Absolute value describes how far a number is from zero on the number line, without considering direction. Because distance cannot be negative, the absolute value is always zero or positive. The notation uses vertical bars, as in |x|.
Absolute value formula
|x| = x when x ≥ 0
|x| = −x when x < 0
This piecewise definition means a non-negative number stays the same, while a negative number changes sign. For example, |7| = 7, |−7| = 7, and |0| = 0.
Rules and examples
- Non-negative: |x| ≥ 0 for every real number x.
- Opposites: |x| = |−x|.
- Product: |ab| = |a||b|.
- Quotient: |a ÷ b| = |a| ÷ |b| when b is not zero.
- Distance: the distance between a and b is |a − b|.
For example, the distance between −3 and 5 is |−3 − 5| = |−8| = 8 units.
How to evaluate absolute value
- Identify the expression inside the vertical bars.
- Evaluate that expression first if it contains arithmetic.
- If the result is negative, take its opposite; otherwise keep it unchanged.
- Check that the final value is not negative.
Absolute value equations
An equation such as |x| = 4 has two solutions, x = 4 and x = −4, because both numbers are four units from zero. An equation |x| = −4 has no real solution because absolute value cannot be negative. For more complicated expressions, consider each possible sign of the expression inside the bars and verify the resulting solutions.
Frequently asked questions
Can absolute value be negative?
No. The absolute value of a real number is always non-negative.
Is |−5| equal to −5?
No. It equals 5 because absolute value measures distance from zero.
How is absolute value used?
It is used to describe distances, magnitudes, differences, and errors without regard to direction.
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