# 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?

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.

1. 3, -${\displaystyle {\tfrac {-1}{3}}}$

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

x is equal to negative one third.

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

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 = ${\displaystyle {\tfrac {5}{4}}}$5/4,${\displaystyle {\tfrac {-1}{4}}}$

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

3|4k - 2| = 9

|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 = ${\displaystyle {\tfrac {-1}{4}}}$