McGraw-Hill Education GRE 2019

Maximization and Minimization with Inequalities

Another common type of inequality question will give you two inequalities and

ask you for the maximum or minimum value of their product. In these cases, it is

essential to consider the extremes for all variables.

If −7 ≤ a ≤ 12 and −11 ≤ b ≤ 5, what is the maximum value of ab?

SOLUTION: The trap here is to multiply the maximum value for a and the

maximum value for b and arrive at 60. However, note that if a and b are both

negative, their product will be positive! Thus it is possible that the product of

the smallest values of a and b will yield a larger value than the product of

the largest values of a and b. And that is exactly what happens here: take the

minimum value for a, −7; and the minimum value for b, −11; and the product

is 77, which is greater than 60.

Let’s look at an example with minimization:

If −12 ≤ q ≤ 9 and 8 ≤ r ≤ 12, then the minimum value of qr =?

SOLUTION: As in the preceding example, it might be tempting to multiply the

smallest value for q and r and arrive at −96. However, note that when you

multiply a negative and positive value, the result becomes smaller as the

positive number becomes larger—for example, −3(9) < −3(7). Thus you will

minimize qr when you multiply the minimum value of q by the maximum

value of r: −12 × 12 = −144.

Absolute Value

In its simplest form, absolute value refers to the distance between a number,

variable, or expression and zero. Absolute value is denoted using brackets, for

example, |x + y| or |−3|.

Since absolute value refers to distance, the absolute value of a number or

expression will always be positive. For example, |−8| = 8 since −8 is 8 units away

from zero.

Absolute Value with Unknowns

When an unknown term or expression is within an absolute value, there will be two

possible values for the unknown. For example, if |x| = 2, then x = 2 or x = −2. Why

are there two values for x? Because absolute value refers to distance! Both 2 and −2

are two units away from zero, so x can equal either of those values. When solving

for an unknown within an absolute value, use the following process:

If 9 + |x + 4| = 28, what are the possible values for x?

304 PART 4 ■ MATH REVIEW

03-GRE-Test-2018_173-312.indd 304 12/05/17 11:56 am