Solutions

  1. 11

    2|x − y| + |z + w|=2|2 − 5| + |−3 + 8| = 2|−3| + |5| = 2(3) + 5 = 11. Note that when you deal with more complicated absolute value expressions, such as |x − y| in this example, you should NEVER change individual signs to “+” signs! For instance, in this problem |x − y| = |2 − 5|, not |2 + 5|.

  2. −18

    In division, use the Same Sign rule. In this case, the signs are not the same. Therefore, 66 ÷ (−33) yields a negative number (−2). Then, multiply by the absolute value of −9, which is 9. To multiply −2 × 9, use the Same Sign rule: the signs are not the same, so the answer is negative. Remember to apply division and multiplication from left to right: first the division, then the multiplication.

  3. −3

    This is a two-step subtraction problem. Use the Same Sign rule for both steps. In the first step, , the signs are different; therefore, the answer is −6. In the second step, , the signs are again different. That result is −3. The final answer is −6 − (−3) = −3.

  4. −2

    The sign of the first product, 20 × (−7), is negative (by the Same Sign rule). The sign of the second product, −35 × (−2), is positive (by the Same Sign rule). Applying the Same Sign rule to the final division problem, the final answer must be negative.

  5. x ≤ 4

    Absolute value brackets can only do one of two things to the expression inside of them: (1) leave the expression unchanged, whenever the expression is 0 or positive, or (2) change the sign of the whole expression, whenever the expression is 0 or negative. (Notice that both outcomes occur when the expression is 0, because “negative 0” and “positive 0” are equal.) In this case, the sign of the whole expression x − 4 is being changed, resulting in −(x − 4) = 4 − x. This will happen only if the expression x − 4 is 0 or negative. Therefore, x − 4 ≤ 0, or x ≤ 4.

  6. Negative

    The product of the first two negative numbers is positive. This positive product times the third negative is negative.

  7. Negative

    By the Same Sign rule, the quotient of a negative and a positive number must be negative.

  8. Cannot Be Determined

    x is negative. However, y could be either positive or negative. Thus, there is no way to determine whether the product xy is positive or negative. Numerical examples are x = −2 and y = 3 or −3, leading to xy = −6 or 6.

  9. Positive

    |x| is positive because absolute value can never be negative, and x ≠ 0 (since xy ≠ 0). Also, y2 is positive because y2 will be either positive × positive or negative × negative (and y ≠ 0). The product of two positive numbers is positive, by the Same Sign rule.

  10. Negative

    Do this problem in two steps: First, a negative number divided by a negative number yields a positive number (by the Same Sign rule). Second, a positive number divided by a negative number yields a negative number (again, by the Same Sign rule).

  11. Negative

    a and b are both negative. Therefore, this problem is a positive number (by the definition of absolute value) divided by a negative number. By the Same Sign rule, the answer will be negative.

  12. Negative

    You do not need to know the sign of d to solve this problem. Because d is within the absolute value symbols, you can treat the expression |d| as a positive number (since you know that d ≠ 0). By the Same Sign rule, a negative number times a positive number yields a negative number.

  13. Positive

    r and s are negative; w and t are positive. Therefore, rst is a positive number. A positive number divided by another positive number yields a positive number.

  14. Negative

    Nonzero numbers raised to even exponents always yield positive numbers. Therefore, h4 and m2 are both positive. Because k is negative, k3 is negative. Therefore, the final product, h4k3m2, is the product of two positives and a negative, which is negative.

  15. Negative

    Simplifying the original fraction yields: .

    If the product xyz is positive, then there are two possible scenarios: (1) all the integers are positive, or (2) two of the integers are negative and the third is positive. Test out both scenarios, using real numbers. In the first case, the end result is negative. In the second case, the two negative integers will essentially cancel each other out. Again, the end result is negative.

  16. (C)

    If xy > 0, x and y have the same sign. You already know that the denominator of both fractions described in the quantities will be positive. The numerator will either be positive for both, or negative for both. If both x and y are positive, the quantities simplify like this:

      xy > 0  
    Quantity A   Quantity B
     

    In this case, both quantities equal 1. If x and y are both negative, the quantities simplify like this:

      xy > 0  
    Quantity A   Quantity B
     

    In this case, both quantities equal −1. Either way, the values in the two quantities are equal.

  17. (A)

    If a is positive, then −a is negative, and Quantity A can be rewritten as (negative) × (negative) × (positive) × (positive), which will result in a positive product.

    If a is negative, then −a is positive, and Quantity A can be rewritten as (positive) × (positive) × (negative) × (negative), which will result in a positive product.

    In either of these situations, the quantities look like this:

    Quantity A Quantity B
    positive −1

    Quantity A will be greater.

    The other possibility is that a is 0. If a is 0, then Quantity A looks like this:

    Quantity A Quantity B
    0 × 0 × 0 × 0 = 0 −1

    Therefore, Quantity A is greater in either scenario.

  18. (D)

    If |x| = |y|, then the two numbers could either be equal (positive or negative) or opposite (one positive and one negative). The following chart shows all the possible arrangements if |x| = |y| = 3:

    x y Quantity A = x + y   Quantity B = 2x
    3 3 6 = 6
    3 −3 0 < 6
    −3 3 0 > −6
    −3 −3 −6 = −6

    Alternatively, you could reason that if x and y are the same sign, then x = y. Substitute x for y in Quantity A:

      |x| = |y|, x ≠ 0  
    Quantity A   Quantity B
    x + (x) = 2x   2x

    If x and y are the same sign, the quantities are equal.

    If x and y have opposite signs, then −x = y. Substitute −x for y in Quantity A:

      |x| = |y|, x ≠ 0  
    Quantity A   Quantity B
    x + (−x) = 0   2x

    If x does not equal 0, then the values in the two quantities will be different. The correct answer is (D). Thus, the relationship cannot be determined from the information given.