McGraw-Hill Education GRE 2019

For example, if you are told that xy > 1, you may be tempted to multiply both

sides by y and arrive at x > y. However, this would be incorrect. Why? Because you

do not know the sign of y. Since you don’t know the sign of y, you do not know

whether the inequality arrow will flip when you multiply. Thus you need to keep

the inequality in its original form.

Manipulating Compound Inequalities

From the introduction to this section, recall that a compound inequality looks like

the following: −7 < a + 3 < 12.

The rules for manipulating compound inequalities are the exact same ones as

those for normal inequalities. Just make sure that you perform the same operation

on all three parts of the inequality.

Let’s solve for a in the preceding example:

−7 < a + 3 < 12

−3 − 3 − 3

−10 < a < 9

Extremes with Inequalities

In some inequality questions, you will be presented with multiple inequalities, or

with an inequality and an equation, and will be asked to draw inferences about

their products. In these examples, choosing extreme values for the variables is often

the optimal approach.

If a = 3 and −6 < b < 12, which of the following can equal ab? (Indicate all

that apply.)

A –18

B 0

C 18

D 24

E 36

SOLUTION: Since you are trying to figure out possible values of ab, you should

consider the greatest value that ab could be and the smallest value that

ab could be. Since you know a = 3, the product will be smallest when b is

smallest. So choose the extreme value for b: in this case, −6. If b = −6, then

ab = −18. However, you know b > −6. Therefore, ab > −18. Now try the upper

bound. If b = 12, then ab = 36. However, you know that b < 12, meaning that

ab < 36. You arrive at the compound inequality: −18 < ab < 36. The answer is

B, C, and D.

CHAPTER 11 ■ ALGEBRA 303

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