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# Elementary Algebra: Absolute Value

Module by: John Redden. E-mail the author

Summary: This module defines absolute value in two ways and demonstrates how to work with them within arithmetic expressions.

## Module Overview

This module defines absolute value and demonstrates how to work with them within arithmetic expressions. (Prerequisites: Working knowledge of real numbers and the order of operations.)

### Objectives

• Geometric Definition of Absolute Value
• Algebraic Definition of Absolute Value
• Simplifying Expressions Containing Absolute Values

## Module Notes

Definition 1: Geometric Definition of Absolute Value
The absolute value of a number a, denoted |a| a , is the distance from a a to zero on the number line.
The absolute value of any non-zero real number will always be positive since it represents the distance to the origin.

### Example 1

|-10| = 10 and |10| = 10 - 10 = 10 and 10 = 10

Both -10 and 10 are ten units from zero on the number line. It is also of interest to note that: |0| = 0 0 = 0 .

Absolute values within an expression have the same level of precedence as parenthesis. Apply the innermost absolute values first as we would with parenthesis in respect to the order of operations.

### Example 2

Simplify: |5| - 7 || 5 - 7 |5| - 7 || = |57| = |-2| = 2 5 - 7 = 5 7 = - 2 = 2

Definition 2: Algebraic Definition of Absolute Value
The absolute value of a number a a is |a| = a if a 0 - a if a < 0 a = a if a 0 - a if a 0 .
If the variable a a is positive a 0 a 0 then the absolute value will be that number. If the variable a a is negative a < 0 a 0 then the absolute value will the the opposite of that number.

### Example 3

|-10| = - - 10 = 10 and |10| = 10 -10 = - - 10 = 10 and 10 = 10

Since -10 is less than zero the absolute value will be the opposite of that number, -(-10) = 10.

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