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Mathematics LibreTexts

7.5E: Exercises

  • Page ID
    30945
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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, \(f(x)=\dfrac{x-2}{x^{2}+6 x+8}\):

    1. Find the domain of the function
    2. Solve \(f(x)=5\)
    3. Find the points on the graph at this function value
    Answer
    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
    Answer
    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. \(\dfrac{c}{r}=2 \pi \text { for } r\)

    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.


    This page titled 7.5E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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