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7.2 Add and Subtract Rational Expressions

Matching denominators, handling denominators that are opposites of each other, and building a least common denominator from fully factored forms.

Intermediate Algebra 2e, 7.2 Add and Subtract Rational Expressions

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65 exercises

7.2 Exercise 1

Before you get started, take this readiness quiz.

Add: 710+815.\frac{7}{10} + \frac{8}{15} .
If you missed this problem, review the reference.

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7.2 Exercise 2

Subtract: 3x4−89.\frac{3 x}{4} - \frac{8}{9} .
If you missed this problem, review the reference.

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7.2 Exercise 3

Subtract: 6(2x+1)−4(x−5).6 \left(2 x + 1\right) - 4 \left(x - 5\right) .
If you missed this problem, review the reference.

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7.2 Exercise 4

Add: 11x+28x+4+x2x+4.\frac{11 x + 28}{x + 4} + \frac{x^{2}}{x + 4} .

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7.2 Exercise 5

Simplify: 9x+14x+7+x2x+7.\frac{9 x + 14}{x + 7} + \frac{x^{2}}{x + 7} .

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7.2 Exercise 6

Simplify: x2+8xx+5+15x+5.\frac{x^{2} + 8 x}{x + 5} + \frac{15}{x + 5} .

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7.2 Exercise 7

Subtract: 5x2−7x+3x2−3x-18−4x2+x−9x2−3x-18.\frac{5 x^{2} - 7 x + 3}{x^{2} - 3 x - 18} - \frac{4 x^{2} + x - 9}{x^{2} - 3 x - 18} .

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7.2 Exercise 8

Subtract: 4x2−11x+8x2−3x+2−3x2+x−3x2−3x+2.\frac{4 x^{2} - 11 x + 8}{x^{2} - 3 x + 2} - \frac{3 x^{2} + x - 3}{x^{2} - 3 x + 2} .

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7.2 Exercise 9

Subtract: 6x2−x+20x2−81−5x2+11x−7x2−81.\frac{6 x^{2} - x + 20}{x^{2} - 81} - \frac{5 x^{2} + 11 x - 7}{x^{2} - 81} .

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7.2 Exercise 11

Subtract: y2−5yy2−4−6y−64−y2.\frac{y^{2} - 5 y}{y^{2} - 4} - \frac{6 y - 6}{4 - y^{2}} .

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7.2 Exercise 12

Subtract: 2n2+8n−1n2−1−n2−7n−11−n2.\frac{2 n^{2} + 8 n - 1}{n^{2} - 1} - \frac{n^{2} - 7 n - 1}{1 - n^{2}} .

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7.2 Exercise 14

ⓐ Find the LCD for the expressions 2x2−x−12,1x2−16\frac{2}{x^{2} - x - 12} , \frac{1}{x^{2} - 16} ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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7.2 Exercise 15

ⓐ Find the LCD for the expressions 3xx2−3x–10,5x2+3x+2\frac{3 x}{x^{2} - 3 x – 10} , \frac{5}{x^{2} + 3 x + 2} ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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7.2 Exercise 17

Add: 2x−2+5x+3.\frac{2}{x - 2} + \frac{5}{x + 3} .

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7.2 Exercise 18

Add:4m+3+3m+4.\frac{4}{m + 3} + \frac{3}{m + 4} .

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7.2 Exercise 20

Add: 1m2−m−2+5mm2+3m+2.\frac{1}{m^{2} - m - 2} + \frac{5 m}{m^{2} + 3 m + 2} .

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7.2 Exercise 21

Add:2nn2−3n−10+6n2+5n+6.\frac{2 n}{n^{2} - 3 n - 10} + \frac{6}{n^{2} + 5 n + 6} .

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7.2 Exercise 23

Subtract: 2xx2−4−1x+2.\frac{2 x}{x^{2} - 4} - \frac{1}{x + 2} .

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7.2 Exercise 24

Subtract: 3z+3−6zz2−9.\frac{3}{z + 3} - \frac{6 z}{z^{2} - 9} .

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7.2 Exercise 26

Subtract :3x−1x2−5x−6−26−x.\frac{3 x - 1}{x^{2} - 5 x - 6} - \frac{2}{6 - x} .

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7.2 Exercise 27

Subtract: −2y−2y2+2y−8−y−12−y.\frac{−2 y - 2}{y^{2} + 2 y - 8} - \frac{y - 1}{2 - y} .

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7.2 Exercise 29

Subtract: 3b2−4b−5−2b2−6b+5.\frac{3}{b^{2} - 4 b - 5} - \frac{2}{b^{2} - 6 b + 5} .

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7.2 Exercise 30

Subtract: 4x2−4−3x2−x−2.\frac{4}{x^{2} - 4} - \frac{3}{x^{2} - x - 2} .

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7.2 Exercise 32

Simplify: vv+1+3v−1−6v2−1.\frac{v}{v + 1} + \frac{3}{v - 1} - \frac{6}{v^{2} - 1} .

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7.2 Exercise 33

Simplify: 3ww+2+2w+7−17w+4w2+9w+14.\frac{3 w}{w + 2} + \frac{2}{w + 7} - \frac{17 w + 4}{w^{2} + 9 w + 14} .

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7.2 Exercise 35

Find R(x)=f(x)−g(x)R \left(x\right) = f \left(x\right) - g \left(x\right) where f(x)=x+1x+3f \left(x\right) = \frac{x + 1}{x + 3} and g(x)=x+17x2−x−12.g \left(x\right) = \frac{x + 17}{x^{2} - x - 12} .

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7.2 Exercise 36

Find R(x)=f(x)+g(x)R \left(x\right) = f \left(x\right) + g \left(x\right) where f(x)=x−4x+3f \left(x\right) = \frac{x - 4}{x + 3} and g(x)=4x+6x2−9.g \left(x\right) = \frac{4 x + 6}{x^{2} - 9} .

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7.2 Exercise 37

Add.

215+715\frac{2}{15} + \frac{7}{15}

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7.2 Exercise 39

Add.

3c4c−5+54c−5\frac{3 c}{4 c - 5} + \frac{5}{4 c - 5}

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7.2 Exercise 41

Add.

2r22r−1+15r−82r−1\frac{2 r^{2}}{2 r - 1} + \frac{15 r - 8}{2 r - 1}

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7.2 Exercise 43

Add.

2w2w2−16+8ww2−16\frac{2 w^{2}}{w^{2} - 16} + \frac{8 w}{w^{2} - 16}

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7.2 Exercise 45

Subtract.

9a23a−7−493a−7\frac{9 a^{2}}{3 a - 7} - \frac{49}{3 a - 7}

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7.2 Exercise 47

Subtract.

3m26m−30−21m−306m−30\frac{3 m^{2}}{6 m - 30} - \frac{21 m - 30}{6 m - 30}

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7.2 Exercise 49

Subtract.

6p2+3p+4p2+4p−5−5p2+p+7p2+4p−5\frac{6 p^{2} + 3 p + 4}{p^{2} + 4 p - 5} - \frac{5 p^{2} + p + 7}{p^{2} + 4 p - 5}

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7.2 Exercise 51

Subtract.

5r2+7r−33r2−49−4r2+5r+30r2−49\frac{5 r^{2} + 7 r - 33}{r^{2} - 49} - \frac{4 r^{2} + 5 r + 30}{r^{2} - 49}

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7.2 Exercise 53

Add or subtract.

10v2v−1+2v+41−2v\frac{10 v}{2 v - 1} + \frac{2 v + 4}{1 - 2 v}

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7.2 Exercise 55

Add or subtract.

10x2+16x−78x−3+2x2+3x−13−8x\frac{10 x^{2} + 16 x - 7}{8 x - 3} + \frac{2 x^{2} + 3 x - 1}{3 - 8 x}

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7.2 Exercise 57

Add or subtract.

z2+6zz2−25−3z+2025−z2\frac{z^{2} + 6 z}{z^{2} - 25} - \frac{3 z + 20}{25 - z^{2}}

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7.2 Exercise 59

Add or subtract.

2b2+30b−13b2−49−2b2−5b−849−b2\frac{2 b^{2} + 30 b - 13}{b^{2} - 49} - \frac{2 b^{2} - 5 b - 8}{49 - b^{2}}

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7.2 Exercise 61

ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

5x2−2x−8,2xx2−x−12\frac{5}{x^{2} - 2 x - 8} , \frac{2 x}{x^{2} - x - 12}

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7.2 Exercise 63

ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

9z2+2z−8,4zz2−4\frac{9}{z^{2} + 2 z - 8} , \frac{4 z}{z^{2} - 4}

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7.2 Exercise 65

ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

4b2+6b+9,2bb2−2b−15\frac{4}{b^{2} + 6 b + 9} , \frac{2 b}{b^{2} - 2 b - 15}

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7.2 Exercise 67

ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

23d2+14d−5,5d3d2−19d+6\frac{2}{3 d^{2} + 14 d - 5} , \frac{5 d}{3 d^{2} - 19 d + 6}

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7.2 Exercise 69

Perform the indicated operations.

710x2y+415xy2\frac{7}{10 x^{2} y} + \frac{4}{15 x y^{2}}

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7.2 Exercise 71

Perform the indicated operations.

3r+4+2r−5\frac{3}{r + 4} + \frac{2}{r - 5}

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7.2 Exercise 73

Perform the indicated operations.

53w−2+2w+1\frac{5}{3 w - 2} + \frac{2}{w + 1}

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7.2 Exercise 75

Perform the indicated operations.

2yy+3+3y−1\frac{2 y}{y + 3} + \frac{3}{y - 1}

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7.2 Exercise 77

Perform the indicated operations.

5ba2b−2a2+2bb2−4\frac{5 b}{a^{2} b - 2 a^{2}} + \frac{2 b}{b^{2} - 4}

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7.2 Exercise 79

Perform the indicated operations.

−3m3m−3+5mm2+3m−4\frac{−3 m}{3 m - 3} + \frac{5 m}{m^{2} + 3 m - 4}

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7.2 Exercise 81

Perform the indicated operations.

3rr2+7r+6+9r2+4r+3\frac{3 r}{r^{2} + 7 r + 6} + \frac{9}{r^{2} + 4 r + 3}

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7.2 Exercise 83

Perform the indicated operations.

tt−6−t−2t+6\frac{t}{t - 6} - \frac{t - 2}{t + 6}

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7.2 Exercise 85

Perform the indicated operations.

5aa+3−a+2a+6\frac{5 a}{a + 3} - \frac{a + 2}{a + 6}

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7.2 Exercise 87

Perform the indicated operations.

6m+6−12mm2−36\frac{6}{m + 6} - \frac{12 m}{m^{2} - 36}

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7.2 Exercise 89

Perform the indicated operations.

−9p−17p2−4p−21−p+17−p\frac{−9 p - 17}{p^{2} - 4 p - 21} - \frac{p + 1}{7 - p}

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7.2 Exercise 91

Perform the indicated operations.

−2r−16r2+6r−16−52−r\frac{−2 r - 16}{r^{2} + 6 r - 16} - \frac{5}{2 - r}

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7.2 Exercise 93

Perform the indicated operations.

2x+710x−1+3\frac{2 x + 7}{10 x - 1} + 3

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7.2 Exercise 95

Perform the indicated operations.

3x2−3x−4−2x2−5x+4\frac{3}{x^{2} - 3 x - 4} - \frac{2}{x^{2} - 5 x + 4}

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7.2 Exercise 97

Perform the indicated operations.

5x2+8x−9−4x2+10x+9\frac{5}{x^{2} + 8 x - 9} - \frac{4}{x^{2} + 10 x + 9}

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7.2 Exercise 99

Perform the indicated operations.

5aa−2+9a−2a+18a2−2a\frac{5 a}{a - 2} + \frac{9}{a} - \frac{2 a + 18}{a^{2} - 2 a}

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7.2 Exercise 101

Perform the indicated operations.

cc+2+5c−2−10cc2−4\frac{c}{c + 2} + \frac{5}{c - 2} - \frac{10 c}{c^{2} - 4}

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7.2 Exercise 103

Perform the indicated operations.

3dd+2+4d−d+8d2+2d\frac{3 d}{d + 2} + \frac{4}{d} - \frac{d + 8}{d^{2} + 2 d}

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7.2 Exercise 105

Find ⓐ R(x)=f(x)+g(x)R \left(x\right) = f \left(x\right) + g \left(x\right) ⓑ R(x)=f(x)−g(x).R \left(x\right) = f \left(x\right) - g \left(x\right) .

f(x)=−5x−5x2+x−6f \left(x\right) = \frac{−5 x - 5}{x^{2} + x - 6} and
g(x)=x+12−xg \left(x\right) = \frac{x + 1}{2 - x}

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7.2 Exercise 107

Find ⓐ R(x)=f(x)+g(x)R \left(x\right) = f \left(x\right) + g \left(x\right) ⓑ R(x)=f(x)−g(x).R \left(x\right) = f \left(x\right) - g \left(x\right) .

f(x)=6xx2−64f \left(x\right) = \frac{6 x}{x^{2} - 64} and
g(x)=3x−8g \left(x\right) = \frac{3}{x - 8}

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7.2 Exercise 109

Donald thinks that 3x+4x\frac{3}{x} + \frac{4}{x} is 72x.\frac{7}{2 x} . Is Donald correct? Explain.

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7.2 Exercise 111

Felipe thinks 1x+1y\frac{1}{x} + \frac{1}{y} is 2x+y.\frac{2}{x + y} . ⓐ Choose numerical values for x and y and evaluate 1x+1y.\frac{1}{x} + \frac{1}{y} . ⓑ Evaluate 2x+y\frac{2}{x + y} for the same values of x and y you used in part ⓐ. ⓒ Explain why Felipe is wrong. ⓓ Find the correct expression for 1x+1y.\frac{1}{x} + \frac{1}{y} .

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Adapted from Intermediate Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Intermediate Algebra 2e. Imported at revision 77a933c.