Solutions

  1.             If x = 2:                         If x = −2:            
          −3x2 = −3(4) = −12     −3x2 = −3(4) = −12
          −3x3 = −3(8) = −24 −3x3 = −3(−8) = 24
            3x2 = 3(4) = 12        3x2 = 3(4) = 12
      (−3x)2 = (−6)2 = 36       (−3x)2 = 62 = 36
          (−3x)3 = (−6)3 = −216         (−3x)3 = 63 = 216

    Remember that exponents are evaluated before multiplication. Watch not only the order of operations, but also the signs in these problems.

  2. −3

  3. (a): The parentheses around 52 are unnecessary, because this exponent is performed before the negation (which counts as multiplying by −1) and before the subtraction. The other parentheses are necessary because they cause the right-hand subtraction to be performed before the left-hand subtraction. Without them, the two subtractions would be performed from left to right.

    (b): The first and last pairs of parentheses are unnecessary. The addition is performed before the neighboring subtraction by default, because addition and subtraction are performed from left to right. The multiplication is the first operation to be performed, so the right-hand parentheses are completely unnecessary. The middle parentheses are necessary to ensure that the addition is performed before the subtraction that comes to the left of it.

  5. 2x − 3

    Do not forget to reverse the signs of every term in a subtracted expression:
    x − (3 − x) = x − 3 + x = 2x − 3

  6. −5y + 10 (or 10 − 5y)

    Do not forget to reverse the signs of every term in a subtracted expression.
    (4 − y) − 2(2y − 3) = 4 − y − 4y + 6 = −5y + 10 (or 10 − 5y)

  7. −1

  9. (B)

    Evaluate Quantity A first:

    3 × (5 + 6) ÷ −1
    3 × (11) ÷ −1 Simplify the parentheses.
    33 ÷ −1 Multiply and divide in order from left to right.
    −33
    Now evaluate Quantity B:
    3 × 5 + 6 ÷ −1
    15 + −6 Multiply and divide in order from left to right.
    9 Add.
    Quantity A Quantity B
    −33 9

    Quantity B is greater.

  10. (A)

    Simplify the given equation to solve for x:

    Quantity A Quantity B
    x = 1 −4

    Quantity A is greater.

  11. (B)

    Use substitution to solve for the values of x and y:

    2x + y = 10 → y = 10 − 2x Isolate y in the first equation.
    3x − 2y = 1 → 3x − 2(10 − 2x) = 1 Substitute (10 − 2x) for y in the second equation.
    3x − 20 + 4x = 1 Distribute
    7x = 21 Group like terms (3x and 4x) and add 20 to both sides.
    x = 3 Divide both sides by 7.
    2x + y = 10 → 2(3) + y = 10 Substitute 3 for x in the first equation.
    6 + y = 10
    y = 4
    Quantity A Quantity B
    x = 3 y = 4

    Quantity B is greater.