
# 7.3E: Exercises


## Practice Makes Perfect

Add and Subtract Rational Expressions with a Common Denominator

1. $$\dfrac{2}{15}+\dfrac{7}{15}$$

$$\dfrac{3}{5}$$

2. $$\dfrac{7}{24}+\dfrac{11}{24}$$

3. $$\dfrac{3c}{4c−5}+\dfrac{5}{4c−5}$$

$$\dfrac{3c+5}{4c−5}$$

4. $$\dfrac{7m}{2m+n}+\dfrac{4}{2m+n}$$

5. $$\dfrac{2r^2}{2r−1}+\dfrac{15r−8}{2r−1}$$

$$r+8$$

6. $$\dfrac{3s^2}{3s−2}+\dfrac{13s−10}{3s−2}$$

7. $$\dfrac{2w^2}{w^2−16}+\dfrac{8w}{w^2−16}$$

$$\dfrac{2w}{w−4}$$

8. $$\dfrac{7x^2}{x^2−9}+\dfrac{21x}{x^2−9}$$

In the following exercises, subtract.

9. $$\dfrac{9a^2}{3a−7}−\dfrac{49}{3a−7}$$

$$3a+7$$

10. $$\dfrac{25b^2}{5b−6}−\dfrac{36}{5b−6}$$

11. $$\dfrac{3m^2}{6m−30}−\dfrac{21m−30}{6m−30}$$

$$\dfrac{m−2}{2}$$

12. $$\dfrac{2n^2}{4n−32}−\dfrac{18n−16}{4n−32}$$

13. $$\dfrac{6p^2+3p+4}{p^2+4p−5}−\dfrac{5p^2+p+7}{p^2+4p−5}$$

$$\dfrac{p+3}{p+5}$$

14. $$\dfrac{5q^2+3q−9}{q^2+6q+8}−\dfrac{4q^2+9q+7}{q^2+6q+8}$$

15. $$\dfrac{5r^2+7r−33}{r^2−49}−\dfrac{4r^2+5r+30}{r^2−49}$$

$$\dfrac{r+9}{r+7}$$

16. $$\dfrac{7t^2−t−4}{t^2−25}−\dfrac{6t^2+12t−44}{t^2−25}$$

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

17. $$\dfrac{10v}{2v−1}+\dfrac{2v+4}{1−2v}$$

$$4$$

18. $$\dfrac{20w}{5w−2}+\dfrac{5w+6}{2−5w}$$

19. $$\dfrac{10x^2+16x−7}{8x−3}+\dfrac{2x^2+3x−1}{3−8x}$$

$$x+2$$

20. $$\dfrac{6y^2+2y−11}{3y−7}+\dfrac{3y^2−3y+17}{7−3y}$$

21. $$\dfrac{z^2+6z}{z^2−25}−\dfrac{3z+20}{25−z^2}$$

$$\dfrac{z+4}{z−5}$$

22. $$\dfrac{a^2+3a}{a^2−9}−\dfrac{3a−27}{9−a^2}$$

23. $$\dfrac{2b^2+30b−13}{b^2−49}−\dfrac{2b^2−5b−8}{49−b^2}$$

$$\dfrac{4b−3}{b−7}$$

24. $$\dfrac{c^2+5c−10}{c^2−16}−\dfrac{c^2−8c−10}{16−c^2}$$

Find the Least Common Denominator of Rational Expressions

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

25. $$\dfrac{5}{x^2−2x−8},\dfrac{2x}{x^2−x−12}$$

a. $$(x+2)(x−4)(x+3)$$
b. $$\dfrac{5x+15}{(x+2)(x−4)(x+3)}$$,
$$\dfrac{2x^2+4x}{(x+2)(x−4)(x+3)}$$

26. $$\dfrac{8}{y^2+12y+35},\dfrac{3y}{y^2+y−42}$$

27. $$\dfrac{9}{z^2+2z−8},\dfrac{4z}{z^2−4}$$

a. $$(z−2)(z+4)(z−4)$$
b. $$\dfrac{9z−36}{(z−2)(z+4)(z−4)}$$,
$$\dfrac{4z^2−8z}{(z−2)(z+4)(z−4)}$$

28. $$\dfrac{6}{a^2+14a+45},\dfrac{5a}{a^2−81}$$

29. $$\dfrac{4}{b^2+6b+9},\dfrac{2b}{b^2−2b−15}$$

a. $$(b+3)(b+3)(b−5)$$
b. $$\dfrac{4b−20}{(b+3)(b+3)(b−5)}$$,
$$\dfrac{2b^2+6b}{(b+3)(b+3)(b−5)}$$

30. $$\dfrac{5}{c^2−4c+4},\dfrac{3c}{c^2−7c+10}$$

31. $$\dfrac{2}{3d^2+14d−5},\dfrac{5d}{3d^2−19d+6}$$

a. $$(d+5)(3d−1)(d−6)$$
b. $$\dfrac{2d−12}{(d+5)(3d−1)(d−6)}$$,
$$\dfrac{5d^2+25d}{(d+5)(3d−1)(d−6)}$$

32. $$\dfrac{3}{5m^2−3m−2},\dfrac{6m}{5m^2+17m+6}$$

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

33. $$\dfrac{7}{10x^2y}+\dfrac{4}{15xy^2}$$

$$\dfrac{21y+8x}{30x^2y^2}$$

34. $$\dfrac{1}{12a^3b^2}+\dfrac{5}{9a^2b^3}$$

35. $$\dfrac{3}{r+4}+\dfrac{2}{r−5}$$

$$\dfrac{5r−7}{(r+4)(r−5)}$$

36. $$\dfrac{4}{s−7}+\dfrac{5}{s+3}$$

37. $$\dfrac{5}{3w−2}+\dfrac{2}{w+1}$$

$$\dfrac{11w+1}{(3w−2)(w+1)}$$

38. $$\dfrac{4}{2x+5}+\dfrac{2}{x−1}$$

39. $$\dfrac{2y}{y+3}+\dfrac{3}{y−1}$$

$$\dfrac{2y^2+y+9}{(y+3)(y−1)}$$

40. $$\dfrac{3z}{z−2}+\dfrac{1}{z+5}$$

41. $$\dfrac{5b}{a^2b−2a^2}+\dfrac{2b}{b^2−4}$$

$$\dfrac{b(5b+10+2a^2)}{a^2(b−2)(b+2)}$$

42. $$\dfrac{4}{cd+3c}+\dfrac{1}{d^2−9}$$

43. $$\dfrac{−3m}{3m−3}+\dfrac{5m}{m^2+3m−4}$$

$$-\dfrac{m}{m+4}$$

44. $$\dfrac{8}{4n+4}+\dfrac{6}{n^2−n−2}$$

45. $$\dfrac{3r}{r^2+7r+6}+\dfrac{9}{r^2+4r+3}$$

$$\dfrac{3(r^2+6r+18)}{(r+1)(r+6)(r+3)}$$

46. $$\dfrac{2s}{s^2+2s−8}+\dfrac{4}{s^2+3s−10}$$

47. $$\dfrac{t}{t−6}−\dfrac{t−2}{t+6}$$

$$\dfrac{2(7t−6)}{(t−6)(t+6)}$$

48. $$\dfrac{x−3}{x+6}−\dfrac{x}{x+3}$$

49. $$\dfrac{5a}{a+3}−\dfrac{a+2}{a+6}$$

$$\dfrac{4a^2+25a−6}{(a+3)(a+6)}$$

50. $$\dfrac{3b}{b−2}−\dfrac{b−6}{b−8}$$

51. $$\dfrac{6}{m+6}−\dfrac{12m}{m^2−36}$$

$$\dfrac{−6}{m−6}$$

52. $$\dfrac{4}{n+4}−\dfrac{8n}{n^2−16}$$

53. $$\dfrac{−9p−17}{p^2−4p−21}−\dfrac{p+1}{7−p}$$

$$\dfrac{p+2}{p+3}$$

54. $$\dfrac{−13q−8}{q^2+2q−24}−\dfrac{q+2}{4−q}$$

55. $$\dfrac{−2r−16}{r^2+6r−16}−\dfrac{5}{2−r}$$

$$\dfrac{3}{r−2}$$

56. $$\dfrac{2t−30}{t^2+6t−27}−\dfrac{2}{3−t}$$

57. $$\dfrac{2x+7}{10x−1}+3$$

$$\dfrac{4(8x+1)}{10x−1}$$

58. $$\dfrac{8y−4}{5y+2}−6$$

59. $$\dfrac{3}{x^2−3x−4}−\dfrac{2}{x^2−5x+4}$$

$$\dfrac{x−5}{(x−4)(x+1)(x−1)}$$

60. $$\dfrac{4}{x^2−6x+5}−\dfrac{3}{x^2−7x+10}$$

61. $$\dfrac{5}{x^2+8x−9}−\dfrac{4}{x^2+10x+9}$$

$$\dfrac{1}{(x−1)(x+1)}$$

62. $$\dfrac{3}{2x^2+5x+2}−\dfrac{1}{2x^2+3x+1}$$

63. $$\dfrac{5a}{a−2}+\dfrac{9}{a}−\dfrac{2a+18}{a^2−2a}$$

$$\dfrac{5a^2+7a−36}{a(a−2)}$$

64. $$\dfrac{2b}{b−5}+\dfrac{3}{2b}−\dfrac{2b−15}{2b^2−10b}$$

65. $$\dfrac{c}{c+2}+\dfrac{5}{c−2}−\dfrac{10c}{c^2−4}$$

$$\dfrac{c−5}{c+2}$$

66. $$\dfrac{6d}{d−5}+\dfrac{1}{d+4}+\dfrac{7d−5}{d^2−d−20}$$

67. $$\dfrac{3d}{d+2}+\dfrac{4}{d}−\dfrac{d+8}{d^2+2d}$$

$$\dfrac{3(d+1)}{d+2}$$

68. $$\dfrac{2q}{q+5}+\dfrac{3}{q−3}−\dfrac{13q+15}{q^2+2q−15}$$

In the following exercises, find a. $$R(x)=f(x)+g(x)$$ b. $$R(x)=f(x)−g(x)$$.

69. $$f(x)=\dfrac{−5x−5}{x^2+x−6}$$ and $$g(x)=\dfrac{x+1}{2−x}$$

a. $$R(x)=−\dfrac{(x+8)(x+1)}{(x−2)(x+3)}$$
b. $$R(x)=\dfrac{x+1}{x+3}$$

70. $$f(x)=\dfrac{−4x−24}{x^2+x−30}$$ and $$g(x)=\dfrac{x+7}{5−x}$$

71. $$f(x)=\dfrac{6x}{x^2−64}$$ and $$g(x)=\dfrac{3}{x−8}$$

a. $$R(x)=\dfrac{3(3x+8)}{(x−8)(x+8)}$$
b. $$R(x)=\dfrac{3}{x+8}$$

72. $$f(x)=\dfrac{5}{x+7}$$ and $$g(x)=\dfrac{10x}{x^2−49}$$

## Writing Exercises

73. Donald thinks that $$\dfrac{3}{x}+\dfrac{4}{x}$$ is $$\dfrac{7}{2x}$$. Is Donald correct? Explain.

74. Explain how you find the Least Common Denominator of $$x^2+5x+4$$ and $$x^2−16$$.

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

d. $$\dfrac{x+y}{x}$$

76. Simplify the expression $$\dfrac{4}{n^2+6n+9}−\dfrac{1}{n^2−9}$$ and explain all your steps.

## Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. After reviewing this checklist, what will you do to become confident for all objectives?

7.3E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.