Absolute Value
The absolute value of a real number is its distance from zero on the number line, always expressed as a non-negative quantity. It strips the sign from a number, so |x| = x if x ≥ 0 and |x| = −x if x < 0. Absolute value is fundamental in measuring distances, errors, and deviations in mathematics, physics, and engineering.
Key Formula
|x| = x if x ≥ 0, and |x| = -x if x < 0
LaTeX: |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}
| Symbol | Meaning | Unit |
|---|---|---|
| x | Any real number | dimensionless |
| |x| | Absolute value of x (non-negative) | dimensionless |
Worked Example
Problem
Solve |2x − 4| = 10.
Solution
Step 1: Set up two cases. Case 1: 2x − 4 = 10 → 2x = 14 → x = 7. Case 2: 2x − 4 = −10 → 2x = −6 → x = −3. Step 2: Both solutions are valid.
Answer
x = 7 or x = −3.
Properties of Absolute Value
| Property | Formula | Example |
|---|---|---|
| Non-negativity | |x| ≥ 0 | |−5| = 5 ≥ 0 |
| Definiteness | |x| = 0 iff x = 0 | |0| = 0 |
| Multiplicativity | |xy| = |x||y| | |2·(−3)| = 6 |
| Triangle inequality | |x+y| ≤ |x|+|y| | |3+(−5)| ≤ 3+5 |
| Symmetry | |−x| = |x| | |−7| = |7| = 7 |
Interactive Tools
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Related Terms
Linear Inequality
A linear inequality is a mathematical statement that compares two linear expressions using an inequality symbol such as <, >, ≤, or ≥. Unlike a linear equation, it defines a range of values that satisfy the condition rather than a single solution. Linear inequalities are widely used in optimisation problems, budgeting, and real-world constraints.
Distance Formula
The distance formula gives the straight-line (Euclidean) distance between two points in the Cartesian plane using their coordinates. It is derived directly from the Pythagorean theorem by treating the horizontal and vertical separations as legs of a right triangle. The formula extends naturally to three dimensions as d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) and is fundamental in analytic geometry, physics, and data science.
The word "absolute" derives from Latin "absolutus" (set free, unrestricted). The vertical bar notation |x| was introduced by Karl Weierstrass in 1841 to denote the modulus of a complex number, later adopted for real numbers.