Absolute Value Definition

How to absolute value definition, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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.

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})

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Frequently Asked Questions

What is absolute value simple definition?

Absolute value is a number’s distance from zero on the number line, no matter which direction. It is written with two vertical bars, so |−5| is read as the absolute value of negative five. Because a distance can’t be negative, |−5| = 5 and |5| = 5. On the GED, many absolute value questions become easy once you picture that distance.

What is absolute value definition and example?

The absolute value of x, written |x|, equals x when x is zero or positive, and equals −x (the opposite of x) when x is negative. Examples: |12| = 12, |−3.5| = 3.5 and |0| = 0. When there is an expression inside the bars, simplify it first: |4 − 10| = |−6| = 6. Only then take the distance from zero.

Can absolute value be positive?

Yes. The absolute value of any nonzero number is always positive, and the absolute value of 0 is 0. It can never be negative, because it measures distance. Watch out for expressions like −|8|: the bars give 8 first, then the negative sign outside makes the result −8. A sign outside the bars is not part of the absolute value.

What is an absolute value function?

An absolute value function, such as f(x) = |x|, outputs how far the input is from zero. Its graph is a V shape with its lowest point, the vertex, at (0, 0). For example, f(−3) = 3 and f(3) = 3, so both points sit at the same height. A version like f(x) = |x − 2| + 1 moves the V 2 units right and 1 unit up.

Is absolute value important?

Yes. Absolute value shows up whenever only the size of a difference matters, such as distance, temperature change or how far a measurement is from a target. For instance, the change from −4°F to 9°F is |9 − (−4)| = 13 degrees. On GED Math it appears in simple evaluations, number line comparisons and word problems, so it is worth practicing.

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