7.5E: Exercises

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Solve Rational Equations

In the following exercises, solve each rational equation.

1. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)

Answer: \(a=10\)

2. \(\dfrac{6}{3}-\dfrac{2}{d}=\dfrac{4}{9}\)

3. \(\dfrac{4}{5}+\dfrac{1}{4}=\dfrac{2}{v}\)

Answer: \(v=\dfrac{40}{21}\)

4. \(\dfrac{3}{8}+\dfrac{2}{y}=\dfrac{1}{4}\)

5. \(1-\dfrac{2}{m}=\dfrac{8}{m^{2}}\)

Answer: \(m=-2,\; m=4\)

6. \(1+\dfrac{4}{n}=\dfrac{21}{n^{2}}\)

7. \(1+\dfrac{9}{p}=\dfrac{-20}{p^{2}}\)

Answer: \(p=-5, \; p=-4\)

8. \(1-\dfrac{7}{q}=\dfrac{-6}{q^{2}}\)

9. \(\dfrac{5}{3 v-2}=\dfrac{7}{4 v}\)

Answer: \(v=14\)

10. \(\dfrac{8}{2 w+1}=\dfrac{3}{w}\)

11. \(\dfrac{3}{x+4}+\dfrac{7}{x-4}=\dfrac{8}{x^{2}-16}\)

Answer: \(x=-\dfrac{4}{5}\)

12. \(\dfrac{5}{y-9}+\dfrac{1}{y+9}=\dfrac{18}{y^{2}-81}\)

13. \(\dfrac{8}{z-10}-\dfrac{7}{z+10}=\dfrac{5}{z^{2}-100}\)

Answer: \(z=-145\)

14. \(\dfrac{9}{a+11}-\dfrac{6}{a-11}=\dfrac{6}{a^{2}-121}\)

15. \(\dfrac{-10}{q-2}-\dfrac{7}{q+4}=1\)

Answer: \(q=-18, \; q=-1\)

16. \(\dfrac{2}{s+7}-\dfrac{3}{s-3}=1\)

17. \(\dfrac{v-10}{v^{2}-5 v+4}=\dfrac{3}{v-1}-\dfrac{6}{v-4}\)

Answer: no solution

18. \(\dfrac{w+8}{w^{2}-11 w+28}=\dfrac{5}{w-7}+\dfrac{2}{w-4}\)

19. \(\dfrac{x-10}{x^{2}+8 x+12}=\dfrac{3}{x+2}+\dfrac{4}{x+6}\)

Answer: no solution

20. \(\dfrac{y-5}{y^{2}-4 y-5}=\dfrac{1}{y+1}+\dfrac{1}{y-5}\)

21. \(\dfrac{b+3}{3 b}+\dfrac{b}{24}=\dfrac{1}{b}\)

Answer: \(b=-8\)

22. \(\dfrac{c+3}{12 c}+\dfrac{c}{36}=\dfrac{1}{4 c}\)

23. \(\dfrac{d}{d+3}=\dfrac{18}{d^{2}-9}+4\)

Answer: \(d=2\)

24. \(\dfrac{m}{m+5}=\dfrac{50}{m^{2}-25}+6\)

25. \(\dfrac{n}{n+2}-3=\dfrac{8}{n^{2}-4}\)

Answer: \(m=1\)

26. \(\dfrac{p}{p+7}-8=\dfrac{98}{p^{2}-49}\)

27. \(\dfrac{q}{3 q-9}-\dfrac{3}{4 q+12}=\dfrac{7 q^{2}+6 q+63}{24 q^{2}-216}\)

Answer: no solution

28. \(\dfrac{r}{3 r-15}-\dfrac{1}{4 r+20}=\dfrac{3 r^{2}+17 r+40}{12 r^{2}-300}\)

29. \(\dfrac{s}{2 s+6}-\dfrac{2}{5 s+5}=\dfrac{5 s^{2}-3 s-7}{10 s^{2}+40 s+30}\)

Answer: \(s=\dfrac{5}{4}\)

30. \(\dfrac{t}{6 t-12}-\dfrac{5}{2 t+10}=\dfrac{t^{2}-23 t+70}{12 t^{2}+36 t-120}\)

31. \(\dfrac{2}{x^{2}+2 x-8}-\dfrac{1}{x^{2}+9 x+20}=\dfrac{4}{x^{2}+3 x-10}\)

Answer: \(x=-\dfrac{4}{3}\)

32. \(\dfrac{5}{x^{2}+4 x+3}+\dfrac{2}{x^{2}+x-6}=\dfrac{3}{x^{2}-x-2}\)

33. \(\dfrac{3}{x^{2}-5 x-6}+\dfrac{3}{x^{2}-7 x+6}=\dfrac{6}{x^{2}-1}\)

Answer: no solution

34. \(\dfrac{2}{x^{2}+2 x-3}+\dfrac{3}{x^{2}+4 x+3}=\dfrac{6}{x^{2}-1}\)

Solve Rational Equations that Involve Functions

35. For rational function, \(f(x)=\dfrac{x-2}{x^{2}+6 x+8}\):

Find the domain of the function
Solve \(f(x)=5\)
Find the points on the graph at this function value

Answer

The domain is all real numbers except \(x \neq-2\) and \(x \neq-4\)
\(x=-3, x=-\dfrac{14}{5}\)
\((-3,5),\left(-\dfrac{14}{5}, 5\right)\)

36. For rational function, \(f(x)=\dfrac{x+1}{x^{2}-2 x-3}\):

Find the domain of the function
Solve \(f(x)=1\)
Find the points on the graph at this function value

37. For rational function, \(f(x)=\dfrac{2-x}{x^{2}-7 x+10}\):

Find the domain of the function
Solve \(f(x)=2\)
Find the points on the graph at this function value

Answer

The domain is all real numbers except \(x \neq 2\) and \(x \neq 5\)
\(x=\dfrac{9}{2}\)
\(\left(\dfrac{9}{2}, 2\right)\)

38. For rational function, \(f(x)=\dfrac{5-x}{x^{2}+5 x+6}\):

Find the domain of the function
Solve \(f(x)=3\)
Find the points on the graph at this function value

Solve a Rational Equation for a Specific Variable

In the following exercises, solve:

39. \(\dfrac{c}{r}=2 \pi \text { for } r\)

Answer: \(r=\dfrac{C}{2 \pi}\)

40. \(\dfrac{I}{r}=P \text { for } r\)

41. \(\dfrac{v+3}{w-1}=\dfrac{1}{2} \text { for } w\)

Answer: \(w=2 v+7\)

42. \(\dfrac{x+5}{2-y}=\dfrac{4}{3} \text { for } y\)

43. \(a=\dfrac{b+3}{c-2} \text { for } c\)

Answer: \(c=\dfrac{b+3+2 a}{a}\)

44. \(m=\dfrac{n}{2-n} \text { for } n\)

45. \(\dfrac{1}{p}+\dfrac{2}{q}=4 \text { for } p\)

Answer: \(p=\dfrac{q}{4 q-2}\)

46. \(\dfrac{3}{s}+\dfrac{1}{t}=2 \text { for } s\)

47. \(\dfrac{2}{v}+\dfrac{1}{5}=\dfrac{3}{w} \text { for } w\)

Answer: \(w=\dfrac{15 v}{10+v}\)

48. \(\dfrac{6}{x}+\dfrac{2}{3}=\dfrac{1}{y} \text { for } y\)

49. \(\dfrac{m+3}{n-2}=\dfrac{4}{5} \text { for } n\)

Answer: \(n=\dfrac{5 m+23}{4}\)

50. \(r=\dfrac{s}{3-t} \text { for } t\)

51. \(\dfrac{E}{c}=m^{2} \text { for } c\)

Answer: \(c=\dfrac{E}{m^{2}}\)

52. \(\dfrac{R}{T}=W \text { for } T\)

53. \(\dfrac{3}{x}-\dfrac{5}{y}=\dfrac{1}{4} \text { for } y\)

Answer: \(y=\dfrac{20 x}{12-x}\)

54. \(c=\dfrac{2}{a}+\dfrac{b}{5} \text { for } a\)

Writing Exercises

55. Your class mate is having trouble in this section. Write down the steps you would use to explain how to solve a rational equation.

Answer: Answers will vary.

56. Alek thinks the equation \(\dfrac{y}{y+6}=\dfrac{72}{y^{2}-36}+4\) has two solutions, \(y=-6\) and \(y=4\). Explain why Alek is wrong.