Absolute Value Calculator
Calculate absolute value, solve absolute value equations |ax+b| = c, and evaluate expressions with absolute values. Find the distance from zero on the number line.
∑ Algebra📐 |x| = x if x ≥ 0, -x if x < 0🔢 Math
Mode
Value or a (coefficient)
b (constant, for solve mode)
c (right side, for solve mode)
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Absolute Value Calculator | — | |x| = x if x ≥ 0, -x if x < 0 | non-negative |
Step-by-Step Examples
Example 1
Evaluate
|−7| = ?
- −7 < 0, so |−7| = −(−7) = 7
- Distance from 0 is 7
✓ |-7| = 7
Example 2
Solve Equation
|2x + 3| = 5.
- Case 1: 2x+3=5 → x=1
- Case 2: 2x+3=−5 → x=−4
- Both satisfy the equation
✓ x = 1 or x = −4
Example 3
No Solution
|x + 2| = −3.
- Absolute value is always ≥ 0
- −3 < 0: no solution exists
✓ No solution
Real-World Applications
Geometry
Absolute value computes distances, which are always non-negative.
Statistics
Mean absolute deviation uses |xi − mean| for each data point.
Programming
abs() function appears in nearly every programming language.
Error Analysis
Absolute error = |measured − actual|. Relative error = |error|/|actual|.
Common Mistakes to Avoid
⚠️
Absolute value of negative gives positive
|-5| = 5, not -5. The output is always non-negative.
⚠️
Equation |x| = c has two solutions
When c > 0: x = c and x = −c. When c = 0: x = 0 only. When c < 0: no solution.
⚠️
|a + b| ≠ |a| + |b| in general
Triangle inequality: |a+b| ≤ |a|+|b|. Equality only when a and b have the same sign.
Frequently Asked Questions
What is absolute value? ▾
The distance of a number from zero on the number line. Always non-negative. |5| = 5, |−5| = 5, |0| = 0.
What is the geometric meaning? ▾
In 1D, |a−b| is the distance between a and b. In 2D, the equivalent is Euclidean distance.
How do I solve absolute value inequalities? ▾
|ax+b| < c: −c < ax+b < c (compound inequality). |ax+b| > c: ax+b < −c OR ax+b > c (two separate inequalities).
What is the triangle inequality? ▾
|a+b| ≤ |a|+|b|. This generalizes to vectors: |u+v| ≤ |u|+|v|. Equality holds when vectors point in same direction.
What is the absolute value function's graph? ▾
The V-shaped graph with vertex at origin. f(x) = |x| has slope +1 for x>0 and slope −1 for x<0. Not differentiable at x=0.
Related Math Calculators
Formula Explorer connections
Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.