Absolute Value Properties
The absolute value rules and properties including triangle inequality, product rule, and solving absolute value equations.
Definition
The absolute value of a number is its distance from zero on the number line. It is always non-negative. For example, |−7| = 7 and |7| = 7.
Core Properties
| Property | Formula |
|---|---|
| Non-negativity | |a| ≥ 0 for all real a |
| Symmetry | |−a| = |a| |
| Zero property | |a| = 0 if and only if a = 0 |
| Product rule | |ab| = |a| × |b| |
| Quotient rule | |a/b| = |a| / |b| (b ≠ 0) |
| Power rule | |a²| = a² = |a|² |
| Square root connection | √(a²) = |a| |
Triangle Inequality
The absolute value of a sum is at most the sum of the absolute values. Equality holds when a and b have the same sign (or one is zero).
The reverse triangle inequality is also useful:
Solving Absolute Value Equations
For |x| = c where c > 0:
For |x| = c where c < 0: No solution exists (absolute value cannot be negative).
Solving Absolute Value Inequalities
For |x| < c (where c > 0):
For |x| > c (where c > 0):
Example 1
Solve |2x − 5| = 9
Set up two equations: 2x − 5 = 9 or 2x − 5 = −9
First equation: 2x = 14, so x = 7
Second equation: 2x = −4, so x = −2
x = 7 or x = −2
Example 2
Solve |3x + 1| < 8
Rewrite as a compound inequality: −8 < 3x + 1 < 8
Subtract 1: −9 < 3x < 7
Divide by 3: −3 < x < 7/3
Solution: −3 < x < 7/3 (approximately −3 < x < 2.33)
When to Use It
Absolute value appears throughout mathematics wherever distance or magnitude is involved.
- Solving absolute value equations and inequalities in algebra
- Distance problems on a number line or in coordinate geometry
- Error analysis — the absolute error is |measured − actual|
- Defining norms and metrics in linear algebra and analysis
- Complex number modulus is the absolute value generalized to the complex plane