exercises, add. #math.equation(block: true, alt: "the fraction 3 over 8 plus the fraction 2 over 8")[$frac(3, 8) + frac(2, 8)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 8")[$frac(5, 8)$] ] #math.equation(block: true, alt: "the fraction 4 over 5 plus the fraction 1 over 5")[$frac(4, 5) + frac(1, 5)$] #math.equation(block: true, alt: "the fraction 2 over 5 plus the fraction 1 over 5")[$frac(2, 5) + frac(1, 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 5")[$frac(3, 5)$] ] #math.equation(block: true, alt: "the fraction 15 over 32 plus the fraction 9 over 32")[$frac(15, 32) + frac(9, 32)$] #math.equation(block: true, alt: "the fraction x over 10 plus the fraction 7 over 10")[$frac(x, 10) + frac(7, 10)$] #solutionbox[ #math.equation(block: true, alt: "the fraction x plus 7 over 10")[$frac(x + 7, 10)$] ] In the following exercises, subtract. #math.equation(block: true, alt: "the fraction 8 over 11 minus the fraction 6 over 11")[$frac(8, 11) − frac(6, 11)$] #math.equation(block: true, alt: "the fraction 11 over 12 minus the fraction 5 over 12")[$frac(11, 12) − frac(5, 12)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 2")[$frac(1, 2)$] ] #math.equation(block: true, alt: "the fraction 4 over 5 minus the fraction y over 5")[$frac(4, 5) − frac(y, 5)$] #math.equation(block: true, alt: "minus the fraction 31 over 30 minus the fraction 7 over 30")[$− frac(31, 30) − frac(7, 30)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 19 over 15")[$− frac(19, 15)$] ] #math.equation(block: true, alt: "the fraction 3 over 2 minus open parenthesis the fraction 3 over 2 close parenthesis")[$frac(3, 2) − ( frac(3, 2) )$] #math.equation(block: true, alt: "the fraction 11 over 15 minus the fraction 5 over 15 minus open parenthesis minus the fraction 2 over 15 close parenthesis")[$frac(11, 15) − frac(5, 15) − ( − frac(2, 15) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 8 over 15")[$frac(8, 15)$] ] ==== Add and Subtract Fractions with Different Denominators In the following exercises, find the least common denominator. #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] and #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$] #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] and #math.equation(block: false, alt: "the fraction 4 over 5")[$frac(4, 5)$] #solutionbox[ 15 ] #math.equation(block: false, alt: "the fraction 8 over 15")[$frac(8, 15)$] and #math.equation(block: false, alt: "the fraction 11 over 20")[$frac(11, 20)$] #math.equation(block: false, alt: "the fraction 3 over 4 , the fraction 1 over 6 ,")[$frac(3, 4) , frac(1, 6) ,$]and#math.equation(block: false, alt: "the fraction 5 over 10")[$frac(5, 10)$] #solutionbox[ 60 ] In the following exercises, change to equivalent fractions using the given LCD. #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] and #math.equation(block: false, alt: "the fraction 1 over 5 ,")[$frac(1, 5) ,$] LCD #math.equation(block: false, alt: "equals 15")[$= 15$] #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] and #math.equation(block: false, alt: "the fraction 5 over 6 ,")[$frac(5, 6) ,$] LCD #math.equation(block: false, alt: "equals 24")[$= 24$] #solutionbox[ #math.equation(block: false, alt: "the fraction 9 over 24")[$frac(9, 24)$] and #math.equation(block: false, alt: "the fraction 20 over 24")[$frac(20, 24)$] ] #math.equation(block: false, alt: "minus the fraction 9 over 16")[$− frac(9, 16)$] and #math.equation(block: false, alt: "the fraction 5 over 12 ,")[$frac(5, 12) ,$] LCD #math.equation(block: false, alt: "equals 48")[$= 48$] #math.equation(block: false, alt: "the fraction 1 over 3 , the fraction 3 over 4")[$frac(1, 3) , frac(3, 4)$] and #math.equation(block: false, alt: "the fraction 4 over 5 ,")[$frac(4, 5) ,$] LCD #math.equation(block: false, alt: "equals 60")[$= 60$] #solutionbox[ #math.equation(block: false, alt: "the fraction 20 over 60 , the fraction 45 over 60")[$frac(20, 60) , frac(45, 60)$] and #math.equation(block: false, alt: "the fraction 48 over 60")[$frac(48, 60)$] ] In the following exercises, perform the indicated operations and simplify. #math.equation(block: true, alt: "the fraction 1 over 5 plus the fraction 2 over 3")[$frac(1, 5) + frac(2, 3)$] #math.equation(block: true, alt: "the fraction 11 over 12 minus the fraction 2 over 3")[$frac(11, 12) − frac(2, 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] #math.equation(block: true, alt: "minus the fraction 9 over 10 minus the fraction 3 over 4")[$− frac(9, 10) − frac(3, 4)$] #math.equation(block: true, alt: "minus the fraction 11 over 36 minus the fraction 11 over 20")[$− frac(11, 36) − frac(11, 20)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 77 over 90")[$− frac(77, 90)$] ] #math.equation(block: true, alt: "minus the fraction 22 over 25 plus the fraction 9 over 40")[$− frac(22, 25) + frac(9, 40)$] #math.equation(block: true, alt: "the fraction y over 10 minus the fraction 1 over 3")[$frac(y, 10) − frac(1, 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 y minus 10 over 30")[$frac(3 y − 10, 30)$] ] #math.equation(block: true, alt: "the fraction 2 over 5 plus open parenthesis minus the fraction 5 over 9 close parenthesis")[$frac(2, 5) + ( − frac(5, 9) )$] #math.equation(block: true, alt: "the fraction 4 over 11 divided by the fraction 2 over 7 d")[$frac(4, 11) ÷ frac(2, 7 d)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 14 d over 11")[$frac(14 d, 11)$] ] #math.equation(block: true, alt: "the fraction 2 over 5 plus open parenthesis minus the fraction 3 n over 8 close parenthesis open parenthesis minus the fraction 2 over 9 n close parenthesis")[$frac(2, 5) + ( − frac(3 n, 8) ) ( − frac(2, 9 n) )$] #math.equation(block: true, alt: "the fraction open parenthesis the fraction 2 over 3 close parenthesis squared over open parenthesis the fraction 5 over 8 close parenthesis squared")[$frac(attach(( frac(2, 3) ), t: 2), attach(( frac(5, 8) ), t: 2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 256 over 225")[$frac(256, 225)$] ] #math.equation(block: true, alt: "open parenthesis the fraction 11 over 12 plus the fraction 3 over 8 close parenthesis divided by open parenthesis the fraction 5 over 6 minus the fraction 1 over 10 close parenthesis")[$( frac(11, 12) + frac(3, 8) ) ÷ ( frac(5, 6) − frac(1, 10) )$] In the following exercises, evaluate. #math.equation(block: false, alt: "y minus the fraction 4 over 5")[$y − frac(4, 5)$] when + ⓐ #math.equation(block: false, alt: "y equals minus the fraction 4 over 5")[$y = − frac(4, 5)$] + ⓑ #math.equation(block: false, alt: "y equals the fraction 1 over 4")[$y = frac(1, 4)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "minus the fraction 8 over 5")[$− frac(8, 5)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 11 over 20")[$− frac(11, 20)$] ] #math.equation(block: false, alt: "6 m n squared")[$6 m n^(2)$] when #math.equation(block: false, alt: "m equals the fraction 3 over 4")[$m = frac(3, 4)$] and #math.equation(block: false, alt: "n equals minus the fraction 1 over 3")[$n = − frac(1, 3)$] ==== Add and Subtract Mixed Numbers In the following exercises, perform the indicated operation. #math.equation(block: true, alt: "4 the fraction 1 over 3 plus 9 the fraction 1 over 3")[$4 frac(1, 3) + 9 frac(1, 3)$] #solutionbox[ #math.equation(block: true, alt: "13 the fraction 2 over 3")[$13 frac(2, 3)$] ] #math.equation(block: true, alt: "6 the fraction 2 over 5 plus 7 the fraction 3 over 5")[$6 frac(2, 5) + 7 frac(3, 5)$] #math.equation(block: true, alt: "5 the fraction 8 over 11 plus 2 the fraction 4 over 11")[$5 frac(8, 11) + 2 frac(4, 11)$] #solutionbox[ #math.equation(block: true, alt: "8 the fraction 1 over 11")[$8 frac(1, 11)$] ] #math.equation(block: true, alt: "3 the fraction 5 over 8 plus 3 the fraction 7 over 8")[$3 frac(5, 8) + 3 frac(7, 8)$] #math.equation(block: true, alt: "9 the fraction 13 over 20 minus 4 the fraction 11 over 20")[$9 frac(13, 20) − 4 frac(11, 20)$] #solutionbox[ #math.equation(block: true, alt: "5 the fraction 1 over 10")[$5 frac(1, 10)$] ] #math.equation(block: true, alt: "2 the fraction 3 over 10 minus 1 the fraction 9 over 10")[$2 frac(3, 10) − 1 frac(9, 10)$] #math.equation(block: true, alt: "2 the fraction 11 over 12 minus 1 the fraction 7 over 12")[$2 frac(11, 12) − 1 frac(7, 12)$] #solutionbox[ #math.equation(block: true, alt: "1 the fraction 1 over 3")[$1 frac(1, 3)$] ] #math.equation(block: true, alt: "8 the fraction 6 over 11 minus 2 the fraction 9 over 11")[$8 frac(6, 11) − 2 frac(9, 11)$] ==== Solve Equations with Fractions In the following exercises, determine whether the each number is a solution of the given equation. #math.equation(block: true, alt: "x minus the fraction 1 over 2 equals the fraction 1 over 6")[$x − frac(1, 2) = frac(1, 6)$]: + ⓐ #math.equation(block: false, alt: "x equals 1")[$x = 1$] + ⓑ #math.equation(block: false, alt: "x equals the fraction 2 over 3")[$x = frac(2, 3)$] + ⓒ #math.equation(block: false, alt: "x equals minus the fraction 1 over 3")[$x = − frac(1, 3)$] #solutionbox[ + ⓐ no + ⓑ yes + ⓒ no ] #math.equation(block: true, alt: "y plus the fraction 3 over 5 equals the fraction 5 over 9")[$y + frac(3, 5) = frac(5, 9)$]: + ⓐ #math.equation(block: false, alt: "y equals the fraction 1 over 2")[$y = frac(1, 2)$] + ⓑ #math.equation(block: false, alt: "y equals the fraction 52 over 45")[$y = frac(52, 45)$] + ⓒ #math.equation(block: false, alt: "y equals minus the fraction 2 over 45")[$y = − frac(2, 45)$] In the following exercises, solve the equation. #math.equation(block: true, alt: "n plus the fraction 9 over 11 equals the fraction 4 over 11")[$n + frac(9, 11) = frac(4, 11)$] #solutionbox[ #math.equation(block: true, alt: "n equals minus the fraction 5 over 11")[$n = − frac(5, 11)$] ] #math.equation(block: true, alt: "x minus the fraction 1 over 6 equals the fraction 7 over 6")[$x − frac(1, 6) = frac(7, 6)$] #math.equation(block: true, alt: "h minus open parenthesis minus the fraction 7 over 8 close parenthesis equals minus the fraction 2 over 5")[$h − ( − frac(7, 8) ) = − frac(2, 5)$] #solutionbox[ #math.equation(block: true, alt: "h equals minus the fraction 51 over 40")[$h = − frac(51, 40)$] ] #math.equation(block: true, alt: "the fraction x over 5 equals −10")[$frac(x, 5) = −10$] #math.equation(block: true, alt: "minus z equals 23")[$− z = 23$] #solutionbox[ #emph[z] = −23 ] In the following exercises, translate and solve. The sum of two-thirds and #math.equation(block: false, alt: "n")[$n$] is #math.equation(block: false, alt: "minus the fraction 3 over 5 .")[$− frac(3, 5) .$] The difference of #math.equation(block: false, alt: "q")[$q$] and one-tenth is #math.equation(block: false, alt: "the fraction 1 over 2 .")[$frac(1, 2) .$] #solutionbox[ #math.equation(block: true, alt: "q minus the fraction 1 over 10 equals the fraction 1 over 2 ; q equals the fraction 3 over 5")[$q − frac(1, 10) = frac(1, 2) ; q = frac(3, 5)$] ] The quotient of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "−4")[$−4$] is #math.equation(block: false, alt: "−8 .")[$−8 .$] Three-eighths of #math.equation(block: false, alt: "y")[$y$] is #math.equation(block: false, alt: "24 .")[$24 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 8 y equals 24 ; y equals 64")[$frac(3, 8) y = 24 ; y = 64$] ] === Chapter Practice Test Convert the improper fraction to a mixed number. #math.equation(block: true, alt: "the fraction 19 over 5")[$frac(19, 5)$] Convert the mixed number to an improper fraction. #math.equation(block: true, alt: "3 the fraction 2 over 7")[$3 frac(2, 7)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 23 over 7")[$frac(23, 7)$] ] Locate the numbers on a number line. #math.equation(block: false, alt: "the fraction 1 over 2 , 1 the fraction 2 over 3 , −2 the fraction 3 over 4 ,")[$frac(1, 2) , 1 frac(2, 3) , −2 frac(3, 4) ,$] and #math.equation(block: false, alt: "the fraction 9 over 4")[$frac(9, 4)$] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction 5 over 20")[$frac(5, 20)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] #math.equation(block: true, alt: "the fraction 18 r over 27 s")[$frac(18 r, 27 s)$] #math.equation(block: true, alt: "the fraction 1 over 3 times the fraction 3 over 4")[$frac(1, 3) · frac(3, 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] #math.equation(block: true, alt: "the fraction 3 over 5 times 15")[$frac(3, 5) · 15$] #math.equation(block: true, alt: "−36 u open parenthesis minus the fraction 4 over 9 close parenthesis")[$−36 u ( − frac(4, 9) )$] #solutionbox[ 16#emph[u] ] #math.equation(block: true, alt: "−5 the fraction 7 over 12 times 4 the fraction 4 over 11")[$−5 frac(7, 12) · 4 frac(4, 11)$] #math.equation(block: true, alt: "minus the fraction 5 over 6 divided by the fraction 5 over 12")[$− frac(5, 6) ÷ frac(5, 12)$] #solutionbox[ −2 ] #math.equation(block: true, alt: "the fraction 7 over 11 divided by open parenthesis minus the fraction 7 over 11 close parenthesis")[$frac(7, 11) ÷ ( − frac(7, 11) )$] #math.equation(block: true, alt: "the fraction 9 a over 10 divided by the fraction 15 a over 8")[$frac(9 a, 10) ÷ frac(15 a, 8)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 12 over 25")[$frac(12, 25)$] ] #math.equation(block: true, alt: "−6 the fraction 2 over 5 divided by 4")[$−6 frac(2, 5) ÷ 4$] #math.equation(block: true, alt: "open parenthesis −15 the fraction 5 over 6 close parenthesis divided by open parenthesis −3 the fraction 1 over 6 close parenthesis")[$( −15 frac(5, 6) ) ÷ ( −3 frac(1, 6) )$] #solutionbox[ 5 ] #math.equation(block: true, alt: "the fraction −6 over the fraction 6 over 11")[$frac(−6, frac(6, 11))$] #math.equation(block: true, alt: "the fraction the fraction p over 2 over the fraction q over 5")[$frac(#h(0.2em) frac(p, 2) #h(0.2em), #h(0.2em) frac(q, 5) #h(0.2em))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 p over 2 q")[$frac(5 p, 2 q)$] ] #math.equation(block: true, alt: "the fraction minus the fraction 4 over 15 over −2 the fraction 2 over 3")[$frac(− frac(4, 15), −2 frac(2, 3))$] #math.equation(block: true, alt: "the fraction 9 squared minus 4 squared over 9 minus 4")[$frac(9^(2) − 4^(2), 9 − 4)$] #solutionbox[ 13 ] #math.equation(block: true, alt: "the fraction 2 over d plus the fraction 9 over d")[$frac(2, d) + frac(9, d)$] #math.equation(block: true, alt: "minus the fraction 3 over 13 plus open parenthesis minus the fraction 4 over 13 close parenthesis")[$− frac(3, 13) + ( − frac(4, 13) )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 7 over 13")[$− frac(7, 13)$] ] #math.equation(block: true, alt: "minus the fraction 22 over 25 plus the fraction 9 over 40")[$− frac(22, 25) + frac(9, 40)$] #math.equation(block: true, alt: "the fraction 2 over 5 plus open parenthesis minus the fraction 7 over 5 close parenthesis")[$frac(2, 5) + ( − frac(7, 5) )$] #solutionbox[ −1 ] #math.equation(block: true, alt: "minus the fraction 3 over 10 plus open parenthesis minus the fraction 5 over 8 close parenthesis")[$− frac(3, 10) + ( − frac(5, 8) )$] #math.equation(block: true, alt: "minus the fraction 3 over 4 divided by the fraction x over 3")[$− frac(3, 4) ÷ frac(x, 3)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 9 over 4 x")[$− frac(9, 4 x)$] ] #math.equation(block: true, alt: "the fraction 2 cubed minus 2 squared over open parenthesis the fraction 3 over 4 close parenthesis squared")[$frac(2^(3) − 2^(2), attach(( frac(3, 4) ), t: 2))$] #math.equation(block: true, alt: "the fraction the fraction 5 over 14 plus the fraction 1 over 8 over the fraction 9 over 56")[$frac(frac(5, 14) + frac(1, 8), frac(9, 56))$] #solutionbox[ 3 ] Evaluate. #math.equation(block: false, alt: "x plus the fraction 1 over 3")[$x + frac(1, 3)$] when + ⓐ #math.equation(block: false, alt: "x equals the fraction 2 over 3")[$x = frac(2, 3)$] + ⓑ #math.equation(block: false, alt: "x equals minus the fraction 5 over 6")[$x = − frac(5, 6)$] In the following exercises, solve the equation. #math.equation(block: true, alt: "y plus the fraction 3 over 5 equals the fraction 7 over 5")[$y + frac(3, 5) = frac(7, 5)$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 4 over 5")[$y = frac(4, 5)$] ] #math.equation(block: true, alt: "a minus the fraction 3 over 10 equals minus the fraction 9 over 10")[$a − frac(3, 10) = − frac(9, 10)$] #math.equation(block: true, alt: "f plus open parenthesis minus the fraction 2 over 3 close parenthesis equals the fraction 5 over 12")[$f + ( − frac(2, 3) ) = frac(5, 12)$] #solutionbox[ #math.equation(block: true, alt: "f equals the fraction 13 over 12")[$f = frac(13, 12)$] ] #math.equation(block: true, alt: "the fraction m over −2 equals −16")[$frac(m, −2) = −16$] #math.equation(block: true, alt: "minus the fraction 2 over 3 c equals 18")[$− frac(2, 3) c = 18$] #solutionbox[ #emph[c] = −27 ] Translate and solve: The quotient of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "−4")[$−4$] is #math.equation(block: false, alt: "−8 .")[$−8 .$] Solve for #math.equation(block: false, alt: "p .")[$p .$]