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# 7.5E: Exercises

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## Solve Rational Equations

In the following exercises, solve each rational equation.

1. $$\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}$$

$$a=10$$

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

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

$$v=\dfrac{40}{21}$$

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

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

$$m=-2,\; m=4$$

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

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

$$p=-5, \; p=-4$$

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

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

$$v=14$$

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

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

$$x=-\dfrac{4}{5}$$

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

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

$$z=-145$$

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

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

$$q=-18, \; q=-1$$

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

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

no solution

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

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

no solution

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

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

$$b=-8$$

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

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

$$d=2$$

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

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

$$m=1$$

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

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

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

$$s=\dfrac{5}{4}$$

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

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

$$x=-\dfrac{4}{3}$$

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

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

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, $$f(x)=\dfrac{x-2}{x^{2}+6 x+8}$$:

1. Find the domain of the function
2. Solve $$f(x)=5$$
3. Find the points on the graph at this function value
1. The domain is all real numbers except $$x \neq-2$$ and $$x \neq-4$$
2. $$x=-3, x=-\dfrac{14}{5}$$
3. $$(-3,5),\left(-\dfrac{14}{5}, 5\right)$$

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

1. Find the domain of the function
2. Solve $$f(x)=1$$
3. Find the points on the graph at this function value

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

1. Find the domain of the function
2. Solve $$f(x)=2$$
3. Find the points on the graph at this function value
1. The domain is all real numbers except $$x \neq 2$$ and $$x \neq 5$$
2. $$x=\dfrac{9}{2}$$
3. $$\left(\dfrac{9}{2}, 2\right)$$

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

1. Find the domain of the function
2. Solve $$f(x)=3$$
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. $$\dfrac{c}{r}=2 \pi \text { for } r$$

$$r=\dfrac{C}{2 \pi}$$

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

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

$$w=2 v+7$$

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

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

$$c=\dfrac{b+3+2 a}{a}$$

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

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

$$p=\dfrac{q}{4 q-2}$$

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

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

$$w=\dfrac{15 v}{10+v}$$

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

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

$$n=\dfrac{5 m+23}{4}$$

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

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

$$c=\dfrac{E}{m^{2}}$$

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

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

$$y=\dfrac{20 x}{12-x}$$

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.

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.