Check Your Skills Answer Key

  1. Yes

    The absolute values of 5 and −5 are both 5, and 5 + 5 + 5 = 15.

  2. 1

    This question is easy as long as you pay close attention to the signs. The left side of the equation will have a double negative (in front of the 4), so it simplifies to 4y − x + 4. The right side has no double negative (in fact, the parentheses are unnecessary), so it simplifies to 4x − y + 4.

    The equation now reads:

    4y − x + 4 = 4x − y + 4.

    Continue to simplify: 5y = 5x, or x = y. If x and y are equal, then .

  1. Negative

    Here there are three negative numbers and three positive numbers. The product of two of the negative numbers will be positive, and the third negative number will make the final product negative.

  2. No

    Because the two negative signs multiply to a positive, you can say that (−x)(−y) = xy. However, always be careful when dealing with variables, because x or y could themselves be negative. If x and y are both positive or both negative, their product will be positive. But if x is positive and y is negative, the product will be negative.

  1. (C)

    What’s more fun than making a chart? Nothing. Try looking at the four possible situations with x and y that maintain the requirement that the absolute value of x is greater than the absolute value of y:

    Scenario x y xy x + y x2 y2
    1 5 3 15 8 25 9
    2 −5 3 −15 −2 25 9
    3 5 −3 −15 2 25 9
    4 −5 −3 15 −8 25 9

    The second scenario here gets rid of both choice (A) and choice (B), because xy and x + y are both negative. However, in every case, x2 > y2, so choice (C) alone is necessarily true. You could also solve this by looking at each statement while keeping in mind what you know if the absolute value of x is greater than the absolute value of y. You don’t know if xy is positive because x or y could be negative.

  2. (C)

    Though a bit tricky, the given information here tells you everything you need to know about a and b. Instead of testing numbers to get through the answer choices (as above), test numbers to make sense of the given information. First, if ab < 0, then the two variables must have opposite signs. Second, if a > b, then a must be the positive number, and b the negative number. Finally, if a > −b, a must have a larger absolute value than b (if a = 4 and b = −5, then −b > a, which is the opposite of what you want).

    Now, it should be enough just to walk through the answer choices.

    1. Untrue: If a and b have opposite signs, then the quotient will be negative.

    2. a + b < 0

      Untrue: You know that a is positive and has a larger absolute value than b. No matter what a and b are, their sum has to be positive.

    3. b − (−a) > 0

      True: This actually simplifies to look like the equation in answer choice (B), though with the sign switched. You already know this has to be true.

    4. .

      Untrue: The quotient of a positive and a negative number must be negative, so can never be 1.

    5. a − b < 0

      Untrue: If you subtract a negative number from a positive number, you’ll be left with an even bigger positive number.