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Mathematics LibreTexts

1.3e: Exercises - Rational Equations

  • Page ID
    38265
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    A: Rational Expression or Equation?

    Exercise \(\PageIndex{A}\)

    \( \bigstar \) Simplify or solve, whichever is appropriate

    1. \(\dfrac { 1 } { x } + \dfrac { 2 } { x - 3 } = - \dfrac { 2 } { 3 } \\[4pt] \)
    2. \(\dfrac { 1 } { x - 3 } - \dfrac { 3 } { 4 } = \dfrac { 1 } { x } \\[4pt] \)
    1. \(\dfrac { x - 2 } { 3 x - 1 } - \dfrac { 2 - x } { x } \\[4pt] \)
    2. \(\dfrac { 5 } { 2 } + \dfrac { x } { 2 x - 1 } - \dfrac { 1 } { 2 x } \\[4pt] \)
    1. \(\dfrac { x - 1 } { 3 x } + \dfrac { 2 } { x + 1 } - \dfrac { 5 } { 6 } \\[4pt] \)
    2. \(\dfrac { x - 1 } { 3 x } + \dfrac { 2 } { x + 1 } = \dfrac { 5 } { 6 } \\[4pt] \)
    1. \(\dfrac { 2 x + 1 } { 2 x - 3 } + 2 = \dfrac { 1 } { 2 x } \\[4pt] \)
    2. \(5 - \dfrac { 3 x + 1 } { 2 x } + \dfrac { 1 } { x + 1 } \\[4pt] \)
    Answers to Odd Exercises:
    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

    1. \(\dfrac { 3 } { x } + 2 = \dfrac { 1 } { 3 x } \\[4pt] \)
    2. \(5 - \dfrac { 1 } { 2 x } = - \dfrac { 1 } { x } \\[4pt] \)
    3. \(\dfrac { 7 } { x ^ { 2 } } + \dfrac { 3 } { 2 x } = \dfrac { 1 } { x ^ { 2 } } \\[4pt] \)
    4. \(\dfrac { 4 } { 3 x ^ { 2 } } + \dfrac { 1 } { 2 x } = \dfrac { 1 } { 3 x ^ { 2 } } \\[4pt] \)
    1. \(\dfrac { 1 } { 6 } + \dfrac { 2 } { 3 x } = \dfrac { 7 } { 2 x ^ { 2 } } \\[4pt] \)
    2. \(\dfrac { 1 } { 12 } - \dfrac { 1 } { 3 x } = \dfrac { 1 } { x ^ { 2 } } \\[4pt] \)
    3. \(2 + \dfrac { 3 } { x } + \dfrac { 7 } { x ( x - 3 ) } = 0 \\[4pt] \)
    4. \(\dfrac { 20 } { x } - \dfrac { x + 44 } { x ( x + 2 ) } = 3 \\[4pt] \)
    1. \(\dfrac { 2 x } { 2 x - 3 } + \dfrac { 4 } { x } = \dfrac { x - 18 } { x ( 2 x - 3 ) } \\[4pt] \)
    2. \(\dfrac { 2 x } { x - 5 } + \dfrac {1 } { x } =  \dfrac { 9x+5 } { x (x-5)} \\[4pt] \)
    3. \(\dfrac { 4 } { 4 x - 1 } - \dfrac { 1 } { x - 1 } = \dfrac { 2 } { 4 x - 1 } \\[4pt] \)
    4. \(\dfrac { 5 } { 2 x - 3 } - \dfrac { 1 } { x + 3 } = \dfrac { 2 } { 2 x - 3 } \\[4pt] \)
    Answers to Odd Exercises:
    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

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

    \( \bigstar \) Solve

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

    \( \bigstar \) Solve

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

    \( \bigstar \) Solve

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

    \( \bigstar \)

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