Absolute Value Definition

Absolute value is one of the most straightforward concepts in mathematics, yet it is tested on the GED in multiple ways — from simple evaluations to comparisons and real-world problems. Understanding what absolute value really means (distance from zero) makes every absolute-value question easy to handle.

What Is Absolute Value?

The absolute value of a number is its distance from zero on the number line, regardless of direction. Distance is always non-negative, so the absolute value of any number is always zero or positive.

GED Math for Beginners 2026 The Ultimate Step by Step Guide to Preparing for the GED Math Test

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Rated 4.33 out of 5 based on 105 customer ratings

Satisfied 91 Students

The absolute value of a number \(\color{blue}{x}\) is written as \(\color{blue}{|x|}\) (the number between two vertical bars).

• \(\color{blue}{|7| = 7}\)   (7 is 7 units from zero)
• \(\color{blue}{|-7| = 7}\)   (−7 is also 7 units from zero)
• \(\color{blue}{|0| = 0}\)   (0 is 0 units from zero)

How to Evaluate Absolute Value Expressions

Rule for positive numbers and zero

If the number inside the bars is positive or zero, the absolute value equals the number itself:

\(\color{blue}{|x| = x}\)   when \(\color{blue}{x \ge 0}\)

Rule for negative numbers

If the number inside the bars is negative, the absolute value equals the opposite (positive version) of that number:

\(\color{blue}{|x| = -x}\)   when \(\color{blue}{x < 0}\)

(Note: \(\color{blue}{-x}\) here means “the opposite of x” — when x is negative, −x is positive.)

Expressions inside the bars

When there is an expression inside the absolute value bars, evaluate the expression first, then take the absolute value.

• \(\color{blue}{|3 – 8| = |-5| = 5}\)
• \(\color{blue}{|2 + (-9)| = |-7| = 7}\)

Step-by-Step Summary

1. Evaluate any arithmetic inside the absolute value bars first.
2. If the result is positive or zero, write it as-is.
3. If the result is negative, write its positive opposite.
4. The absolute value is \(\color{blue}{\text{ always } \ge 0}\).

Watch: Absolute Value Definition (Video Lesson)

Math Antics explains absolute value visually using a number line:

Worked Examples

Example 1: Find \(\color{blue}{|-15|}\)

−15 is 15 units from zero. So \(\color{blue}{|-15| = 15}\).

Example 2: Find \(\color{blue}{|4 – 10|}\)

Evaluate inside first: \(\color{blue}{4 – 10 = -6}\). Then \(\color{blue}{|-6| = 6}\).

Example 3: Compare \(\color{blue}{|-8|}\) and \(\color{blue}{|5|}\).

\(\color{blue}{|-8| = 8}\) and \(\color{blue}{|5| = 5}\). Since \(\color{blue}{8 > 5}\), we have \(\color{blue}{|-8| > |5|}\).

Example 4: Evaluate \(\color{blue}{|-3| + |7 – 12|}\)

\(\color{blue}{|-3| = 3}\) and \(\color{blue}{|7 – 12| = |-5| = 5}\). Sum: \(\color{blue}{3 + 5 = 8}\).

More Practice: Absolute Value and Number Lines

Khan Academy reinforces the connection between absolute value and the number line with additional examples:

Exercises

1. Find \(\color{blue}{|-23|}\)
2. Find \(\color{blue}{|0|}\)
3. Evaluate \(\color{blue}{|6 – 14|}\)
4. Evaluate \(\color{blue}{|-5| + |-9|}\)
5. Which is greater: \(\color{blue}{|-12|}\) or \(\color{blue}{|11|}\)?
6. Evaluate \(\color{blue}{2 \times |-4| – |3 – 10|}\)

Answers

1. \(\color{blue}{23}\)
2. \(\color{blue}{0}\)
3. \(\color{blue}{|-8| = 8}\)
4. \(\color{blue}{5 + 9 = 14}\)
5. \(\color{blue}{|-12| = 12 > 11 = |11|}\)
6. \(\color{blue}{2 \times 4 – 7 = 8 – 7 = 1}\)

Pre-Algebra for Beginners 2026 The Ultimate Step by Step Guide to Preparing for the Pre-Algebra Test

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Rated 4.30 out of 5 based on 105 customer ratings

Satisfied 92 Students

Frequently Asked Questions

Can absolute value ever be negative?

No. Absolute value measures distance, and distance is always zero or positive. So \(\color{blue}{|x| \ge 0}\) for every real number \(\color{blue}{x}\).

What is the difference between absolute value and the opposite of a number?

The opposite of a positive number is negative (\(\color{blue}{-3}\) is the opposite of 3). Absolute value always gives a non-negative result: \(\color{blue}{|3| = 3}\) and \(\color{blue}{|-3| = 3}\). They are not the same thing.

Why does absolute value matter on the GED?

Absolute value appears in real-world contexts such as distance traveled, temperature changes, and financial gains or losses — situations where only the magnitude of a change matters, not its direction.

Related Topics

• Absolute Value of Rational Numbers
• Solve Word Problems of Absolute Value and Integers
• Opposite Integers
• Using Number Lines to Represent Integers
• Using Number Lines to Compare and Order Rational Numbers