# 1.3e: Exercises - Rational Equations

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### A: Rational Expression or Equation?

Exercise $$\PageIndex{A}$$

$$\bigstar$$ Simplify or solve, whichever is appropriate

 $$\dfrac { 1 } { x } + \dfrac { 2 } { x - 3 } = - \dfrac { 2 } { 3 } \\[4pt]$$ $$\dfrac { 1 } { x - 3 } - \dfrac { 3 } { 4 } = \dfrac { 1 } { x } \\[4pt]$$ $$\dfrac { x - 2 } { 3 x - 1 } - \dfrac { 2 - x } { x } \\[4pt]$$ $$\dfrac { 5 } { 2 } + \dfrac { x } { 2 x - 1 } - \dfrac { 1 } { 2 x } \\[4pt]$$ $$\dfrac { x - 1 } { 3 x } + \dfrac { 2 } { x + 1 } - \dfrac { 5 } { 6 } \\[4pt]$$ $$\dfrac { x - 1 } { 3 x } + \dfrac { 2 } { x + 1 } = \dfrac { 5 } { 6 } \\[4pt]$$ $$\dfrac { 2 x + 1 } { 2 x - 3 } + 2 = \dfrac { 1 } { 2 x } \\[4pt]$$ $$5 - \dfrac { 3 x + 1 } { 2 x } + \dfrac { 1 } { x + 1 } \\[4pt]$$
 1. Solve; $$- 3 , \dfrac { 3 } { 2 }$$ 3. Simplify; $$\dfrac { ( 4 x - 1 ) ( x - 2 ) } { x ( 3 x - 1 ) }$$ 5. Simplify; $$- \dfrac { ( x - 2 ) ( 3 x - 1 ) } { 6 x ( x + 1 ) }$$ 7. Solve; $$\dfrac{1}{2}$$

### B: Solve Rational Equations

Exercise $$\PageIndex{B}$$

$$\bigstar$$ Solve

 $$\dfrac { 3 } { x } + 2 = \dfrac { 1 } { 3 x } \\[4pt]$$ $$5 - \dfrac { 1 } { 2 x } = - \dfrac { 1 } { x } \\[4pt]$$ $$\dfrac { 7 } { x ^ { 2 } } + \dfrac { 3 } { 2 x } = \dfrac { 1 } { x ^ { 2 } } \\[4pt]$$ $$\dfrac { 4 } { 3 x ^ { 2 } } + \dfrac { 1 } { 2 x } = \dfrac { 1 } { 3 x ^ { 2 } } \\[4pt]$$ $$\dfrac { 1 } { 6 } + \dfrac { 2 } { 3 x } = \dfrac { 7 } { 2 x ^ { 2 } } \\[4pt]$$ $$\dfrac { 1 } { 12 } - \dfrac { 1 } { 3 x } = \dfrac { 1 } { x ^ { 2 } } \\[4pt]$$ $$2 + \dfrac { 3 } { x } + \dfrac { 7 } { x ( x - 3 ) } = 0 \\[4pt]$$ $$\dfrac { 20 } { x } - \dfrac { x + 44 } { x ( x + 2 ) } = 3 \\[4pt]$$ $$\dfrac { 2 x } { 2 x - 3 } + \dfrac { 4 } { x } = \dfrac { x - 18 } { x ( 2 x - 3 ) } \\[4pt]$$ $$\dfrac { 2 x } { x - 5 } + \dfrac {1 } { x } = \dfrac { 9x+5 } { x (x-5)} \\[4pt]$$ $$\dfrac { 4 } { 4 x - 1 } - \dfrac { 1 } { x - 1 } = \dfrac { 2 } { 4 x - 1 } \\[4pt]$$ $$\dfrac { 5 } { 2 x - 3 } - \dfrac { 1 } { x + 3 } = \dfrac { 2 } { 2 x - 3 } \\[4pt]$$
 11. $$−\dfrac{4}{3}$$ 13. $$−4$$ 15. $$−7, 3$$ 17. $$−\dfrac{1}{2} , 2$$ 19. $$−2, −\dfrac{3}{2}$$ 21. $$−\dfrac{1}{2}$$

$$\bigstar$$ Solve

 $$\dfrac { 4 x } { x - 3 } + \dfrac { 4 } { x ^ { 2 } - 2 x - 3 } = - \dfrac { 1 } { x + 1 } \\[4pt]$$ $$\dfrac { x } { x - 2 } - \dfrac { 2 } { x + 4 } = \dfrac { 12 } { x ^ { 2 } + 2 x - 8 } \\[4pt]$$ $$\dfrac { x } { x - 8 } - \dfrac { 8 } { x - 1 } = \dfrac { 56 } { x ^ { 2 } - 9 x + 8 } \\[4pt]$$ $$\dfrac { 2 x } { x - 1 } + \dfrac { 9 } { 3 x - 1 } + \dfrac { 11 } { 3 x ^ { 2 } - 4 x + 1 } = 0 \\[4pt]$$ $$\dfrac { 3 x } { x - 2 } - \dfrac { 14 } { 2 x ^ { 2 } - x - 6 } = \dfrac { 2 } { 2 x + 3 } \\[4pt]$$ $$\dfrac { x } { x - 4 } - \dfrac { 4 } { x - 5 } = - \dfrac { 4 } { x ^ { 2 } - 9 x + 20 } \\[4pt]$$ $$\dfrac { 2 x } { 5 + x } - \dfrac { 1 } { 5 - x } = \dfrac { 2 x } { x ^ { 2 } - 25 } \\[4pt]$$ $$\dfrac { 2 x } { 2 x + 3 } - \dfrac { 1 } { 2 x - 3 } = \dfrac { 6 } { 9 - 4 x ^ { 2 } } \\[4pt]$$ $$1 + \dfrac { 1 } { x + 1 } = \dfrac { 8 } { x - 1 } - \dfrac { 16 } { x ^ { 2 } - 1 } \\[4pt]$$ $$1 - \dfrac { 1 } { 3 x + 5 } = \dfrac { 2 x } { 3 x - 5 } - \dfrac { 2 ( 6 x + 5 ) } { 9 x ^ { 2 } - 25 } \\[4pt]$$
 23. $$−\dfrac{1}{4}$$ 25. $$Ø$$ 27. $$−2, \dfrac{5}{6}$$ 29. $$\dfrac{1}{2}$$ 31. $$6$$

$$\bigstar$$ Solve

 $$2 x ^ { - 1 } = 2 x ^ { - 2 } - x ^ { - 1 } \\[4pt]$$ $$3 + x ( x + 1 ) ^ { - 1 } = 2 ( x + 1 ) ^ { - 1 } \\[4pt]$$ $$x ^ { - 2 } - 64 = 0 \\[4pt]$$ $$1 - 4 x ^ { - 2 } = 0 \\[4pt]$$ $$x - ( x + 2 ) ^ { - 1 } = - 2 \\[4pt]$$ $$2 x - 9 ( 2 x - 1 ) ^ { - 1 } = 1 \\[4pt]$$ $$2 x ^ { - 2 } + ( x - 12 ) ^ { - 1 } = 0 \\[4pt]$$ $$- 2 x ^ { - 2 } + 3 ( x + 4 ) ^ { - 1 } = 0 \\[4pt]$$
 33. $$\dfrac{2}{3}$$ 35. $$\pm \dfrac { 1 } { 8 }$$ 37. $$- 3 , - 1$$ 39. $$- 6,4$$

$$\bigstar$$ Solve

 $$\dfrac { 5 } { n } = - \dfrac { 3 } { n - 2 } \\[4pt]$$ $$\dfrac { 2 n - 1 } { 2 n } = - \dfrac { 1 } { 2 } \\[4pt]$$ $$- 3 = \dfrac { 5 n + 2 } { 3 n } \\[4pt]$$ $$\dfrac { n + 1 } { 2 n - 1 } = \dfrac { 1 } { 3 } \\[4pt]$$ $$\dfrac { x + 2 } { x - 5 } = \dfrac { x + 4 } { x - 2 } \\[4pt]$$ $$\dfrac { x + 1 } { x + 5 } = \dfrac { x - 5 } { x } \\[4pt]$$ $$\dfrac { 2 x + 1 } { 6 x - 1 } = \dfrac { x + 5 } { 3 x - 2 } \\[4pt]$$ $$\dfrac { 6 ( 2 x + 3 ) } { 4 x - 1 } = \dfrac { 3 x } { x + 2 } \\[4pt]$$ $$\dfrac { 3 ( x + 1 ) } { 1 - x } = \dfrac { x + 3 } { x + 1 } \\[4pt]$$ $$\dfrac { 8 ( x - 2 ) } { x + 1 } = \dfrac { 5 - x } { x - 2 } \\[4pt]$$ $$\dfrac { x + 3 } { x + 7 } = \dfrac { x + 3 } { 3 ( 5 - x ) } \\[4pt]$$ $$\dfrac { x + 1 } { x + 4 } = \dfrac { - 8 ( x + 4 ) } { x + 7 } \\[4pt]$$
 41. $$\dfrac{5}{4}$$ 43. $$-\dfrac{1}{7}$$ 45. $$-16$$ 47. $$\dfrac{1}{10}$$ 49. $$-2,0$$ 51. $$-3,2$$

$$\bigstar$$ Solve

 $$\dfrac { x } { x - 2 } - \dfrac { 3 } { x + 8 } = \dfrac { x + 2 } { x + 8 } + \dfrac { 5 ( x + 3 ) } { x ^ { 2 } + 6 x - 16 } \\[4pt]$$ $$\dfrac { 2 x } { x - 10 } + \dfrac { 1 } { x - 3 } = \dfrac { x + 3 } { x - 10 } + \dfrac { x ^ { 2 } - 5 x + 5 } { x ^ { 2 } - 13 x + 30 } \\[4pt]$$ $$\dfrac { 5 } { x ^ { 2 } + 9 x + 18 } + \dfrac { x + 3 } { x ^ { 2 } + 7 x + 6 } = \dfrac { 5 } { x ^ { 2 } + 4 x + 3 } \\[4pt]$$ $$\dfrac { 1 } { x ^ { 2 } + 4 x - 60 } + \dfrac { x - 6 } { x ^ { 2 } + 16 x + 60 } = \dfrac { 1 } { x ^ { 2 } - 36 } \\[4pt]$$ $$\dfrac { 4 } { x ^ { 2 } + 10 x + 21 } + \dfrac { 2 ( x + 3 ) } { x ^ { 2 } + 6 x - 7 } = \dfrac { x + 7 } { x ^ { 2 } + 2 x - 3 } \\[4pt]$$ $$\dfrac { x - 1 } { x ^ { 2 } - 11 x + 28 } + \dfrac { x - 1 } { x ^ { 2 } - 5 x + 4 } = \dfrac { x - 4 } { x ^ { 2 } - 8 x + 7 } \\[4pt]$$ $$\dfrac { 5 } { x ^ { 2 } + 5 x + 4 } + \dfrac { x + 1 } { x ^ { 2 } + 3 x - 4 } = \dfrac { 5 } { x ^ { 2 } - 1 } \\[4pt]$$ $$\dfrac { 1 } { x ^ { 2 } - 2 x - 63 } + \dfrac { x - 9 } { x ^ { 2 } + 10 x + 21 } = \dfrac { 1 } { x ^ { 2 } - 6 x - 27 } \\[4pt]$$ $$\dfrac { 4 } { x ^ { 2 } - 4 } + \dfrac { 2 ( x - 2 ) } { x ^ { 2 } - 4 x - 12 } = \dfrac { x + 2 } { x ^ { 2 } - 8 x + 12 } \\[4pt]$$ $$\dfrac { x + 2 } { x ^ { 2 } - 5 x + 4 } + \dfrac { x + 2 } { x ^ { 2 } + x - 2 } = \dfrac { x - 1 } { x ^ { 2 } - 2 x - 8 } \\[4pt]$$
 53. $$Ø$$ 55. $$−8, 2$$ 57. $$5$$ 59. $$−6, 4$$ 61. $$10$$

$$\bigstar$$

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