Free Compass Math Practice Questions
on Reducing Rational Expressions with Solutions and Explanations - Sample 7

Solutions with detailed explanations to compass math test practice questions in sample 7.

  1. For all x not equal to 1, which of the following is equivalent to the rational expression below?

    x2 + 5x - 6
    ----------------
    x - 1


    A) x - 6
    B) x - 1
    C) x + 6
    D) - x - 6
    E) 6 - x

    Solution

    Factor the numerator
    x2 + 5x - 6
    -------------------------
    x - 1
    (x - 1)(x + 6)
    = ----------------
    x - 1
    For x not equal to 1, the given rational expression could be simplified to

    = x + 6

  2. Which of the following is a simplified expression equal to

    5 - x
    ---------
    2x - 10


    for all x not equal to 5?

    A) -1/2
    B) 1 / (x - 5)
    C) -2
    D) - 1 / (x - 5)
    E) 1/2

    Solution

    Factor denominator 2x - 10 in the given expression as follows
    5 - x
    ---------
    2x - 10
    5 - x
    = ---------
    -2(5 - x)
    For x not equal to 5, the above could be simplified to

    = - 1 / 2

  3. For all x not equal to - 4, which of the given expressions is equivalent to the expression below?

    16 - x2
    ------------
    x + 4


    A) x - 4
    B) 16 - 1
    C) x + x
    D) - x - 4
    E) 4 - x

    Solution

    Factor the numerator in the given expression as follows
    16 - x2
    ------------
    x + 4
    (4 - x)(4 + x)
    = -------------------
    x + 4
    For x not equal to -4, the above could be simplified to

    = 4 - x


  4. Simplify the following rational expression.

    x + 2
    ------------ =
    x2 + 2x


    Solution

    Factor the denominator in the given expression as follows
    x + 2
    ------------ =
    x2 + 2x
    x + 2
    ------------------
    x(x + 2)
    For x not equal to -2, the above could be simplified to

    = 1/x

  5. For all x not equal to 3, which of the given expressions is equivalent to the expression below?

    3 - x
    ------------
    x2- x - 6


    A) - 1 / (x + 2)
    B) 1 / (x - 2)
    C) -1 / (x - 2)
    D) 1 / (x - 3)
    E) -1 / (x - 3)

    Solution

    Factor the numerator and denominator in the given expression as follows
    3 - x
    ------------
    x2- x - 6
    -(x - 3)
    = ------------
    (x + 2)(x - 3)
    For x not equal to 3, the above could be simplified to

    = - 1 / (x + 2)


  6. x3 - x
    ------------ =
    x2 - 1


    Solution

    Factor the numerator and denominator in the given expression as follows
    x3 - x
    ------------ =
    x2 - 1
    x(x2 - 1)
    ------------------
    x2 - 1
    Since x2 - 1 = 0 for x = 1 or x = -1, for all x not equal to 1 or -1, then the above could be simplified to

    = x


  7. x2 - 4
    ---------------- =
    x2 + 4x - 12


    Solution

    Factor the numerator and denominator in the given expression as follows
    x2 - 4
    ---------------- =
    x2 + 4x - 12
    (x - 2)(x + 2)
    = ------------------
    (x - 2)(x + 6)
    For all x not equal to 2, we can simplify the above to

    = (x + 2) / (x + 6)

  8. Simplify the following rational expression.

    x2 + 1
    ---------------
    x3 + x


    Solution

    Factor the denominator in the given expression as follows
    x2 + 1
    ---------------
    x3 + x
    x2 + 1
    = ---------------
    x(x2 + 1)
    For reall x, x2 + 1 is never zero, hence the above could be simplified to

    = 1 / x , for all x


  9. x2 + 2x - 3
    ---------------- =
    2x2 + 3x - 5


    Solution

    Factor the numerator and denominator in the given expression as follows
    x2 + 2x - 3
    ---------------- =
    2x2 + 3x - 5
    (x - 1)(x + 3)
    = ---------------- =
    (x - 1)(2x + 5)
    For all x not equal to 1, the above could be simplified to

    = (x + 3) / (2x + 5)

  10. For all x not equal to 1, which of the following is equivalent to the rational expression below?

    x - 1
    ----------------
    (x2 - 1)(x + 3)


    A) 1 / (x + 3)
    B) 1 / (x2 + 4 x + 3)
    C) 1 / (x + 1)
    D) 1 / x
    E) 1 / (x - 1)

    Solution

    Factor the term x2 - 1 in the denominator as follows
    x - 1
    ----------------
    (x2 - 1)(x + 3)
    x - 1
    = ---------------------------
    (x - 1)(x + 1)(x + 3)
    For all x not equal to 1, we can simplify the above to to

    = 1 / [(x + 1)(x + 3)]

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