Absolute Value
Reference for absolute value |x|.
Covers equations |ax+b|=c, inequalities, distance on the number line, complex modulus, and properties with examples.
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The Formula
|x| = x if x ≥ 0, |x| = -x if x < 0
The absolute value of a number is its distance from zero, regardless of direction. It strips away the sign, always giving a non-negative result.
Variables
| Symbol | Meaning |
|---|---|
| |x| | Absolute value of x (always ≥ 0) |
| x | Any real number |
Example 1
Solve |2x - 6| = 10
Case 1: 2x - 6 = 10 → 2x = 16 → x = 8
Case 2: 2x - 6 = -10 → 2x = -4 → x = -2
x = 8 or x = -2
Example 2
Find the distance between -7 and 4 on the number line
Distance = |(-7) - 4| = |-11|
Distance = 11
When to Use It
Use absolute value when:
- Finding the distance between two numbers
- Expressing error or deviation (always positive)
- Solving equations and inequalities with absolute values
- Working with magnitudes regardless of direction
Key Notes
- |x| = c has two solutions (x = ±c) when c > 0, one solution (x = 0) when c = 0, and no solution when c < 0
- |x| < a means −a < x < a (a bounded interval); |x| > a means x < −a or x > a (two separate rays)
- f(x) = |x| is continuous everywhere but not differentiable at x = 0 — the graph has a sharp corner (cusp) at the origin
- In complex numbers, |z| = √(a² + b²) is the modulus — the distance from the origin in the complex plane
Key Notes
- Definition: |x| = x if x ≥ 0; |x| = −x if x < 0: The absolute value always returns a non-negative result. |−5| = 5; |5| = 5; |0| = 0. It removes the sign — measuring the magnitude, not the direction.
- Distance interpretation: |a − b| is the distance between a and b: On the number line, the distance between any two points is their absolute difference. |3 − 7| = |−4| = 4; |7 − 3| = |4| = 4. This interpretation extends to absolute value equations and inequalities.
- Solving |x − a| < r: This means x is within distance r of a: a − r < x < a + r. Solving |x − a| > r gives the complement: x < a − r or x > a + r. Absolute value inequalities always split into two cases based on the definition.
- Triangle inequality: |a + b| ≤ |a| + |b|: The magnitude of a sum never exceeds the sum of magnitudes. Equality holds when a and b have the same sign. This inequality appears in analysis, vector geometry, and metric space theory.
- Applications: Absolute value is used in error analysis (|measured − true|), magnitude of complex numbers (|a + bi| = √(a² + b²)), distance functions in machine learning and statistics (mean absolute error), signal amplitude, and solving equations and inequalities in algebra.