Solutions

  1. (B)

    Notice that in each of the quantities, you can factor an x out of one of the expressions:

    0 < x < 1
    Quantity A Quantity B
    (x3 − x)(4x + 3) =
    x(x2 − 1)(4x + 3)     		(x2 + 1)(4x2 + 3x) =
    (x2 + 1)(4x + 3)x

    Because you know that x is not 0, you can use the Invisible Inequality to divide away the common terms from both quantities (x and (4x + 3)):

    0 < x < 1
    Quantity A Quantity B
    x(x2 − 1)(4x + 3) =
    x2 − 1     		(x2 + 1)(4x + 3)x =
    x2 + 1

    Now the comparison is easy to make. Because x2 will always be positive, (x2 + 1) will always be bigger than (x2 − 1).

  2. (D)

    Try to prove (D) by testing the boundaries of the range for m.

    If m = 6, then Quantity A is equal to 9 − (6) = 3, and Quantity B is equal to (6) − 9 = −3. Eliminate answer choices (B) and (C).

    If m = 12, then Quantity A is equal to 9 − (12) = −3, and Quantity B is equal to (12) − 9 = 3. Eliminate answer choice (A).

    The answer is (D).

  3. (D)

    Never leave a complex fraction in place; that is, simplify in order to find a direct comparison. First, multiply both sides by 2:

    Multiply both sides by 2 again:

    Divide by −7 in order to make the left side of the equation 3m (Quantity A):

    −21m = 14n  →  3m = −2n

    Since 3m = −2n, you can substitute −2n for 3m in Quantity A. The problem now reads:

    mn ≠ 0
    Quantity A Quantity B
    −2n −n

    If n is positive, Quantity B is bigger. If n is negative, Quantity A is bigger.

    The answer is (D).

  4. (D)

    If there are no constraints on a variable, try to prove (D).

    If x = 0, Quantity A is equal to 0 and Quantity B is equal to −4. Eliminate answer choices (B) and (C).
    If x = 10, Quantity A is equal to 10 and Quantity B is equal to 26. Eliminate answer choice (A).

    The answer is (D).

  5. (A)

    This question contains a quadratic equation in the common information. The first thing to note here is that there will be two possible values for x. But you should not jump to conclusions and assume the answer will be (D). To make sure you get the right answer, solve for both values of x and plug them BOTH into the quantities:

    Now the problem reads:

    x = −7 or 6
    Quantity A Quantity B
    |x + 1| 5

    If x = −7, Quantity A is equal to the absolute value of −6, which is 6, and Quantity B will still be equal to 5.

    If x = 6, Quantity A is equal to the absolute value of 7, which is 7, and Quantity B will still be equal to 5.

    In either case, Quantity A is bigger.

  6. (B)

    If a QC question with a strange symbol formula contains numbers, plug in the numbers and evaluate the formula.

    In Quantity A, @(10) = (10)2 − 4 = 96.

    In Quantity B, work outwards from the “inner core.” @(4) = 42 − 4 = 12. Now evaluate @(12).

    @(12) = 122 − 4 = 140.

    Quantity B is bigger.

    The answer is (B).

  7. (C)

    Remember, if you are given a strange symbol on the GRE, the exam will have to define that strange symbol for you. Since you are not given numbers to plug in, you should evaluate the formula using the variable itself—otherwise, if you plugged in numbers, you would have no way of knowing whether you would have to try more numbers to try to prove (D).

    Quantity A asks for ♣(♣x). They want you to plug the function into itself. So, plug in for x:

    Combine the two terms in the denominator:

    Remember, if a fraction is under a 1, just flip it over:

    The answer is (C).

  8. (B)

    As with all absolute value equations or inequalities, here you must solve twice:

    Even better, you could express the possible values of x on a number line:

    Quantity A is equal to the minimum possible value of |x − 3.5|. Another way to think of |x − 3.5| is the distance on a number line from x to 3.5. Look at 3.5 on the number line above and note the nearest possible distance that is greater than 5 (x may not be exactly 5, but it could be 5.000001, for instance, since there is no requirement that it be an integer). Therefore, since the distance from 3.5 to greater than 5 is greater than 1.5, Quantity A is equal to greater than 1.5. That is, the ­minimum possible value of |x − 3.5| is 1.5 plus any very small amount—for instance, 1.5000001 would be a legal value.

    Quantity B can be conceived as the smallest distance from x to 1.5. Look at 1.5 on the number line—the nearest value is less than −1, which is more than 2.5 units away. Thus, the minimum ­possible value of |x − 1.5| is greater than 2.5.

    If Quantity A’s minimum is just greater than 1.5 and Quantity B’s minimum is just greater than 2.5, Quantity B is larger.

    The answer is (B).

    You could also solve this problem by plugging in values, rather than using a number line. First, solve the inequality as above to get x > 5 or x < −1. Now try plugging in greater than 5, less than −1, as well as very small and very large numbers—that is, the extremes of both ranges for x (although you may be able to use a bit of logic beforehand to tell that you only want values very close to 1.5 and 3.5)—to make sure that you generate the smallest possible value for each quantity. Quantity A’s smallest value will be smaller than Quantity B’s smallest value.

    The answer is (B).

  9. (A)

    When variables are ordered from least to greatest, look for the pattern. Notice that a and b are negative, and c and d are positive. Try working only with positives and negatives first, before considering more specific numbers.

    In Quantity A, abc is a negative times a negative times a positive—that is, Quantity A is a positive value.

    In Quantity B, c − d is a positive minus a positive. Now, a positive minus a positive can yield either a positive or a negative value (for instance 10 minus 1 versus 1 minus 10). So look back up at the ­common information to see that d is greater than c. Thus, c − d is an instance of subtracting a larger positive from a smaller positive, which yields a negative.

    Quantity A is positive and Quantity B is negative.

    The answer is (A).

  10. (A)

    When variables are ordered from least to greatest, look for the pattern. You can also use the technique of the invisible inequality—and since all of the variables are positive, you can cross-multiply across that invisible inequality:

    ?
    abc2 ? abc

    Since you know a, b, and c are positive, go ahead and divide out abc:

    c ? 1

    You were directly told in the common information that 1 < c, so Quantity A is bigger.

    The answer is (A).