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Math 20-1

Math 20-1 · Worksheets

Rational expressions

Ten questions of mixed difficulty, covering Rational expressions. Print it, or work through it on screen — the answer key starts on its own page.

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Rational expressions

Math 20-1 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. (x² − 1)/(x + 2) ÷ (x − 1)/(x + 2) equals

    1. a) x − 1
    2. b) 1
    3. c) (x² − 1)/(x − 1)
    4. d) x + 1

  2. (x² + 5x + 6)/(x² − 4) simplifies to

    1. a) (x + 3)/(x + 2), x ≠ ±2
    2. b) (x + 3)/(x − 2)
    3. c) (x + 3)/(x − 2), x ≠ ±2
    4. d) (5x + 6)/(−4)

  3. The non-permissible values of x/(x² − 4) are

    1. a) x = 0
    2. b) x = 2 only
    3. c) x = 4
    4. d) x = 2 and x = −2

  4. The solution to 3/(x − 2) = x is

    1. a) x = 3 only
    2. b) no solution
    3. c) x = 3 or x = −1
    4. d) x = 2 or x = −1

  5. The non-permissible value of (x + 2)/(x − 5) is

    1. a) x = −5
    2. b) x = −2
    3. c) x = 5
    4. d) x = 2

  6. 3/x + 4/x equals

    1. a) 7/(2x)
    2. b) 12/x²
    3. c) 7/x²
    4. d) 7/x

  7. Amira can paint a room in 3 hours and Ben can paint the same room in 6 hours. Working together, they take

    1. a) 2 hours
    2. b) 9 hours
    3. c) 4.5 hours
    4. d) 1.5 hours

  8. (x/3) · (6/x²) equals

    1. a) 2/x²
    2. b) 2x
    3. c) 6x/(3x²)
    4. d) 2/x, x ≠ 0

  9. (x² − 9)/(x + 3) simplifies to

    1. a) x + 3, x ≠ 3
    2. b) x − 9, x ≠ −3
    3. c) x − 3, x ≠ −3
    4. d) x − 3

  10. 1/x + 1/(x + 1) equals

    1. a) 2/(2x + 1)
    2. b) (2x + 1)/(x + 1)
    3. c) (2x + 1)/[x(x + 1)], x ≠ 0, −1
    4. d) 1/(2x + 1)

Answer key · Rational expressions

Math 20-1 · maddyhelps.com

  1. d) x + 1 — Dividing is multiplying by the reciprocal: [(x − 1)(x + 1)/(x + 2)] · [(x + 2)/(x − 1)]. The (x + 2) and the (x − 1) both cancel, leaving x + 1.
  2. c) (x + 3)/(x − 2), x ≠ ±2 — Factor both: (x + 3)(x + 2)/[(x − 2)(x + 2)]. The (x + 2) cancels, leaving (x + 3)/(x − 2). Both original restrictions survive — x ≠ 2 and x ≠ −2 — even though only one of them is still visible.
  3. d) x = 2 and x = −2 — Factor the denominator: x² − 4 = (x − 2)(x + 2), which is zero at x = 2 and at x = −2. A quadratic denominator usually gives two restrictions, so factor before you look.
  4. c) x = 3 or x = −1 — Multiply both sides by (x − 2): 3 = x² − 2x, so x² − 2x − 3 = 0 and (x − 3)(x + 1) = 0. Both x = 3 and x = −1 are allowed, since neither is the non-permissible value x = 2.
  5. c) x = 5 — A rational expression is undefined when its denominator is zero. Set x − 5 = 0 to get x = 5. The numerator being zero is fine — it just makes the whole expression zero.
  6. d) 7/x — Same denominator, so add the numerators and leave the denominator alone: (3 + 4)/x = 7/x. Adding the denominators as well is the standard slip.
  7. a) 2 hours — Work in rates, not times. Amira does 1/3 of the room per hour and Ben 1/6, so together 1/3 + 1/6 = 1/2 of the room per hour, which is one room in 2 hours. Averaging the two times gives 4.5 hours and is wrong — together must be faster than either alone.
  8. d) 2/x, x ≠ 0 — Multiply straight across: 6x/(3x²), then reduce to 2/x. Cancelling before multiplying is faster and gives the same thing: the 3 into the 6 and one x from x².
  9. c) x − 3, x ≠ −3 — Factor first: (x − 3)(x + 3)/(x + 3), and the (x + 3) cancels to leave x − 3. The restriction stays: the original was undefined at x = −3, and cancelling does not change that, so it must be written down.
  10. c) (2x + 1)/[x(x + 1)], x ≠ 0, −1 — The common denominator is x(x + 1). Rewriting gives (x + 1)/[x(x+1)] + x/[x(x+1)] = (2x + 1)/[x(x + 1)]. Adding the tops and bottoms separately is the mistake that produces 2/(2x + 1).