A Guide to the GRE/Absolute Value

= Absolute Value =

The concept of absolute value - meaning a number's distance from zero - is tested on nearly every GRE.

Rule
Absolute value makes a negative positive, but otherwise does nothing.

“|  |” designates absolute value. For example, if | x + 3 | = 5, there are two possible values for x:


 * x + 3 = 5, meaning x is 2
 * x + 3 = -5, meaning x is -8

On an absolute value questions, split the value into two equations as seen above.

Practice
1. If | 3x - 4 | = 5, then what could be the value of x?

2. If | a | > a, then what is the greatest integer that a could be?

3. If 3|4k - 2| - 12 = -3, what is the value of k?

Comments
Absolute value tends to be tested in the quantity comparison section of the test, often with a variable modified by a constant within the absolute value. (e.g. | q + 7 | = 5) Solve these by writing out both of the potential values for the variable, and remember that either one could be the value. For example, in the prior equation, q could equal either -2 or -12, so it is unclear whether it is greater or less than -5.

Answers to Practice Questions
1. 3, $$\tfrac{-1}{3}$$

If | 3x - 4 | = 5 then

3x - 4 = 5


 * or

3x - 4 = -5

3x - 4 = 5
 * Take the first equation and solve it. First, add 4 to both sides.

3x = 9
 * Now divide both sides by 3.

x = 3
 * x is equal to 3. But remember, this is just one solution - you still need to solve the other equation.

3x - 4 = -5
 * Now take the second equation and solve it. Add 4 to both sides.

3x = -1
 * Now divide both sides by 3.

x = $$\tfrac{-1}{3}$$


 * x is equal to negative one third.

This means that x = 3 or $$\tfrac{-1}{3}$$

2. -1

Absolute value makes a positive negative, but otherwise does nothing - if the absolute value of a number is greater than that number itself, the number must be negative. The greatest negative number is -1.

3. k = $$\tfrac{5}{4}$$5/4,$$\tfrac{-1}{4}$$

If 3|4k - 2| - 12 = -3, then

3|4k - 2| = 9
 * Add 12 to both sides.


 * 4k - 2| = 3
 * Divide both sides by 3.

4k - 2 = 3

Split both possibilities.

4k - 2 = -3
 * Add 2 to to both sides

4k = 5			.
 * Divide both sides by 5.

k = $$\tfrac{-1}{4}$$