# Absolute Value - GMAT Math Study Guide

## Definitions

• Absolute Value - The value of a number without regard to its sign.
For example, the absolute value of -3, denoted |-3|, is 3. Similarly, |-2| = 2 = |2|

### A Deeper Understanding of Absolute Value

One means to see the concept of absolute value is graphically. The following graph depicts two equations:
y = x [Blue Line]
y = |x| [Red Line] x |x|
-7 7
-6 6
-5 5
-4 4
-3 3
-2 2
-1 1
0 0
1 1
2 2
3 3
4 4
5 5

The absolute value of a number is its distance from the origin.

Another means to understand absolute value is as the distance from the origin. For example, at the value of x = -7, the distance from the origin, where x = 0, is the absolute value of x=-7, which is 7.

## Solving Equations With Absolute Values

An equation with an absolute value term within it often has two solutions. In order to solve an absolute value equation: (1) isolate the absolute value term (2) Solve the absolute value term for two circumstances: (i) the value inside the absolute value brackets is positive (ii) the value within the absolute value brackets is negative.

To summarize: 1.) Isolate the absolute value term. 2.) Set the expression opposite the absolute value as negative (i.e., multiply it by negative one).

|x| + 10 = 20
Isolate the Absolute Value:
|x| + 10 - 10 = 20 - 10
|x| = 10
X is Positive: x = 10
X is Negative: x = -10 [Make the expression opposite the absolute value negative]

Another Example:

|x + 5| + 20 = 60
Isolate the Absolute Value:
|x + 5| + 20 - 20 = 60 - 20
|x + 5| = 40
X is Positive: x + 5 = 40
x + 5 - 5 = 40 - 5
x = 35
X is Negative: x + 5 = -40 [Make the expression opposite the absolute value negative]
x = -45

Another Example:

|10x + 8| - 4 > 5 + 2x
Isolate the Absolute Value:
|10x + 8| > 9 + 2x
X is Positive: 10x + 8 > 9 + 2x
8x + 8 > 9
8x > 1
x > 1/8
X is Negative: 10x + 8 < (-1)(9 + 2x)
10x + 8 < -9 - 2x
12x + 8 < -9
12x < -17
x < -17/12

## Working With Multiple Absolute Values

Multiple absolute values should be dealt with by working with the innermost one and moving outwards. For example:

| 3 - (| -5|) |
= |3 - 5|
= |-2|
= 2

## Types of GMAT Problems

1. Solving Equations With Absolute Values

Begin by isolating the absolute value. Rewrite the new equation without the absolute value brackets and solve for the variable. Rewrite the new equation without the absolute value brackets and multiply one side of the equation by -1. Solve this new equation for the variable. The solution should consist of two different answers, usually one being negative and the other being positive (although this is not always true).

Solve for x, |x+5|-10 = 0
 A) x=5 or x=-15 B) x=2 or x=-18 C) x=3 or x=-17 D) x=6 or x=-14 E) x=1 or x=-17
1. Isolate the absolute value.
|x+5|-10 = 0
|x+5| = 10; (add 10 to both sides)
2. x+5 can equal +10 or -10 due to absolute value
3. Rewrite the equation without the brackets.
x+5 = 10
4. Solve for x by subtracting 5 from both sides.
x = 10 - 5
x=5
5. Rewrite |x+5| = 10 without the brackets and multiply the right side by -1 since the expression in the absolute value brackets could be negative due to the effect of the absolute value.
x+5 = (10)(-1)
x+5 = -10
6. Solve for x by subtracting 5 from both sides.
x = -10 - 5
x=-15
7. x = 5 or x = -15
2. Solving Inequalities With Absolute Values

Begin by isolating the absolute value. Rewrite the new equation without the absolute value brackets and solve for the variable. Rewrite the new equation without the absolute value brackets, flip the inequality, and multiply the side that did not have the absolute value by -1. Solve this new equation for the variable. The solution will correspond to a section (or sections) of the number line.

Solve for x, |3x-5|-5 < -2x
 A) x>1 and x<-3 B) 02 and x<-5 E) x>1 and x<0
1. Rewrite the inequality so that the absolute value is isolated.
|3x-5|-5 < -2x
|3x-5| < -2x+5
2. Remove the brackets.
3x-5 < -2x+5
3. Solve for x.
3x+2x < 5+5 [added 2x to both sides; added 5 to both sides]
5x < 10
x < 2
4. Rewrite |3x-5| < -2x+5 without the brackets and multiply the right side by -1.
Since you are multiplying an inequality by a negative number, you must flip the inequality sign.
3x-5 > (-2x+5)(-1)
3x-5 > 2x-5
5. Solve for x.
3x-2x > -5+5
x > 0
6. Combine the two solutions to get: 0<x<2