Using Extreme Values

One effective technique for solving GRE inequality problems is to focus on the Extreme Values of a given inequality. This is particularly helpful when solving the following types of inequality problems:

1. Problems with multiple inequalities where the question involves the potential range of values for variables in the problem
2. Problems involving both equations and inequalities

Inequalities with Ranges

Whenever a question asks about the possible range of values for a problem, consider using Extreme Values:

• If 0 ≤ x ≤ 3 and y < 8, which of the following could NOT be the value of xy?

  1. 0

  2. 8

  3. 12

  4. 16

  5. 24

To solve this problem, consider the Extreme Values of each variable:

Extreme Values for x	Extreme Values for y
The lowest value for x is 0. The highest value for x is 3.	The lowest value for y is negative infinity. The highest value for y is less than 8.

(y cannot be 8, therefore, this upper limit is termed “less than 8” or “LT8” for shorthand.)

What is the lowest value for xy? Plug in the lowest values for both x and y. In this problem, y has no lower limit, so there is no lower limit to xy.

What is the highest value for xy? Plug in the highest values for both x and y. In this problem, the highest value for x is 3, and the highest value for y is LT8.

Multiplying these two extremes together yields: 3 × LT8 = LT24. Notice that you can multiply LT8 by another number (if that other number is positive) just as though it were 8. You just have to remember to include the “LT” tag on the result.

Because the upper extreme for xy is less than 24, xy CANNOT be 24, and the answer is (E).

Notice that you would run into trouble if x did not have to be non-negative. Consider this slight variation:

If −1 ≤ x ≤ 3 and y < 8, what is the possible range of values for xy?

Because x could be negative and because y could be a very negative number, there is no longer an upper extreme on xy. For example, if x = −1 and y = −1,000, then xy = 1,000. Obviously, even greater positive results are possible for xy if both x and y are very negative. Likewise, because x can be positive and y can be infinitely negative, xy can be infinitely negative. Therefore, xy can equal any number.

Check Your Skills

27. If −4 < a < 4 and −2 < b < −1, which of the following could NOT be the value of ab?

    1. −3
    2. 0
    3. 4
    4. 6
    5. 9