Absolute Value — Complete Explanation
The absolute value |x| of a number x is its distance from zero on the number line — always a non-negative quantity. |3| = 3 and |−3| = 3, because both are 3 units from zero. This simple idea has powerful consequences in solving equations, inequalities, and in defining concepts like error, deviation, and distance in mathematics and physics.
This guide explains the definition, geometric meaning, algebraic properties, and step-by-step methods for solving equations and inequalities involving absolute values, plus the shape of the absolute value function and common applications.
Formula
|x| = x if x ≥ 0, −x if x < 0 | |x−a| = distance from x to a on the number line
Definition and Geometric Meaning
The absolute value of x is defined piecewise: |x| = x when x ≥ 0, and |x| = −x when x < 0. The −x for negative x makes the result positive: |−5| = −(−5) = 5.
Geometric meaning: |x| is the distance from x to 0 on the number line. More generally, |x − a| is the distance between x and a on the number line. |x − 3| = 5 means x is exactly 5 units from 3, so x = 8 or x = −2.
Examples: |7| = 7. |−7| = 7. |0| = 0. |π − 3| ≈ 0.14. |−√2| = √2.
The sign function: Every nonzero x can be written as x = |x| · sgn(x), where sgn(x) = +1 if x > 0 and sgn(x) = −1 if x < 0. The absolute value strips the sign; the sign function carries it.
Key Properties of Absolute Value
Six fundamental properties govern all absolute value computations:
| Property | Formula | Example |
|---|---|---|
| Non-negativity | |x| ≥ 0 | |−5| = 5 ≥ 0 |
| Identity | |x| = 0 ↔ x = 0 | The only number with |x|=0 is 0 |
| Symmetry | |x| = |−x| | |3| = |−3| = 3 |
| Product | |xy| = |x||y| | |−2 × 3| = |−2||3| = 6 |
| Quotient | |x/y| = |x|/|y| | |−6/2| = 6/2 = 3 |
| Triangle inequality | |x+y| ≤ |x|+|y| | |3+(−5)| = 2 ≤ 3+5 = 8 |
| Power even | |x²| = x² = |x|² | Always non-negative |
| Square root | √(x²) = |x| | √(9) = 3 = |±3| |
Solving Absolute Value Equations — |expression| = c
An equation |expression| = c has two cases (provided c ≥ 0):
Case 1: expression = c
Case 2: expression = −c
If c < 0, there is no solution (absolute value is never negative).
Example 1: |x − 3| = 5. Case 1: x − 3 = 5 → x = 8. Case 2: x − 3 = −5 → x = −2. Solutions: x = 8 or x = −2.
Example 2: |2x + 1| = 7. Case 1: 2x + 1 = 7 → x = 3. Case 2: 2x + 1 = −7 → x = −4. Solutions: x = 3 or x = −4.
Example 3 — No solution: |x + 2| = −3. No solution — absolute value cannot be negative.
Example 4 — Compound: |x² − 4| = 0. Only solution: x² − 4 = 0 → x = ±2.
Verify: Always substitute both solutions back into the original equation.
Absolute Value Inequalities — |x| < a (AND type)
The inequality |x| < a (or |x| ≤ a) means x is within distance a of 0 — between −a and a.
Rule: |expression| < a ↔ −a < expression < a.
Example 1: |x − 2| < 5. Rewrite: −5 < x − 2 < 5. Add 2: −3 < x < 7. Solution: (−3, 7).
Example 2: |2x + 3| ≤ 1. Rewrite: −1 ≤ 2x + 3 ≤ 1. Subtract 3: −4 ≤ 2x ≤ −2. Divide by 2: −2 ≤ x ≤ −1. Solution: [−2, −1].
Geometric interpretation: |x − a| < r means x is within radius r of a — an open interval of width 2r centred at a.
Absolute Value Inequalities — |x| > a (OR type)
The inequality |x| > a means x is farther than a from 0 — either x > a or x < −a. The solution is two separate intervals.
Rule: |expression| > a ↔ expression > a OR expression < −a.
Example 1: |x + 1| > 4. Case 1: x + 1 > 4 → x > 3. Case 2: x + 1 < −4 → x < −5. Solution: (−∞, −5) ∪ (3, ∞).
Example 2: |3x − 6| ≥ 9. Case 1: 3x − 6 ≥ 9 → x ≥ 5. Case 2: 3x − 6 ≤ −9 → x ≤ −1. Solution: (−∞, −1] ∪ [5, ∞).
Memory trick: < (less than) → a single interval "between" (AND). > (greater than) → two separate intervals "outside" (OR).
The Absolute Value Function f(x) = |x|
The function f(x) = |x| produces a V-shaped graph with vertex at the origin. For x ≥ 0 the graph coincides with y = x (slope +1); for x < 0 it coincides with y = −x (slope −1).
Transformations:
f(x) = |x − h| + k: shifts the vertex to (h, k).
f(x) = a|x − h| + k: scales the V by factor |a|. If a < 0, the V opens downward.
Example: f(x) = 2|x − 3| − 1. Vertex at (3, −1), slopes ±2, opens upward.
Domain: (−∞, ∞) — absolute value is defined for all real x.
Range: [0, ∞) for f(x) = |x|; [k, ∞) if a > 0 for general form; (−∞, k] if a < 0.
Not differentiable at vertex: f(x) = |x| has a "corner" at x = 0 — it is continuous but not differentiable there. This is important in calculus.
Applications of Absolute Value
Error and tolerance in manufacturing: A part must be 50 mm ± 0.5 mm. Acceptable range: |x − 50| ≤ 0.5, which gives 49.5 ≤ x ≤ 50.5.
Distance between points on a number line: The distance between a and b is |a − b|. Distance between −4 and 7: |−4 − 7| = |−11| = 11.
Mean absolute deviation (MAD): In statistics, MAD = (1/n)Σ|xᵢ − x̄| measures average distance from the mean. Less sensitive to outliers than variance.
Absolute error: If the true value is T and measured value is M, absolute error = |M − T|.
L1 norm in machine learning: LASSO regression minimises the sum of absolute values of coefficients, producing sparse models.