#set document(title: "8.4 Add and Subtract Rational Expressions with Unlike Denominators", 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")) == 8.4#h(0.6em)Add and Subtract Rational Expressions with Unlike Denominators #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. Add: #math.equation(block: false, alt: "the fraction 7 over 10 plus the fraction 8 over 15 .")[$frac(7, 10) + frac(8, 15) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "the fraction 37 over 30")[$frac(37, 30)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "6 open parenthesis 2 x plus 1 close parenthesis minus 4 open parenthesis x minus 5 close parenthesis .")[$6 ( 2 x + 1 ) − 4 ( x − 5 ) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "8 x plus 26")[$8 x + 26$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the Greatest Common Factor of #math.equation(block: false, alt: "9 x squared y cubed")[$9 x^(2) y^(3)$] and #math.equation(block: false, alt: "12 x y to the power 5")[$12 x y^(5)$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "3 x y cubed")[$3 x y^(3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Factor completely #math.equation(block: false, alt: "−48 n minus 12")[$−48 n − 12$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus 12 4 n plus 1")[$− 12 4 n + 1$] ] ] === Find the Least Common Denominator of Rational Expressions When we add or subtract rational expressions with unlike denominators we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions. Let’s look at the example #math.equation(block: false, alt: "the fraction 7 over 12 plus the fraction 5 over 18")[$frac(7, 12) + frac(5, 18)$] from Foundations. Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator. To find the LCD of 12 and 18, we factored each number into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD. #math.equation(block: true, alt: "the fraction 12 equals 2 times 2 times 3; 18 equals 2 times 3 times 3 over LCD equals 2 times 2 times 3 times 3; LCD equals 36")[$frac(12 = 2 · 2 · 3 \ 18 = 2 · 3 · 3, #h(0.15em) "LCD" = 2 · 2 · 3 · 3 \ #h(0.15em) "LCD" = 36)$]We do the same thing for rational expressions. However, we leave the LCD in factored form. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find the least common denominator of rational expressions.] + Factor each expression completely. + List the factors of each expression. Match factors vertically when possible. + Bring down the columns. + Multiply the factors. Remember, we always exclude values that would make the denominator zero. What values of #emph[x] should we exclude in this next example? ] #examplebox("Example 1")[][ Find the LCD for #math.equation(block: false, alt: "the fraction 8 over x squared minus 2 x minus 3 , the fraction 3 x over x squared plus 4 x plus 3")[$frac(8, x^(2) − 2 x − 3) , frac(3 x, x^(2) + 4 x + 3)$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "Find the LCD for the fraction 8 over x squared minus 2 x minus 3 , the fraction 3 x over x squared plus 4 x plus 3 .")[$"Find the LCD for" #h(0.2em) frac(8, x^(2) − 2 x − 3) , frac(3 x, x^(2) + 4 x + 3) .$]]), [Factor each expression completely, lining #linebreak() up common factors. #linebreak() Bring down the columns.], [#math.equation(block: false, alt: "the fraction x squared minus 2 x minus 3 equals open parenthesis x plus 1 close parenthesis open parenthesis x minus 3 close parenthesis; x squared plus 4 x plus 3 equals open parenthesis x plus 1 close parenthesis open parenthesis x plus 3 close parenthesis over LCD equals open parenthesis x plus 1 close parenthesis open parenthesis x minus 3 close parenthesis open parenthesis x plus 3 close parenthesis")[$frac(x^(2) − 2 x − 3 = ( x + 1 ) ( x − 3 ) \ x^(2) + 4 x + 3 = ( x + 1 ) #h(1em) ( x + 3 ), #h(4.45em) "LCD" = ( x + 1 ) ( x − 3 ) ( x + 3 ))$]], [Multiply the factors.], [#math.equation(block: false, alt: "The LCD is open parenthesis x plus 1 close parenthesis open parenthesis x minus 3 close parenthesis open parenthesis x plus 3 close parenthesis .")[$"The LCD is" #h(0.2em) ( x + 1 ) ( x − 3 ) ( x + 3 ) .$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the LCD for #math.equation(block: false, alt: "the fraction 2 over x squared minus x minus 12 , the fraction 1 over x squared minus 16")[$frac(2, x^(2) − x − 12) , frac(1, x^(2) − 16)$]. #solutionbox[ #math.equation(block: true, alt: "open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis open parenthesis x plus 3 close parenthesis")[$( x − 4 ) ( x + 4 ) ( x + 3 )$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the LCD for #math.equation(block: false, alt: "the fraction x over x squared plus 8 x plus 15 , the fraction 5 over x squared plus 9 x plus 18")[$frac(x, x^(2) + 8 x + 15) , frac(5, x^(2) + 9 x + 18)$]. #solutionbox[ #math.equation(block: true, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x plus 6 close parenthesis open parenthesis x plus 5 close parenthesis")[$( x + 3 ) ( x + 6 ) ( x + 5 )$] ] ] === Find Equivalent Rational Expressions When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD. #figure(figph[The above image shows how to find the LCD (least common denominator) when adding numerical fractions in the example seven-twelfths plus five-eighteenths. The image shows 7 times 3 divided by 12 times 3 plus 5 times 2 plus 18 times 2. Below this is 21 divided by 36 plus 10 divided by 36. The image next to this shows that 12 equals 2 times 2 times 3. Below this shows 18 equals 2 times 3 times 3. A line is drawn. Below it is LCD equals 2 times 2 times 3 times 3. The line below this shows that the LCD equals 36.], alt: "The above image shows how to find the LCD (least common denominator) when adding numerical fractions in the example seven-twelfths plus five-eighteenths. The image shows 7 times 3 divided by 12 times 3 plus 5 times 2 plus 18 times 2. Below this is 21 divided by 36 plus 10 divided by 36. The image next to this shows that 12 equals 2 times 2 times 3. Below this shows 18 equals 2 times 3 times 3. A line is drawn. Below it is LCD equals 2 times 2 times 3 times 3. The line below this shows that the LCD equals 36.", caption: none) We will do the same thing for rational expressions. #examplebox("Example 2")[][ Rewrite as equivalent rational expressions with denominator #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis open parenthesis x minus 3 close parenthesis open parenthesis x plus 3 close parenthesis")[$( x + 1 ) ( x − 3 ) ( x + 3 )$]: #math.equation(block: false, alt: "the fraction 8 over x squared minus 2 x minus 3 , the fraction 3 x over x squared plus 4 x plus 3 .")[$frac(8, x^(2) − 2 x − 3) , frac(3 x, x^(2) + 4 x + 3) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[Two algebraic fractions are shown: 8 over (x^2 - 2x - 3) and 3x over (x^2 + 4x + 3).], alt: "Two algebraic fractions are shown: 8 over (x^2 - 2x - 3) and 3x over (x^2 + 4x + 3).", caption: none)]), [Factor each denominator.], [#figure(figph[Two algebraic fractions are shown: 8 over (x+1)(x-3) and 3x over (x+1)(x+3).], alt: "Two algebraic fractions are shown: 8 over (x+1)(x-3) and 3x over (x+1)(x+3).", caption: none)], [Find the LCD.  #figure(figph[Quadratic expressions x^2-2x-3 and x^2+4x+3 are factored, leading to the calculation of their Least Common Denominator (LCD) as (x+1)(x-3)(x+3).], alt: "Quadratic expressions x^2-2x-3 and x^2+4x+3 are factored, leading to the calculation of their Least Common Denominator (LCD) as (x+1)(x-3)(x+3).", caption: none)], [], [Multiply each denominator by the 'missing' factor and multiply each numerator by the same factor.], [#figure(figph[Two rational expressions are shown: 8(x+3)/((x+1)(x-3)(x+3)) and 3x(x-3)/((x+1)(x+3)(x-3)). Key terms are highlighted in red.], alt: "Two rational expressions are shown: 8(x+3)/((x+1)(x-3)(x+3)) and 3x(x-3)/((x+1)(x+3)(x-3)). Key terms are highlighted in red.", caption: none)], [Simplify the numerators.], [#figure(figph[Two algebraic fractions displayed. The first is 8x+24 over (x+1)(x-3)(x+3), and the second is 3x^2-9x over (x+1)(x+3)(x-3). Both fractions have common denominators.], alt: "Two algebraic fractions displayed. The first is 8x+24 over (x+1)(x-3)(x+3), and the second is 3x^2-9x over (x+1)(x+3)(x-3). Both fractions have common denominators.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Rewrite as equivalent rational expressions with denominator #math.equation(block: false, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x minus 4 close parenthesis open parenthesis x plus 4 close parenthesis")[$( x + 3 ) ( x − 4 ) ( x + 4 )$]: #linebreak() #math.equation(block: false, alt: "the fraction 2 over x squared minus x minus 12 , the fraction 1 over x squared minus 16 .")[$frac(2, x^(2) − x − 12) , frac(1, x^(2) − 16) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 x plus 8 over open parenthesis x minus 4 close parenthesis open parenthesis x plus 3 close parenthesis open parenthesis x plus 4 close parenthesis ,")[$frac(2 x + 8, ( x − 4 ) ( x + 3 ) ( x + 4 )) ,$] #linebreak() #math.equation(block: true, alt: "the fraction x plus 3 over open parenthesis x minus 4 close parenthesis open parenthesis x plus 3 close parenthesis open parenthesis x plus 4 close parenthesis")[$frac(x + 3, ( x − 4 ) ( x + 3 ) ( x + 4 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Rewrite as equivalent rational expressions with denominator #math.equation(block: false, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x plus 5 close parenthesis open parenthesis x plus 6 close parenthesis")[$( x + 3 ) ( x + 5 ) ( x + 6 )$]: #linebreak() #math.equation(block: false, alt: "the fraction x over x squared plus 8 x plus 15 , the fraction 5 over x squared plus 9 x plus 18 .")[$frac(x, x^(2) + 8 x + 15) , frac(5, x^(2) + 9 x + 18) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x squared plus 6 x over open parenthesis x plus 3 close parenthesis open parenthesis x plus 5 close parenthesis open parenthesis x plus 6 close parenthesis ,")[$frac(x^(2) + 6 x, ( x + 3 ) ( x + 5 ) ( x + 6 )) ,$] #linebreak() #math.equation(block: true, alt: "the fraction 5 x plus 25 over open parenthesis x plus 3 close parenthesis open parenthesis x plus 5 close parenthesis open parenthesis x plus 6 close parenthesis")[$frac(5 x + 25, ( x + 3 ) ( x + 5 ) ( x + 6 ))$] ] ] === Add Rational Expressions with Different Denominators Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first. #examplebox("Example 3")[][ Add: #math.equation(block: false, alt: "the fraction 7 over 12 plus the fraction 5 over 18 .")[$frac(7, 12) + frac(5, 18) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image shows the fraction addition problem '7 over 12 plus 5 over 18'.], alt: "The image shows the fraction addition problem '7 over 12 plus 5 over 18'.", caption: none)]), [Find the LCD of 12 and 18.  #figure(figph[This image illustrates the calculation of the Least Common Denominator (LCD) for the numbers 12 and 18. It shows their prime factorizations as 12 = 2x2x3 and 18 = 2x3x3, leading to an LCD of 2x2x3x3, which equals 36.], alt: "This image illustrates the calculation of the Least Common Denominator (LCD) for the numbers 12 and 18. It shows their prime factorizations as 12 = 2x2x3 and 18 = 2x3x3, leading to an LCD of 2x2x3x3, which equals 36.", caption: none)], [], [Rewrite each fraction as an equivalent fraction with the LCD.], [#figure(figph[A math problem displaying the sum of two fractions: (7 \* 3) / (12 \* 3) + (5 \* 2) / (18 \* 2). The numbers 3 and 2 are highlighted in red, suggesting multiplication to achieve common denominators.], alt: "A math problem displaying the sum of two fractions: (7 * 3) / (12 * 3) + (5 * 2) / (18 * 2). The numbers 3 and 2 are highlighted in red, suggesting multiplication to achieve common denominators.", caption: none)], [Add the fractions.], [#figure(figph[A mathematical expression showing the sum of two fractions with a common denominator: 21/36 + 10/36.], alt: "A mathematical expression showing the sum of two fractions with a common denominator: 21/36 + 10/36.", caption: none)], [The fraction cannot be simplified.], [#figure(figph[A fraction is displayed with 31 as the numerator, a horizontal line, and 36 as the denominator.], alt: "A fraction is displayed with 31 as the numerator, a horizontal line, and 36 as the denominator.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 11 over 30 plus the fraction 7 over 12 .")[$frac(11, 30) + frac(7, 12) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 19 over 20")[$frac(19, 20)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 3 over 8 plus the fraction 9 over 20 .")[$frac(3, 8) + frac(9, 20) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 33 over 40")[$frac(33, 40)$] ] ] Now we will add rational expressions whose denominators are monomials. #examplebox("Example 4")[][ Add: #math.equation(block: false, alt: "the fraction 5 over 12 x squared y plus the fraction 4 over 21 x y squared .")[$frac(5, 12 x^(2) y) + frac(4, 21 x y^(2)) .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([], [], [#figure(figph[An algebraic expression showing the sum of two fractions: 5 over 12x squared y, plus 4 over 21xy squared.], alt: "An algebraic expression showing the sum of two fractions: 5 over 12x squared y, plus 4 over 21xy squared.", caption: none)]), [Find the LCD of #math.equation(block: false, alt: "12 x squared y")[$12 x^(2) y$] and #math.equation(block: false, alt: "21 x y squared")[$21 x y^(2)$].  #figure(figph[Step-by-step calculation of the Least Common Denominator (LCD) for 12x²y and 21xy². Prime factorization leads to an LCD of 84x²y².], alt: "Step-by-step calculation of the Least Common Denominator (LCD) for 12x²y and 21xy². Prime factorization leads to an LCD of 84x²y².", caption: none)], [], [], [], [], [#figure(figph[A mathematical expression showing the sum of two algebraic fractions: 5/(12x^2y) + 4/(21xy^2).], alt: "A mathematical expression showing the sum of two algebraic fractions: 5/(12x^2y) + 4/(21xy^2).", caption: none)], [Rewrite each rational expression as an equivalent fraction with the LCD.], [], [#figure(figph[An algebraic expression displaying the addition of two rational terms: (5 \* 7y) / (12x^2y \* 7y) and (4 \* 4x) / (21xy^2 \* 4x), illustrating a step in finding a common denominator for two fractions.], alt: "An algebraic expression displaying the addition of two rational terms: (5 * 7y) / (12x^2y * 7y) and (4 * 4x) / (21xy^2 * 4x), illustrating a step in finding a common denominator for two fractions.", caption: none)], [Simplify.], [], [#figure(figph[A mathematical expression showing the sum of two fractions: 35y over 84x^2y^2 plus 16x over 84x^2y^2. Both fractions share a common denominator.], alt: "A mathematical expression showing the sum of two fractions: 35y over 84x^2y^2 plus 16x over 84x^2y^2. Both fractions share a common denominator.", caption: none)], [Add the rational expressions.], [], [#figure(figph[A fraction with the numerator 16x + 35y and the denominator 84x^2y^2 is displayed on a white background, representing a mathematical algebraic expression.], alt: "A fraction with the numerator 16x + 35y and the denominator 84x^2y^2 is displayed on a white background, representing a mathematical algebraic expression.", caption: none)], [There are no factors common to the numerator and denominator. The fraction cannot be simplified.], [], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 2 over 15 a squared b plus the fraction 5 over 6 a b squared .")[$frac(2, 15 a^(2) b) + frac(5, 6 a b^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 b plus 25 a over 30 a squared b squared")[$frac(4 b + 25 a, 30 a^(2) b^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 5 over 16 c plus the fraction 3 over 8 c d squared .")[$frac(5, 16 c) + frac(3, 8 c d^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 d squared plus 6 over 16 c d squared")[$frac(5 d^(2) + 6, 16 c d^(2))$] ] ] Now we are ready to tackle polynomial denominators. #examplebox("Example 5")[How to Add Rational Expressions with Different Denominators][ Add: #math.equation(block: false, alt: "the fraction 3 over x minus 3 plus the fraction 2 over x minus 2 .")[$frac(3, x − 3) + frac(2, x − 2) .$] #solutionbox[ #figure(figph[The above image shows the steps to add fractions whose denominators are monomials for the example 5 divided by 12 x squared y plus 4 divided by 21 x y squared. Find the LCD of 12 x squared y and 21 x y squared. To the right of this expression is 12 x squared y equals 2 times 2 times 3 times x times x times y. Below that is 21 x y squared equals 3 times 7 times x times y times y. A line is drawn. Below that is LCD equals 2 times 2 times 3 times 7 times x times x times y times y. Below that is LCD equals 84 x squared y squared. Rewrite each rational expression as an equivalent fraction with the LCD. The original equation is shown. Below that is 5 times 7 y divided by 12 x squared y times 7 y plus 4 times 4 x divided by 21 x y squared times 4 x. Simplify to get 35 y divided by 84 x squared y squared plus 16 x divided by x squared y squared. Add the rational expressions 16 x plus 35 y divided by 84 x squared y squared. There are no factors common to the numeration and denominator. The fraction cannot be simplified.], alt: "The above image shows the steps to add fractions whose denominators are monomials for the example 5 divided by 12 x squared y plus 4 divided by 21 x y squared. Find the LCD of 12 x squared y and 21 x y squared. To the right of this expression is 12 x squared y equals 2 times 2 times 3 times x times x times y. Below that is 21 x y squared equals 3 times 7 times x times y times y. A line is drawn. Below that is LCD equals 2 times 2 times 3 times 7 times x times x times y times y. Below that is LCD equals 84 x squared y squared. Rewrite each rational expression as an equivalent fraction with the LCD. The original equation is shown. Below that is 5 times 7 y divided by 12 x squared y times 7 y plus 4 times 4 x divided by 21 x y squared times 4 x. Simplify to get 35 y divided by 84 x squared y squared plus 16 x divided by x squared y squared. Add the rational expressions 16 x plus 35 y divided by 84 x squared y squared. There are no factors common to the numeration and denominator. The fraction cannot be simplified.", caption: none) #figure(figph[Step 2 is to add the rational expression. Then, add the numerators and place the sum over the common denominator to get 3 x minus 6 plus 2 x minus 6 divided by x minus 3 times x minus 2.], alt: "Step 2 is to add the rational expression. Then, add the numerators and place the sum over the common denominator to get 3 x minus 6 plus 2 x minus 6 divided by x minus 3 times x minus 2.", caption: none) #figure(figph[Step 3 is to simplify, if possible. Because 5 x minus 12 cannot be factored, the answer is simplified to 5 x minus 12 divided by x minus 3 times x minus 2.], alt: "Step 3 is to simplify, if possible. Because 5 x minus 12 cannot be factored, the answer is simplified to 5 x minus 12 divided by x minus 3 times x minus 2.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 2 over x minus 2 plus the fraction 5 over x plus 3 .")[$frac(2, x − 2) + frac(5, x + 3) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 x minus 4 over open parenthesis x plus 3 close parenthesis open parenthesis x minus 2 close parenthesis")[$frac(7 x − 4, ( x + 3 ) ( x − 2 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 4 over m plus 3 plus the fraction 3 over m plus 4 .")[$frac(4, m + 3) + frac(3, m + 4) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 m plus 25 over open parenthesis m plus 3 close parenthesis open parenthesis m plus 4 close parenthesis")[$frac(7 m + 25, ( m + 3 ) ( m + 4 ))$] ] ] The steps to use to add rational expressions are summarized in the following procedure box. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Add rational expressions.] + Determine if the expressions have a common denominator. #linebreak() #strong[Yes] – go to step 2. #linebreak() #strong[No] – Rewrite each rational expression with the LCD. #linebreak() Find the LCD. #linebreak() Rewrite each rational expression as an equivalent rational expression with the LCD. + Add the rational expressions. + Simplify, if possible. ] #examplebox("Example 6")[][ Add: #math.equation(block: false, alt: "the fraction 2 a over 2 a b plus b squared plus the fraction 3 a over 4 a squared minus b squared .")[$frac(2 a, 2 a b + b^(2)) + frac(3 a, 4 a^(2) − b^(2)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression showing the sum of two fractions: 2a divided by (2ab + b^2) plus 3a divided by (4a^2 - b^2).], alt: "A mathematical expression showing the sum of two fractions: 2a divided by (2ab + b^2) plus 3a divided by (4a^2 - b^2).", caption: none)]), [Do the expressions have a common denominator? No. #linebreak() Rewrite each expression with the LCD.], [], [Find the LCD.  #figure(figph[Algebraic factorization of \`2ab + b^2\` and \`4a^2 - b^2\`, demonstrating how to find the Least Common Denominator (LCD) of these expressions, which is \`b(2a + b)(2a - b)\`.], alt: "Algebraic factorization of `2ab + b^2` and `4a^2 - b^2`, demonstrating how to find the Least Common Denominator (LCD) of these expressions, which is `b(2a + b)(2a - b)`.", caption: none)], [], [Rewrite each rational expression as an equivalent rational expression with the LCD.], [#figure(figph[An algebraic expression displaying the sum of two fractions with a common denominator, prior to simplification. Numerators are 2a(2a-b) and 3ab, while the common denominator is b(2a+b)(2a-b).], alt: "An algebraic expression displaying the sum of two fractions with a common denominator, prior to simplification. Numerators are 2a(2a-b) and 3ab, while the common denominator is b(2a+b)(2a-b).", caption: none)], [Simplify the numerators.], [#figure(figph[An algebraic expression showing the sum of two fractions with a common denominator of b(2a+b)(2a-b). The first numerator is 4a^2-2ab, and the second numerator is 3ab.], alt: "An algebraic expression showing the sum of two fractions with a common denominator of b(2a+b)(2a-b). The first numerator is 4a^2-2ab, and the second numerator is 3ab.", caption: none)], [Add the rational expressions.], [#figure(figph[A mathematical expression showing a fraction with 4a^2 - 2ab + 3ab in the numerator and b(2a + b)(2a - b) in the denominator.], alt: "A mathematical expression showing a fraction with 4a^2 - 2ab + 3ab in the numerator and b(2a + b)(2a - b) in the denominator.", caption: none)], [Simplify the numerator.], [#figure(figph[A mathematical fraction is shown, with the numerator as 4a^2 + ab and the denominator as b(2a + b)(2a - b).], alt: "A mathematical fraction is shown, with the numerator as 4a^2 + ab and the denominator as b(2a + b)(2a - b).", caption: none)], [Factor the numerator.], [#figure(figph[A mathematical fraction with the numerator a(4a + b) and the denominator b(2a + b)(2a - b).], alt: "A mathematical fraction with the numerator a(4a + b) and the denominator b(2a + b)(2a - b).", caption: none)], [There are no factors common to the numerator and denominator. The fraction cannot be simplified.], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 5 x over x y minus y squared plus the fraction 2 x over x squared minus y squared .")[$frac(5 x, x y − y^(2)) + frac(2 x, x^(2) − y^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction x open parenthesis 5 x plus 7 y close parenthesis over y open parenthesis x minus y close parenthesis open parenthesis x plus y close parenthesis")[$frac(x ( 5 x + 7 y ), y ( x − y ) ( x + y ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 7 over 2 m plus 6 plus the fraction 4 over m squared plus 4 m plus 3 .")[$frac(7, 2 m + 6) + frac(4, m^(2) + 4 m + 3) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 m plus 15 over 2 open parenthesis m plus 3 close parenthesis open parenthesis m plus 1 close parenthesis")[$frac(7 m + 15, 2 ( m + 3 ) ( m + 1 ))$] ] ] Avoid the temptation to simplify too soon! In the example above, we must leave the first rational expression as #math.equation(block: false, alt: "the fraction 2 a open parenthesis 2 a minus b close parenthesis over b open parenthesis 2 a plus b close parenthesis open parenthesis 2 a minus b close parenthesis")[$frac(2 a ( 2 a − b ), b ( 2 a + b ) ( 2 a − b ))$] to be able to add it to #math.equation(block: false, alt: "the fraction 3 a times b over open parenthesis 2 a plus b close parenthesis open parenthesis 2 a minus b close parenthesis times b")[$frac(3 a · b, ( 2 a + b ) ( 2 a − b ) · b)$]. Simplify only after you have combined the numerators. #examplebox("Example 7")[][ Add: #math.equation(block: false, alt: "the fraction 8 over x squared minus 2 x minus 3 plus the fraction 3 x over x squared plus 4 x plus 3 .")[$frac(8, x^(2) − 2 x − 3) + frac(3 x, x^(2) + 4 x + 3) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[An algebraic expression featuring the sum of two rational fractions: 8/(x^2 - 2x - 3) + 3x/(x^2 + 4x + 3).], alt: "An algebraic expression featuring the sum of two rational fractions: 8/(x^2 - 2x - 3) + 3x/(x^2 + 4x + 3).", caption: none)]), [Do the expressions have a common denominator? No. #linebreak() Rewrite each expression with the LCD.], [], [Find the LCD.  #figure(figph[Factoring quadratic expressions x^2 - 2x - 3 and x^2 + 4x + 3 to determine their Least Common Denominator (LCD), which is (x+1)(x-3)(x+3).], alt: "Factoring quadratic expressions x^2 - 2x - 3 and x^2 + 4x + 3 to determine their Least Common Denominator (LCD), which is (x+1)(x-3)(x+3).", caption: none)], [], [Rewrite each rational expression as an equivalent fraction with the LCD.], [#figure(figph[An algebraic expression showing the sum of two rational fractions, where common factors in the numerators and denominators are highlighted in red, indicating potential simplification steps.], alt: "An algebraic expression showing the sum of two rational fractions, where common factors in the numerators and denominators are highlighted in red, indicating potential simplification steps.", caption: none)], [Simplify the numerators.], [#figure(figph[An algebraic expression demonstrating the sum of two rational functions with a common denominator. The numerators are (8x + 24) and (3x^2 - 9x).], alt: "An algebraic expression demonstrating the sum of two rational functions with a common denominator. The numerators are (8x + 24) and (3x^2 - 9x).", caption: none)], [Add the rational expressions.], [#figure(figph[A mathematical fraction displaying the expression (3x^2 - x + 24) divided by (x + 1)(x - 3)(x + 3).], alt: "A mathematical fraction displaying the expression (3x^2 - x + 24) divided by (x + 1)(x - 3)(x + 3).", caption: none)], [Simplify the numerator.], [#figure(figph[A mathematical fraction displaying the expression (3x^2 - x + 24) divided by (x + 1)(x - 3)(x + 3).], alt: "A mathematical fraction displaying the expression (3x^2 - x + 24) divided by (x + 1)(x - 3)(x + 3).", caption: none)], [The numerator is prime, so there are no common factors.], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 1 over m squared minus m minus 2 plus the fraction 5 m over m squared plus 3 m plus 2 .")[$frac(1, m^(2) − m − 2) + frac(5 m, m^(2) + 3 m + 2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 m squared minus 9 m plus 2 over open parenthesis m minus 2 close parenthesis open parenthesis m plus 1 close parenthesis open parenthesis m plus 2 close parenthesis")[$frac(5 m^(2) − 9 m + 2, ( m − 2 ) ( m + 1 ) ( m + 2 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 2 n over n squared minus 3 n minus 10 plus the fraction 6 over n squared plus 5 n plus 6 .")[$frac(2 n, n^(2) − 3 n − 10) + frac(6, n^(2) + 5 n + 6) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 open parenthesis n squared plus 6 n minus 15 close parenthesis over open parenthesis n plus 2 close parenthesis open parenthesis n minus 5 close parenthesis open parenthesis n plus 3 close parenthesis")[$frac(2 ( n^(2) + 6 n − 15 ), ( n + 2 ) ( n − 5 ) ( n + 3 ))$] ] ] === Subtract Rational Expressions with Different Denominators The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators. #examplebox("Example 8")[How to Subtract Rational Expressions with Different Denominators][ Subtract: #math.equation(block: false, alt: "the fraction x over x minus 3 minus the fraction x minus 2 over x plus 3 .")[$frac(x, x − 3) − frac(x − 2, x + 3) .$] #solutionbox[ #figure(figph[The above image has 3 columns. It shows the steps on how to subtract rational expressions with different denominators for x divided by x minus three minus x plus x minus 3. Step 1 is to Determine if the expressions have a common denominator. Yes – go to step 2. No – Rewrite each rational expression with the LCD. Find the LCD. Rewrite each rational expression as an equivalent rational expression with the LCD. In the above expression, the answer is no. Find the LCD of x minus 3, x plus 3. To the right of this is x – 3: x – 3. Below that is x – 2: x – 2. A line is drawn. Below that is written the LCD is x – 3 times x plus 3. Rewrite as x times x plus 3 divided by x minus 3 times x plus 3 minus x minus 2 times x minus 3 divided by x plus 3 times x minus 3. Keep the denominators factored! Factor to get x squared plus 3 x divided by x minus 3 times x plus 3 minus x squared minus 5 x plus 6 divided by x minus 3 times x plus 3.], alt: "The above image has 3 columns. It shows the steps on how to subtract rational expressions with different denominators for x divided by x minus three minus x plus x minus 3. Step 1 is to Determine if the expressions have a common denominator. Yes – go to step 2. No – Rewrite each rational expression with the LCD. Find the LCD. Rewrite each rational expression as an equivalent rational expression with the LCD. In the above expression, the answer is no. Find the LCD of x minus 3, x plus 3. To the right of this is x – 3: x – 3. Below that is x – 2: x – 2. A line is drawn. Below that is written the LCD is x – 3 times x plus 3. Rewrite as x times x plus 3 divided by x minus 3 times x plus 3 minus x minus 2 times x minus 3 divided by x plus 3 times x minus 3. Keep the denominators factored! Factor to get x squared plus 3 x divided by x minus 3 times x plus 3 minus x squared minus 5 x plus 6 divided by x minus 3 times x plus 3.", caption: none) #figure(figph[Step 2 is to subtract the rational expressions. Subtract the numerators and place the difference over the common denominator to get x 2 plus 3 x minus x squared minus 5 x plus 6 divided by x minus 3 times x plus 3. Then to x squared plus 3 x minus x squared plus 5 x minus 6 divided by x minus 3 times x plus 3. Be careful with the signs! Then to 8 x minus 6 divided by x minus 3 times x plus 3.], alt: "Step 2 is to subtract the rational expressions. Subtract the numerators and place the difference over the common denominator to get x 2 plus 3 x minus x squared minus 5 x plus 6 divided by x minus 3 times x plus 3. Then to x squared plus 3 x minus x squared plus 5 x minus 6 divided by x minus 3 times x plus 3. Be careful with the signs! Then to 8 x minus 6 divided by x minus 3 times x plus 3.", caption: none) #figure(figph[Step 3 is to simplify, if possible. The numerator and denominator have no factors in common. The answer is simplified to 2 times 4 x minus 3 divided by x minus 3 times x plus 3.], alt: "Step 3 is to simplify, if possible. The numerator and denominator have no factors in common. The answer is simplified to 2 times 4 x minus 3 divided by x minus 3 times x plus 3.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "the fraction y over y plus 4 minus the fraction y minus 2 over y minus 5 .")[$frac(y, y + 4) − frac(y − 2, y − 5) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction −7 y plus 8 over open parenthesis y plus 4 close parenthesis open parenthesis y minus 5 close parenthesis")[$frac(−7 y + 8, ( y + 4 ) ( y − 5 ))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "the fraction z plus 3 over z plus 2 minus the fraction z over z plus 3 .")[$frac(z + 3, z + 2) − frac(z, z + 3) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 z plus 9 over open parenthesis z plus 2 close parenthesis open parenthesis z plus 3 close parenthesis")[$frac(4 z + 9, ( z + 2 ) ( z + 3 ))$] ] ] The steps to take to subtract rational expressions are listed below. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Subtract rational expressions.] + Determine if they have a common denominator. #linebreak() #strong[Yes] – go to step 2. #linebreak() #strong[No] – Rewrite each rational expression with the LCD. #linebreak() Find the LCD. #linebreak() Rewrite each rational expression as an equivalent rational expression with the LCD. + Subtract the rational expressions. + Simplify, if possible. ] #examplebox("Example 9")[][ Subtract: #math.equation(block: false, alt: "the fraction 8 y over y squared minus 16 minus the fraction 4 over y minus 4 .")[$frac(8 y, y^(2) − 16) − frac(4, y − 4) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression showing the subtraction of two algebraic fractions: 8y divided by the quantity y squared minus 16, minus 4 divided by the quantity y minus 4.], alt: "A mathematical expression showing the subtraction of two algebraic fractions: 8y divided by the quantity y squared minus 16, minus 4 divided by the quantity y minus 4.", caption: none)]), [Do the expressions have a common denominator? No. #linebreak() Rewrite each expression with the LCD.], [], [Find the LCD.  #figure(figph[Factoring y^2 - 16 into (y-4)(y+4) and identifying y-4, then calculating the Least Common Denominator (LCD) as (y-4)(y+4).], alt: "Factoring y^2 - 16 into (y-4)(y+4) and identifying y-4, then calculating the Least Common Denominator (LCD) as (y-4)(y+4).", caption: none)], [], [Rewrite each rational expression as an equivalent rational expression with the LCD.], [#figure(figph[A mathematical expression showing the subtraction of two fractions with a common denominator of (y-4)(y+4). The numerators are 8y and 4(y+4), with parts of the second fraction highlighted in red.], alt: "A mathematical expression showing the subtraction of two fractions with a common denominator of (y-4)(y+4). The numerators are 8y and 4(y+4), with parts of the second fraction highlighted in red.", caption: none)], [Simplify the numerators.], [#figure(figph[An algebraic expression showing the subtraction of two fractions with a common denominator of (y-4)(y+4). The numerators are 8y and 4y+16.], alt: "An algebraic expression showing the subtraction of two fractions with a common denominator of (y-4)(y+4). The numerators are 8y and 4y+16.", caption: none)], [Subtract the rational expressions.], [#figure(figph[An algebraic expression shown as a fraction with 8y - 4y - 16 in the numerator and (y - 4)(y + 4) in the denominator.], alt: "An algebraic expression shown as a fraction with 8y - 4y - 16 in the numerator and (y - 4)(y + 4) in the denominator.", caption: none)], [Simplify the numerators.], [#figure(figph[A mathematical expression showing the fraction (4y - 16) / ((y - 4)(y + 4)).], alt: "A mathematical expression showing the fraction (4y - 16) / ((y - 4)(y + 4)).", caption: none)], [Factor the numerator to look for common factors.], [#figure(figph[A mathematical expression showing the fraction 4(y-4) over (y-4)(y+4).], alt: "A mathematical expression showing the fraction 4(y-4) over (y-4)(y+4).", caption: none)], [Remove common factors.], [#figure(figph[A mathematical expression showing the fraction 4(y-4) over (y-4)(y+4), with the common factor (y-4) in both the numerator and denominator crossed out, indicating cancellation.], alt: "A mathematical expression showing the fraction 4(y-4) over (y-4)(y+4), with the common factor (y-4) in both the numerator and denominator crossed out, indicating cancellation.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression shows the fraction 4 over (y + 4).], alt: "A mathematical expression shows the fraction 4 over (y + 4).", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "the fraction 2 x over x squared minus 4 minus the fraction 1 over x plus 2 .")[$frac(2 x, x^(2) − 4) − frac(1, x + 2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over x minus 2")[$frac(1, x − 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "the fraction 3 over z plus 3 minus the fraction 6 z over z squared minus 9 .")[$frac(3, z + 3) − frac(6 z, z^(2) − 9) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction −3 over z minus 3")[$frac(−3, z − 3)$] ] ] There are lots of negative signs in the next example. Be extra careful! #examplebox("Example 10")[][ Subtract: #math.equation(block: false, alt: "the fraction −3 n minus 9 over n squared plus n minus 6 minus the fraction n plus 3 over 2 minus n .")[$frac(−3 n − 9, n^(2) + n − 6) − frac(n + 3, 2 − n) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[An algebraic expression showing the subtraction of two rational expressions: (-3n - 9) / (n^2 + n - 6) - (n + 3) / (2 - n).], alt: "An algebraic expression showing the subtraction of two rational expressions: (-3n - 9) / (n^2 + n - 6) - (n + 3) / (2 - n).", caption: none)]), [Factor the denominator.], [#figure(figph[A mathematical expression featuring the subtraction of two rational expressions: (-3n - 9)/((n-2)(n+3)) - (n+3)/(2-n).], alt: "A mathematical expression featuring the subtraction of two rational expressions: (-3n - 9)/((n-2)(n+3)) - (n+3)/(2-n).", caption: none)], [Since #math.equation(block: false, alt: "n minus 2")[$n − 2$] and #math.equation(block: false, alt: "2 minus n")[$2 − n$] are opposites, we will mutliply the second rational expression by#math.equation(block: false, alt: "the fraction −1 over −1")[$frac(−1, −1)$].], [#figure(figph[Subtraction of two rational expressions: (-3n-9)/((n-2)(n+3)) - ((-1)(n+3))/((-1)(2-n)).], alt: "Subtraction of two rational expressions: (-3n-9)/((n-2)(n+3)) - ((-1)(n+3))/((-1)(2-n)).", caption: none)], [Simplify.], [#figure(figph[An algebraic expression showing the sum of two rational functions: ((-3n - 9) / ((n - 2)(n + 3))) + ((n + 3) / (n - 2)).], alt: "An algebraic expression showing the sum of two rational functions: ((-3n - 9) / ((n - 2)(n + 3))) + ((n + 3) / (n - 2)).", caption: none)], [Do the expressions have a common denominator? No.], [], [Find the LCD.  #figure(figph[A mathematical solution showing how to find the Least Common Denominator (LCD) of two algebraic expressions. It factors n^2 + n - 6 into (n - 2)(n + 3) and uses n - 2 to determine the LCD is (n - 2)(n + 3).], alt: "A mathematical solution showing how to find the Least Common Denominator (LCD) of two algebraic expressions. It factors n^2 + n - 6 into (n - 2)(n + 3) and uses n - 2 to determine the LCD is (n - 2)(n + 3).", caption: none)], [], [Rewrite each rational expression as an equivalent rational expression with the LCD.], [#figure(figph[An algebraic expression showing the sum of two fractions with a common denominator of (n-2)(n+3). The numerators are (-3n - 9) and (n+3)(n+3), with an (n+3) term highlighted in red.], alt: "An algebraic expression showing the sum of two fractions with a common denominator of (n-2)(n+3). The numerators are (-3n - 9) and (n+3)(n+3), with an (n+3) term highlighted in red.", caption: none)], [Simplify the numerators.], [#figure(figph[A mathematical expression showing the addition of two rational algebraic expressions. Both fractions have a common denominator of (n-2)(n+3). The numerators are -3n-9 and n^2+6n+9 respectively.], alt: "A mathematical expression showing the addition of two rational algebraic expressions. Both fractions have a common denominator of (n-2)(n+3). The numerators are -3n-9 and n^2+6n+9 respectively.", caption: none)], [Simplify the rational expressions.], [#figure(figph[A mathematical fraction. The numerator is -3n - 9 + n^2 + 6n + 9, and the denominator is the product of (n-2) and (n+3).], alt: "A mathematical fraction. The numerator is -3n - 9 + n^2 + 6n + 9, and the denominator is the product of (n-2) and (n+3).", caption: none)], [Simplify the numerator.], [#figure(figph[A mathematical expression showing a fraction with n squared plus 3n in the numerator and the product of (n minus 2) and (n plus 3) in the denominator.], alt: "A mathematical expression showing a fraction with n squared plus 3n in the numerator and the product of (n minus 2) and (n plus 3) in the denominator.", caption: none)], [Factor the numerator to look for common factors.], [#figure(figph[A mathematical expression showing the fraction n(p+3) / ((n-2)(p+3)), where the (p+3) terms in both the numerator and denominator are crossed out, indicating they are being canceled.], alt: "A mathematical expression showing the fraction n(p+3) / ((n-2)(p+3)), where the (p+3) terms in both the numerator and denominator are crossed out, indicating they are being canceled.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression displaying the fraction n over (n-2).], alt: "A mathematical expression displaying the fraction n over (n-2).", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "the fraction 3 x minus 1 over x squared minus 5 x minus 6 minus the fraction 2 over 6 minus x .")[$frac(3 x − 1, x^(2) − 5 x − 6) − frac(2, 6 − x) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 x plus 1 over x minus 6 x plus 1")[$frac(5 x + 1, x − 6 x + 1)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "the fraction −2 y minus 2 over y squared plus 2 y minus 8 minus the fraction y minus 1 over 2 minus y .")[$frac(−2 y − 2, y^(2) + 2 y − 8) − frac(y − 1, 2 − y) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction y plus 3 over y plus 4")[$frac(y + 3, y + 4)$] ] ] When one expression is not in fraction form, we can write it as a fraction with denominator 1. #examplebox("Example 11")[][ Subtract: #math.equation(block: false, alt: "the fraction 5 c plus 4 over c minus 2 minus 3 .")[$frac(5 c + 4, c − 2) − 3 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression displays a fraction '5c + 4' over 'c - 2', followed by a subtraction of 3. The expression is (5c + 4)/(c - 2) - 3.], alt: "A mathematical expression displays a fraction '5c + 4' over 'c - 2', followed by a subtraction of 3. The expression is (5c + 4)/(c - 2) - 3.", caption: none)]), [Write #math.equation(block: false, alt: "3")[$3$] as #math.equation(block: false, alt: "the fraction 3 over 1")[$frac(3, 1)$] to have 2 rational expressions.], [#figure(figph[A mathematical expression showing the subtraction of two fractions: (5c + 4) / (c - 2) - 3/1.], alt: "A mathematical expression showing the subtraction of two fractions: (5c + 4) / (c - 2) - 3/1.", caption: none)], [Do the rational expressions have a common denominator? No.], [], [Find the LCD of #math.equation(block: false, alt: "c minus 2")[$c − 2$] and #math.equation(block: false, alt: "1 .")[$1 .$] LCD = #math.equation(block: false, alt: "c minus 2 .")[$c − 2 .$]], [], [Rewrite #math.equation(block: false, alt: "the fraction 3 over 1")[$frac(3, 1)$] as an equivalent rational expression with the LCD.], [#figure(figph[A mathematical expression displaying the subtraction of two algebraic fractions. The first term is (5c+4)/(c-2), and the second term is 3(c-2) over 1(c-2), with factors in red.], alt: "A mathematical expression displaying the subtraction of two algebraic fractions. The first term is (5c+4)/(c-2), and the second term is 3(c-2) over 1(c-2), with factors in red.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing the subtraction of two fractions with a common denominator (c-2): (5c+4)/(c-2) - (3c-6)/(c-2).], alt: "A mathematical expression showing the subtraction of two fractions with a common denominator (c-2): (5c+4)/(c-2) - (3c-6)/(c-2).", caption: none)], [Subtract the rational expressions.], [#figure(figph[A mathematical expression showing the fraction (5c + 4 - (3c - 6)) divided by (c - 2).], alt: "A mathematical expression showing the fraction (5c + 4 - (3c - 6)) divided by (c - 2).", caption: none)], [Simplify.], [#figure(figph[A fraction with a numerator of 2c + 10 and a denominator of c - 2 is shown on a white background.], alt: "A fraction with a numerator of 2c + 10 and a denominator of c - 2 is shown on a white background.", caption: none)], [Factor to check for common factors.], [#figure(figph[A fraction with 2(c + 5) in the numerator and c - 2 in the denominator.], alt: "A fraction with 2(c + 5) in the numerator and c - 2 in the denominator.", caption: none)], [There are no common factors; the rational expression is simplified.], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "the fraction 2 x plus 1 over x minus 7 minus 3 .")[$frac(2 x + 1, x − 7) − 3 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction − x plus 22 over x minus 7")[$frac("−" x + 22, x − 7)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "the fraction 4 y plus 3 over 2 y minus 1 minus 5 .")[$frac(4 y + 3, 2 y − 1) − 5 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction −2 open parenthesis 3 y minus 4 close parenthesis over 2 y minus 1")[$frac(−2 ( 3 y − 4 ), 2 y − 1)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Add or subtract rational expressions.] + Determine if the expressions have a common denominator. #linebreak() #strong[Yes] – go to step 2. #linebreak() #strong[No] – Rewrite each rational expression with the LCD. #linebreak() Find the LCD. #linebreak() Rewrite each rational expression as an equivalent rational expression with the LCD. + Add or subtract the rational expressions. + Simplify, if possible. ] We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example we will start by factoring all three denominators to find their LCD. #examplebox("Example 12")[][ Simplify: #math.equation(block: false, alt: "the fraction 2 u over u minus 1 plus the fraction 1 over u minus the fraction 2 u minus 1 over u squared minus u .")[$frac(2 u, u − 1) + frac(1, u) − frac(2 u − 1, u^(2) − u) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression featuring three rational terms being added and subtracted: (2u / (u-1)) + (1/u) - ((2u-1) / (u^2-u)).], alt: "A mathematical expression featuring three rational terms being added and subtracted: (2u / (u-1)) + (1/u) - ((2u-1) / (u^2-u)).", caption: none)]), [Do the rational expressions have a common denominator? No.], [], [Find the LCD.  #figure(figph[A mathematical derivation showing algebraic steps: u-1=u-1, followed by u=u. Then, u^2-u is shown equal to u(u-1), concluding that the Least Common Denominator (LCD) is u(u-1).], alt: "A mathematical derivation showing algebraic steps: u-1=u-1, followed by u=u. Then, u^2-u is shown equal to u(u-1), concluding that the Least Common Denominator (LCD) is u(u-1).", caption: none)], [], [Rewrite each rational expression as an equivalent rational expression with the LCD.], [#figure(figph[A multi-term mathematical expression involving fractions with the variable 'u'. It shows three rational terms being added and subtracted, all sharing a common denominator of u(u-1) or (u-1)u.], alt: "A multi-term mathematical expression involving fractions with the variable 'u'. It shows three rational terms being added and subtracted, all sharing a common denominator of u(u-1) or (u-1)u.", caption: none)], [], [#figure(figph[An algebraic expression showing the sum and difference of three rational terms with a common denominator, written as 2u^2/((u-1)u) + (u-1)/(u\*(u-1)) - (2u-1)/(u(u-1)).], alt: "An algebraic expression showing the sum and difference of three rational terms with a common denominator, written as 2u^2/((u-1)u) + (u-1)/(u*(u-1)) - (2u-1)/(u(u-1)).", caption: none)], [Write as one rational expression.], [#figure(figph[A mathematical expression displaying a fraction. The numerator is 2u^2 + u - 1 - 2u + 1, and the denominator is u(u - 1).], alt: "A mathematical expression displaying a fraction. The numerator is 2u^2 + u - 1 - 2u + 1, and the denominator is u(u - 1).", caption: none)], [Simplify.], [#figure(figph[A mathematical expression displaying the fraction 2u squared minus u over u multiplied by the quantity u minus 1, all in black font against a white background.], alt: "A mathematical expression displaying the fraction 2u squared minus u over u multiplied by the quantity u minus 1, all in black font against a white background.", caption: none)], [Factor the numerator, and remove common factors.], [#figure(figph[A mathematical fraction with mu(2u-1) in the numerator and mu(u-1) in the denominator.], alt: "A mathematical fraction with mu(2u-1) in the numerator and mu(u-1) in the denominator.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression displaying the fraction (2u-1) divided by (u-1).], alt: "A mathematical expression displaying the fraction (2u-1) divided by (u-1).", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction v over v plus 1 plus the fraction 3 over v minus 1 minus the fraction 6 over v squared minus 1 .")[$frac(v, v + 1) + frac(3, v − 1) − frac(6, v^(2) − 1) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction v plus 3 over v plus 1")[$frac(v + 3, v + 1)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 3 w over w plus 2 plus the fraction 2 over w plus 7 minus the fraction 17 w plus 4 over w squared plus 9 w plus 14 .")[$frac(3 w, w + 2) + frac(2, w + 7) − frac(17 w + 4, w^(2) + 9 w + 14) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 w over w plus 7")[$frac(3 w, w + 7)$] ] ] === Key Concepts - #strong[Find the Least Common Denominator of Rational Expressions] + Factor each expression completely. + List the factors of each expression. Match factors vertically when possible. + Bring down the columns. + Multiply the factors. - #strong[Add or Subtract Rational Expressions] + Determine if the expressions have a common denominator. #linebreak() #strong[Yes] – go to step 2. #linebreak() #strong[No] – Rewrite each rational expression with the LCD. - Find the LCD. - Rewrite each rational expression as an equivalent rational expression with the LCD. + Add or subtract the rational expressions. + Simplify, if possible. ==== Practice Makes Perfect In the following exercises, find the LCD. #math.equation(block: true, alt: "the fraction 5 over x squared minus 2 x minus 8 , the fraction 2 x over x squared minus x minus 12")[$frac(5, x^(2) − 2 x − 8) , frac(2 x, x^(2) − x − 12)$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis x minus 4 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x plus 3 close parenthesis")[$( x − 4 ) ( x + 2 ) ( x + 3 )$] ] #math.equation(block: true, alt: "the fraction 8 over y squared plus 12 y plus 35 , the fraction 3 y over y squared plus y minus 42")[$frac(8, y^(2) + 12 y + 35) , frac(3 y, y^(2) + y − 42)$] #math.equation(block: true, alt: "the fraction 9 over z squared plus 2 z minus 8 , the fraction 4 z over z squared minus 4")[$frac(9, z^(2) + 2 z − 8) , frac(4 z, z^(2) − 4)$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis z minus 2 close parenthesis open parenthesis z plus 4 close parenthesis open parenthesis z plus 2 close parenthesis")[$( z − 2 ) ( z + 4 ) ( z + 2 )$] ] #math.equation(block: true, alt: "the fraction 6 over a squared plus 14 a plus 45 , the fraction 5 a over a squared minus 81")[$frac(6, a^(2) + 14 a + 45) , frac(5 a, a^(2) − 81)$] #math.equation(block: true, alt: "the fraction 4 over b squared plus 6 b plus 9 , the fraction 2 b over b squared minus 2 b minus 15")[$frac(4, b^(2) + 6 b + 9) , frac(2 b, b^(2) − 2 b − 15)$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis b plus 3 close parenthesis open parenthesis b plus 3 close parenthesis open parenthesis b minus 5 close parenthesis")[$( b + 3 ) ( b + 3 ) ( b − 5 )$] ] #math.equation(block: true, alt: "the fraction 5 over c squared minus 4 c plus 4 , the fraction 3 c over c squared minus 10 c plus 16")[$frac(5, c^(2) − 4 c + 4) , frac(3 c, c^(2) − 10 c + 16)$] #math.equation(block: true, alt: "the fraction 2 over 3 d squared plus 14 d minus 5 , the fraction 5 d over 3 d squared minus 19 d plus 6")[$frac(2, 3 d^(2) + 14 d − 5) , frac(5 d, 3 d^(2) − 19 d + 6)$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis 3 d minus 1 close parenthesis open parenthesis d plus 5 close parenthesis open parenthesis d minus 6 close parenthesis")[$( 3 d − 1 ) ( d + 5 ) ( d − 6 )$] ] #math.equation(block: true, alt: "the fraction 3 over 5 m squared minus 3 m minus 2 , the fraction 6 m over 5 m squared plus 17 m plus 6")[$frac(3, 5 m^(2) − 3 m − 2) , frac(6 m, 5 m^(2) + 17 m + 6)$] In the following exercises, write as equivalent rational expressions with the given LCD. #math.equation(block: false, alt: "the fraction 5 over x squared minus 2 x minus 8 , the fraction 2 x over x squared minus x minus 12")[$frac(5, x^(2) − 2 x − 8) , frac(2 x, x^(2) − x − 12)$] #linebreak() LCD #math.equation(block: false, alt: "open parenthesis x minus 4 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x plus 3 close parenthesis")[$( x − 4 ) ( x + 2 ) ( x + 3 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 x plus 15 over open parenthesis x minus 4 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x plus 3 close parenthesis ,")[$frac(5 x + 15, ( x − 4 ) ( x + 2 ) ( x + 3 )) ,$] #linebreak() #math.equation(block: true, alt: "the fraction 2 x squared plus 4 x over open parenthesis x minus 4 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x plus 3 close parenthesis")[$frac(2 x^(2) + 4 x, ( x − 4 ) ( x + 2 ) ( x + 3 ))$] ] #math.equation(block: false, alt: "the fraction 8 over y squared plus 12 y plus 35 , the fraction 3 y over y squared plus y minus 42")[$frac(8, y^(2) + 12 y + 35) , frac(3 y, y^(2) + y − 42)$] #linebreak() LCD #math.equation(block: false, alt: "open parenthesis y plus 7 close parenthesis open parenthesis y plus 5 close parenthesis open parenthesis y minus 6 close parenthesis")[$( y + 7 ) ( y + 5 ) ( y − 6 )$] #math.equation(block: false, alt: "the fraction 9 over z squared plus 2 z minus 8 , the fraction 4 z over z squared minus 4")[$frac(9, z^(2) + 2 z − 8) , frac(4 z, z^(2) − 4)$] #linebreak() LCD #math.equation(block: false, alt: "open parenthesis z minus 2 close parenthesis open parenthesis z plus 4 close parenthesis open parenthesis z plus 2 close parenthesis")[$( z − 2 ) ( z + 4 ) ( z + 2 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 9 z plus 18 over open parenthesis z minus 2 close parenthesis open parenthesis z plus 4 close parenthesis open parenthesis z plus 2 close parenthesis ,")[$frac(9 z + 18, ( z − 2 ) ( z + 4 ) ( z + 2 )) ,$] #linebreak() #math.equation(block: true, alt: "the fraction 4 z squared plus 16 z over open parenthesis z minus 2 close parenthesis open parenthesis z plus 4 close parenthesis open parenthesis z plus 2 close parenthesis")[$frac(4 z^(2) + 16 z, ( z − 2 ) ( z + 4 ) ( z + 2 ))$] ] #math.equation(block: false, alt: "the fraction 6 over a squared plus 14 a plus 45 , the fraction 5 a over a squared minus 81")[$frac(6, a^(2) + 14 a + 45) , frac(5 a, a^(2) − 81)$] #linebreak() LCD #math.equation(block: false, alt: "open parenthesis a plus 9 close parenthesis open parenthesis a plus 5 close parenthesis open parenthesis a minus 9 close parenthesis")[$( a + 9 ) ( a + 5 ) ( a − 9 )$] #math.equation(block: false, alt: "the fraction 4 over b squared plus 6 b plus 9 , the fraction 2 b over b squared minus 2 b minus 15")[$frac(4, b^(2) + 6 b + 9) , frac(2 b, b^(2) − 2 b − 15)$] #linebreak() LCD #math.equation(block: false, alt: "open parenthesis b plus 3 close parenthesis open parenthesis b plus 3 close parenthesis open parenthesis b minus 5 close parenthesis")[$( b + 3 ) ( b + 3 ) ( b − 5 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 b minus 20 over open parenthesis b plus 3 close parenthesis open parenthesis b plus 3 close parenthesis open parenthesis b minus 5 close parenthesis ,")[$frac(4 b − 20, ( b + 3 ) ( b + 3 ) ( b − 5 )) ,$] #linebreak() #math.equation(block: true, alt: "the fraction 2 b squared plus 6 b over open parenthesis b plus 3 close parenthesis open parenthesis b plus 3 close parenthesis open parenthesis b minus 5 close parenthesis")[$frac(2 b^(2) + 6 b, ( b + 3 ) ( b + 3 ) ( b − 5 ))$] ] #math.equation(block: false, alt: "the fraction 5 over c squared minus 4 c plus 4 , the fraction 3 c over c squared minus 10 c plus 16")[$frac(5, c^(2) − 4 c + 4) , frac(3 c, c^(2) − 10 c + 16)$] #linebreak() LCD #math.equation(block: false, alt: "open parenthesis c minus 2 close parenthesis open parenthesis c minus 2 close parenthesis open parenthesis c minus 8 close parenthesis")[$( c − 2 ) ( c − 2 ) ( c − 8 )$] #math.equation(block: false, alt: "the fraction 2 over 3 d squared plus 14 d minus 5 , the fraction 5 d over 3 d squared minus 19 d plus 6")[$frac(2, 3 d^(2) + 14 d − 5) , frac(5 d, 3 d^(2) − 19 d + 6)$] #linebreak() LCD #math.equation(block: false, alt: "open parenthesis 3 d minus 1 close parenthesis open parenthesis d plus 5 close parenthesis open parenthesis d minus 6 close parenthesis")[$( 3 d − 1 ) ( d + 5 ) ( d − 6 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 d minus 12 over open parenthesis 3 d minus 1 close parenthesis open parenthesis d plus 5 close parenthesis open parenthesis d minus 6 close parenthesis ,")[$frac(2 d − 12, ( 3 d − 1 ) ( d + 5 ) ( d − 6 )) ,$] #linebreak() #math.equation(block: true, alt: "the fraction 5 d squared plus 25 d over open parenthesis 3 d minus 1 close parenthesis open parenthesis d plus 5 close parenthesis open parenthesis d minus 6 close parenthesis")[$frac(5 d^(2) + 25 d, ( 3 d − 1 ) ( d + 5 ) ( d − 6 ))$] ] #math.equation(block: false, alt: "the fraction 3 over 5 m squared minus 3 m minus 2 , the fraction 6 m over 5 m squared plus 17 m plus 6")[$frac(3, 5 m^(2) − 3 m − 2) , frac(6 m, 5 m^(2) + 17 m + 6)$] #linebreak() LCD #math.equation(block: false, alt: "open parenthesis 5 m plus 2 close parenthesis open parenthesis m minus 1 close parenthesis open parenthesis m plus 3 close parenthesis")[$( 5 m + 2 ) ( m − 1 ) ( m + 3 )$] In the following exercises, add. #math.equation(block: true, alt: "the fraction 5 over 24 plus the fraction 11 over 36")[$frac(5, 24) + frac(11, 36)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 37 over 72")[$frac(37, 72)$] ] #math.equation(block: true, alt: "the fraction 7 over 30 plus the fraction 13 over 45")[$frac(7, 30) + frac(13, 45)$] #math.equation(block: true, alt: "the fraction 9 over 20 plus the fraction 11 over 30")[$frac(9, 20) + frac(11, 30)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 49 over 60")[$frac(49, 60)$] ] #math.equation(block: true, alt: "the fraction 8 over 27 plus the fraction 7 over 18")[$frac(8, 27) + frac(7, 18)$] #math.equation(block: true, alt: "the fraction 7 over 10 x squared y plus the fraction 4 over 15 x y squared")[$frac(7, 10 x^(2) y) + frac(4, 15 x y^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 21 y plus 8 x over 30 x squared y squared")[$frac(21 y + 8 x, 30 x^(2) y^(2))$] ] #math.equation(block: true, alt: "the fraction 1 over 12 a cubed b squared plus the fraction 5 over 9 a squared b cubed")[$frac(1, 12 a^(3) b^(2)) + frac(5, 9 a^(2) b^(3))$] #math.equation(block: true, alt: "the fraction 1 over 2 m plus the fraction 7 over 8 m squared n")[$frac(1, 2 m) + frac(7, 8 m^(2) n)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 m n plus 7 over 8 m squared n")[$frac(4 m n + 7, 8 m^(2) n)$] ] #math.equation(block: true, alt: "the fraction 5 over 6 p squared q plus the fraction 1 over 4 p")[$frac(5, 6 p^(2) q) + frac(1, 4 p)$] #math.equation(block: true, alt: "the fraction 3 over r plus 4 plus the fraction 2 over r minus 5")[$frac(3, r + 4) + frac(2, r − 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 r minus 7 over open parenthesis r plus 4 close parenthesis open parenthesis r minus 5 close parenthesis")[$frac(5 r − 7, ( r + 4 ) ( r − 5 ))$] ] #math.equation(block: true, alt: "the fraction 4 over s minus 7 plus the fraction 5 over s plus 3")[$frac(4, s − 7) + frac(5, s + 3)$] #math.equation(block: true, alt: "the fraction 8 over t plus 5 plus the fraction 6 over t minus 5")[$frac(8, t + 5) + frac(6, t − 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 14 t minus 10 over open parenthesis t plus 5 close parenthesis open parenthesis t minus 5 close parenthesis")[$frac(14 t − 10, ( t + 5 ) ( t − 5 ))$] ] #math.equation(block: true, alt: "the fraction 7 over v plus 5 plus the fraction 9 over v minus 5")[$frac(7, v + 5) + frac(9, v − 5)$] #math.equation(block: true, alt: "the fraction 5 over 3 w minus 2 plus the fraction 2 over w plus 1")[$frac(5, 3 w − 2) + frac(2, w + 1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 11 w plus 1 over open parenthesis 3 w minus 2 close parenthesis open parenthesis w plus 1 close parenthesis")[$frac(11 w + 1, ( 3 w − 2 ) ( w + 1 ))$] ] #math.equation(block: true, alt: "the fraction 4 over 2 x plus 5 plus the fraction 2 over x minus 1")[$frac(4, 2 x + 5) + frac(2, x − 1)$] #math.equation(block: true, alt: "the fraction 2 y over y plus 3 plus the fraction 3 over y minus 1")[$frac(2 y, y + 3) + frac(3, y − 1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 y squared plus y plus 9 over open parenthesis y plus 3 close parenthesis open parenthesis y minus 1 close parenthesis")[$frac(2 y^(2) + y + 9, ( y + 3 ) ( y − 1 ))$] ] #math.equation(block: true, alt: "the fraction 3 z over z minus 2 plus the fraction 1 over z plus 5")[$frac(3 z, z − 2) + frac(1, z + 5)$] #math.equation(block: true, alt: "the fraction 5 b over a squared b minus 2 a squared plus the fraction 2 b over b squared minus 4")[$frac(5 b, a^(2) b − 2 a^(2)) + frac(2 b, b^(2) − 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction b open parenthesis 5 b plus 10 plus 2 a squared close parenthesis over a squared open parenthesis b minus 2 close parenthesis open parenthesis b plus 2 close parenthesis")[$frac(b ( 5 b + 10 + 2 a^(2) ), a^(2) ( b − 2 ) ( b + 2 ))$] ] #math.equation(block: true, alt: "the fraction 4 over c d plus 3 c plus the fraction 1 over d squared minus 9")[$frac(4, c d + 3 c) + frac(1, d^(2) − 9)$] #math.equation(block: true, alt: "the fraction 2 m over 3 m minus 3 plus the fraction 5 m over m squared plus 3 m minus 4")[$frac(2 m, 3 m − 3) + frac(5 m, m^(2) + 3 m − 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 m squared plus 23 m over 3 open parenthesis m minus 1 close parenthesis open parenthesis m plus 4 close parenthesis")[$frac(2 m^(2) + 23 m, 3 ( m − 1 ) ( m + 4 ))$] ] #math.equation(block: true, alt: "the fraction 3 over 4 n plus 4 plus the fraction 6 over n squared minus n minus 2")[$frac(3, 4 n + 4) + frac(6, n^(2) − n − 2)$] #math.equation(block: true, alt: "the fraction 3 over n squared plus 3 n minus 18 plus the fraction 4 n over n squared plus 8 n plus 12")[$frac(3, n^(2) + 3 n − 18) + frac(4 n, n^(2) + 8 n + 12)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 n squared minus 9 n plus 6 over open parenthesis n minus 3 close parenthesis open parenthesis n plus 6 close parenthesis open parenthesis n plus 2 close parenthesis")[$frac(4 n^(2) − 9 n + 6, ( n − 3 ) ( n + 6 ) ( n + 2 ))$] ] #math.equation(block: true, alt: "the fraction 6 over q squared minus 3 q minus 10 plus the fraction 5 q over q squared minus 8 q plus 15")[$frac(6, q^(2) − 3 q − 10) + frac(5 q, q^(2) − 8 q + 15)$] #math.equation(block: true, alt: "the fraction 3 r over r squared plus 7 r plus 6 plus the fraction 9 over r squared plus 4 r plus 3")[$frac(3 r, r^(2) + 7 r + 6) + frac(9, r^(2) + 4 r + 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 open parenthesis r squared plus 6 r plus 18 close parenthesis over open parenthesis r plus 1 close parenthesis open parenthesis r plus 6 close parenthesis open parenthesis r plus 3 close parenthesis")[$frac(3 ( r^(2) + 6 r + 18 ), ( r + 1 ) ( r + 6 ) ( r + 3 ))$] ] #math.equation(block: true, alt: "the fraction 2 s over s squared plus 2 s minus 8 plus the fraction 4 over s squared plus 3 s minus 10")[$frac(2 s, s^(2) + 2 s − 8) + frac(4, s^(2) + 3 s − 10)$] In the following exercises, subtract. #math.equation(block: true, alt: "the fraction t over t minus 6 minus the fraction t minus 2 over t plus 6")[$frac(t, t − 6) − frac(t − 2, t + 6)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 open parenthesis 7 t minus 6 close parenthesis over open parenthesis t minus 6 close parenthesis open parenthesis t plus 6 close parenthesis")[$frac(2 ( 7 t − 6 ), ( t − 6 ) ( t + 6 ))$] ] #math.equation(block: true, alt: "the fraction v over v minus 3 minus the fraction v minus 6 over v plus 1")[$frac(v, v − 3) − frac(v − 6, v + 1)$] #math.equation(block: true, alt: "the fraction w plus 2 over w plus 4 minus the fraction w over w minus 2")[$frac(w + 2, w + 4) − frac(w, w − 2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction −4 open parenthesis 1 plus w close parenthesis over open parenthesis w plus 4 close parenthesis open parenthesis w minus 2 close parenthesis")[$frac(−4 ( 1 + w ), ( w + 4 ) ( w − 2 ))$] ] #math.equation(block: true, alt: "the fraction x minus 3 over x plus 6 minus the fraction x over x plus 3")[$frac(x − 3, x + 6) − frac(x, x + 3)$] #math.equation(block: true, alt: "the fraction y minus 4 over y plus 1 minus the fraction 1 over y plus 7")[$frac(y − 4, y + 1) − frac(1, y + 7)$] #solutionbox[ #math.equation(block: true, alt: "the fraction y squared plus 2 y minus 29 over open parenthesis y plus 1 close parenthesis open parenthesis y plus 7 close parenthesis")[$frac(y^(2) + 2 y − 29, ( y + 1 ) ( y + 7 ))$] ] #math.equation(block: true, alt: "the fraction z plus 8 over z minus 3 minus the fraction z over z minus 2")[$frac(z + 8, z − 3) − frac(z, z − 2)$] #math.equation(block: true, alt: "the fraction 5 a over a plus 3 minus the fraction a plus 2 over a plus 6")[$frac(5 a, a + 3) − frac(a + 2, a + 6)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 a squared plus 25 a minus 6 over open parenthesis a plus 3 close parenthesis open parenthesis a plus 6 close parenthesis")[$frac(4 a^(2) + 25 a − 6, ( a + 3 ) ( a + 6 ))$] ] #math.equation(block: true, alt: "the fraction 3 b over b minus 2 minus the fraction b minus 6 over b minus 8")[$frac(3 b, b − 2) − frac(b − 6, b − 8)$] #math.equation(block: true, alt: "the fraction 6 c over c squared minus 25 minus the fraction 3 over c plus 5")[$frac(6 c, c^(2) − 25) − frac(3, c + 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over c minus 5")[$frac(3, c − 5)$] ] #math.equation(block: true, alt: "the fraction 4 d over d squared minus 81 minus the fraction 2 over d plus 9")[$frac(4 d, d^(2) − 81) − frac(2, d + 9)$] #math.equation(block: true, alt: "the fraction 6 over m plus 6 minus the fraction 12 m over m squared minus 36")[$frac(6, m + 6) − frac(12 m, m^(2) − 36)$] #solutionbox[ #math.equation(block: true, alt: "the fraction −6 over m minus 6")[$frac(−6, m − 6)$] ] #math.equation(block: true, alt: "the fraction 4 over n plus 4 minus the fraction 8 n over n squared minus 16")[$frac(4, n + 4) − frac(8 n, n^(2) − 16)$] #math.equation(block: true, alt: "the fraction −9 p minus 17 over p squared minus 4 p minus 21 minus the fraction p plus 1 over 7 minus p")[$frac(−9 p − 17, p^(2) − 4 p − 21) − frac(p + 1, 7 − p)$] #solutionbox[ #math.equation(block: true, alt: "the fraction p plus 2 over p plus 3")[$frac(p + 2, p + 3)$] ] #math.equation(block: true, alt: "the fraction minus 13 q minus 8 over q squared plus 2 q minus 24 minus the fraction q plus 2 over 4 minus q")[$frac(− 13 q − 8, q^(2) + 2 q − 24) − frac(q + 2, 4 − q)$] #math.equation(block: true, alt: "the fraction −2 r minus 16 over r squared plus 6 r minus 16 minus the fraction 5 over 2 minus r")[$frac(−2 r − 16, r^(2) + 6 r − 16) − frac(5, 2 − r)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over r minus 2")[$frac(3, r − 2)$] ] #math.equation(block: true, alt: "the fraction 2 t minus 30 over t squared plus 6 t minus 27 minus the fraction 2 over 3 minus t")[$frac(2 t − 30, t^(2) + 6 t − 27) − frac(2, 3 − t)$] #math.equation(block: true, alt: "the fraction 5 v minus 2 over v plus 3 minus 4")[$frac(5 v − 2, v + 3) − 4$] #solutionbox[ #math.equation(block: true, alt: "the fraction v minus 14 over v plus 3")[$frac(v − 14, v + 3)$] ] #math.equation(block: true, alt: "the fraction 6 w plus 5 over w minus 1 plus 2")[$frac(6 w + 5, w − 1) + 2$] #math.equation(block: true, alt: "the fraction 2 x plus 7 over 10 x minus 1 plus 3")[$frac(2 x + 7, 10 x − 1) + 3$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 open parenthesis 8 x plus 1 close parenthesis over 10 x minus 1")[$frac(4 ( 8 x + 1 ), 10 x − 1)$] ] #math.equation(block: true, alt: "the fraction 8 y minus 4 over 5 y plus 2 minus 6")[$frac(8 y − 4, 5 y + 2) − 6$] In the following exercises, add and subtract. #math.equation(block: true, alt: "the fraction 5 a over a minus 2 plus the fraction 9 over a minus the fraction 2 a plus 18 over a squared minus 2 a")[$frac(5 a, a − 2) + frac(9, a) − frac(2 a + 18, a^(2) − 2 a)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 a squared plus 7 a minus 36 over a open parenthesis a minus 2 close parenthesis")[$frac(5 a^(2) + 7 a − 36, a ( a − 2 ))$] ] #math.equation(block: true, alt: "the fraction 2 b over b minus 5 plus the fraction 3 over 2 b minus the fraction 2 b minus 15 over 2 b squared minus 10 b")[$frac(2 b, b − 5) + frac(3, 2 b) − frac(2 b − 15, 2 b^(2) − 10 b)$] #math.equation(block: true, alt: "the fraction c over c plus 2 plus the fraction 5 over c minus 2 minus the fraction 10 c over c squared minus 4")[$frac(c, c + 2) + frac(5, c − 2) − frac(10 c, c^(2) − 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction c minus 5 over c plus 2")[$frac(c − 5, c + 2)$] ] #math.equation(block: true, alt: "the fraction 6 d over d minus 5 plus the fraction 1 over d plus 4 minus the fraction 7 d minus 5 over d squared minus d minus 20")[$frac(6 d, d − 5) + frac(1, d + 4) − frac(7 d − 5, d^(2) − d − 20)$] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction 6 a over 3 a b plus b squared plus the fraction 3 a over 9 a squared minus b squared")[$frac(6 a, 3 a b + b^(2)) + frac(3 a, 9 a^(2) − b^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 a open parenthesis 6 a minus b close parenthesis over b open parenthesis 3 a plus b close parenthesis open parenthesis 3 a minus b close parenthesis")[$frac(3 a ( 6 a − b ), b ( 3 a + b ) ( 3 a − b ))$] ] #math.equation(block: true, alt: "the fraction 2 c over 2 c plus 10 plus the fraction 7 c over c squared plus 9 c plus 20")[$frac(2 c, 2 c + 10) + frac(7 c, c^(2) + 9 c + 20)$] #math.equation(block: true, alt: "the fraction 6 d over d squared minus 64 minus the fraction 3 over d minus 8")[$frac(6 d, d^(2) − 64) − frac(3, d − 8)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over d plus 8")[$frac(3, d + 8)$] ] #math.equation(block: true, alt: "the fraction 5 over n plus 7 minus the fraction 10 n over n squared minus 49")[$frac(5, n + 7) − frac(10 n, n^(2) − 49)$] #math.equation(block: true, alt: "the fraction 4 m over m squared plus 6 m minus 7 plus the fraction 2 over m squared plus 10 m plus 21")[$frac(4 m, m^(2) + 6 m − 7) + frac(2, m^(2) + 10 m + 21)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 open parenthesis 2 m squared plus 7 m minus 1 close parenthesis over open parenthesis m plus 7 close parenthesis open parenthesis m minus 1 close parenthesis open parenthesis m plus 3 close parenthesis")[$frac(2 ( 2 m^(2) + 7 m − 1 ), ( m + 7 ) ( m − 1 ) ( m + 3 ))$] ] #math.equation(block: true, alt: "the fraction 3 p over p squared plus 4 p minus 12 plus the fraction 1 over p squared plus p minus 30")[$frac(3 p, p^(2) + 4 p − 12) + frac(1, p^(2) + p − 30)$] #math.equation(block: true, alt: "the fraction −5 n minus 5 over n squared plus n minus 6 plus the fraction n plus 1 over 2 minus n")[$frac(−5 n − 5, n^(2) + n − 6) + frac(n + 1, 2 − n)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction n plus 1 n plus 8 over n plus 3 n minus 2")[$− frac(n + 1 n + 8, n + 3 n − 2)$] ] #math.equation(block: true, alt: "the fraction −4 b minus 24 over b squared plus b minus 30 plus the fraction b plus 7 over 5 minus b")[$frac(−4 b − 24, b^(2) + b − 30) + frac(b + 7, 5 − b)$] #math.equation(block: true, alt: "the fraction 7 over 15 p plus the fraction 5 over 18 p q")[$frac(7, 15 p) + frac(5, 18 p q)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 42 q plus 25 over 90 p q")[$frac(42 q + 25, 90 p q)$] ] #math.equation(block: true, alt: "the fraction 3 over 20 a squared plus the fraction 11 over 12 a b squared")[$frac(3, 20 a^(2)) + frac(11, 12 a b^(2))$] #math.equation(block: true, alt: "the fraction 4 over x minus 2 plus the fraction 3 over x plus 5")[$frac(4, x − 2) + frac(3, x + 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 open parenthesis x plus 2 close parenthesis over open parenthesis x minus 2 close parenthesis open parenthesis x plus 5 close parenthesis")[$frac(7 ( x + 2 ), ( x − 2 ) ( x + 5 ))$] ] #math.equation(block: true, alt: "the fraction 6 over m plus 4 plus the fraction 9 over m minus 8")[$frac(6, m + 4) + frac(9, m − 8)$] #math.equation(block: true, alt: "the fraction 2 q plus 7 over q plus 4 minus 2")[$frac(2 q + 7, q + 4) − 2$] #solutionbox[ #math.equation(block: true, alt: "the fraction minus 1 over q plus 4")[$frac(− 1, q + 4)$] ] #math.equation(block: true, alt: "the fraction 3 y minus 1 over y plus 4 minus 2")[$frac(3 y − 1, y + 4) − 2$] #math.equation(block: true, alt: "the fraction z plus 2 over z minus 5 minus the fraction z over z plus 1")[$frac(z + 2, z − 5) − frac(z, z + 1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 4 z plus 1 over open parenthesis z minus 5 close parenthesis open parenthesis z plus 1 close parenthesis")[$frac(2 4 z + 1, ( z − 5 ) ( z + 1 ))$] ] #math.equation(block: true, alt: "the fraction t over t minus 5 minus the fraction t minus 1 over t plus 5")[$frac(t, t − 5) − frac(t − 1, t + 5)$] #math.equation(block: true, alt: "the fraction 3 d over d plus 2 plus the fraction 4 over d minus the fraction d plus 8 over d squared plus 2 d")[$frac(3 d, d + 2) + frac(4, d) − frac(d + 8, d^(2) + 2 d)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 open parenthesis d plus 1 close parenthesis over d plus 2")[$frac(3 ( d + 1 ), d + 2)$] ] #math.equation(block: true, alt: "the fraction 2 q over q plus 5 plus the fraction 3 over q minus 3 minus the fraction 13 q plus 15 over q squared plus 2 q minus 15")[$frac(2 q, q + 5) + frac(3, q − 3) − frac(13 q + 15, q^(2) + 2 q − 15)$] ==== Everyday Math #strong[Decorating cupcakes] Victoria can decorate an order of cupcakes for a wedding in #math.equation(block: false, alt: "t")[$t$] hours, so in 1 hour she can decorate #math.equation(block: false, alt: "the fraction 1 over t")[$frac(1, t)$] of the cupcakes. It would take her sister 3 hours longer to decorate the same order of cupcakes, so in 1 hour she can decorate #math.equation(block: false, alt: "the fraction 1 over t plus 3")[$frac(1, t + 3)$] of the cupcakes. + ⓐ Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions #math.equation(block: false, alt: "the fraction 1 over t plus the fraction 1 over t plus 3 .")[$frac(1, t) + frac(1, t + 3) .$] + ⓑ Evaluate your answer to part (a) when #math.equation(block: false, alt: "t equals 5 .")[$t = 5 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 2 t plus 3 over t open parenthesis t plus 3 close parenthesis")[$frac(2 t + 3, t ( t + 3 ))$] ⓑ #math.equation(block: false, alt: "the fraction 13 over 40")[$frac(13, 40)$] ] #strong[Kayaking] When Trina kayaks upriver, it takes her #math.equation(block: false, alt: "the fraction 5 over 3 minus c")[$frac(5, 3 − c)$] hours to go 5 miles, where #math.equation(block: false, alt: "c")[$c$] is the speed of the river current. It takes her #math.equation(block: false, alt: "the fraction 5 over 3 plus c")[$frac(5, 3 + c)$] hours to kayak 5 miles down the river. + ⓐ Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding #math.equation(block: false, alt: "the fraction 5 over 3 minus c plus the fraction 5 over 3 plus c .")[$frac(5, 3 − c) + frac(5, 3 + c) .$] + ⓑ Evaluate your answer to part (a) when #math.equation(block: false, alt: "c equals 1")[$c = 1$] to find the number of hours it would take Trina if the speed of the river current is 1 mile per hour. ==== Writing Exercises Felipe thinks #math.equation(block: false, alt: "the fraction 1 over x plus the fraction 1 over y")[$frac(1, x) + frac(1, y)$] is #math.equation(block: false, alt: "the fraction 2 over x plus y .")[$frac(2, x + y) .$] + ⓐ Choose numerical values for #emph[x] and #emph[y] and evaluate #math.equation(block: false, alt: "the fraction 1 over x plus the fraction 1 over y .")[$frac(1, x) + frac(1, y) .$] + ⓑ Evaluate #math.equation(block: false, alt: "the fraction 2 over x plus y")[$frac(2, x + y)$] for the same values of #emph[x] and #emph[y] you used in part (a). + ⓒ Explain why Felipe is wrong. + ⓓ Find the correct expression for #math.equation(block: false, alt: "the fraction 1 over x plus the fraction 1 over y .")[$frac(1, x) + frac(1, y) .$] #solutionbox[ Answers may vary. ] Simplify the expression #math.equation(block: false, alt: "the fraction 4 over n squared plus 6 n plus 9 minus the fraction 1 over n squared minus 9")[$frac(4, n^(2) + 6 n + 9) − frac(1, n^(2) − 9)$] and explain all your steps. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “find the least common denominator of rational expressions,” “find equivalent rational expressions,” “add rational expressions with different denominators,” and “subtract rational expressions with different denominators.” The rest of the cells are blank.], alt: "This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “find the least common denominator of rational expressions,” “find equivalent rational expressions,” “add rational expressions with different denominators,” and “subtract rational expressions with different denominators.” The rest of the cells are blank.", caption: none) ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?