Absolute Value Equation & Inequality Solver
Equations · Inequalities · Double Absolute Values · Step-by-Step
Solve |ax+b| = c, less-than and greater-than inequalities, and double absolute value equations with full algebraic steps and a number line visualization.
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Number Line Visualization
Green = solution set
Solution in Multiple Formats
What Is Absolute Value?
The absolute value of a real number x, denoted |x|, measures its distance from zero on the number line — always a non-negative quantity. The formal definition is piecewise:
|x| = x if x ≥ 0 |x| = −x if x < 0
Equivalently, |x| = max(x, −x) = √(x²). The geometric interpretation is direct: |x| is the number of units between x and 0, so |5| = 5 and |−5| = 5. For expressions, |ax + b| measures the distance between ax + b and 0, making it central to tolerance, error analysis, and constraint problems.
Solving |ax + b| = c (Equation)
The key principle is that |expression| = c has solutions only when c ≥ 0. Split into two cases using the definition:
- Case 1 (positive branch): ax + b = c → x = (c − b) / a
- Case 2 (negative branch): ax + b = −c → x = (−c − b) / a
When a = 0 the equation becomes |b| = c, which is true for all x if |b| = c, and false for all x otherwise. Always verify each solution by substituting back.
Solving |ax + b| < c (Less-Than Inequality)
A less-than absolute value inequality means "the expression is within c units of zero." It is equivalent to the compound inequality:
−c < ax + b < c
Solve both parts simultaneously by isolating x. The solution is always a bounded open interval centred at x = −b/a with radius c/|a|. If c ≤ 0, there is no solution. The number line shows a shaded segment between two open endpoints.
Solving |ax + b| > c (Greater-Than Inequality)
A greater-than inequality means "the expression is more than c units from zero." Split into two separate inequalities:
ax + b > c OR ax + b < −c
The solution is a union of two rays: (−∞, x₁) ∪ (x₂, +∞). If c < 0, every real number is a solution since absolute value is always ≥ 0. The number line shows shading extending outward in both directions from two open points.
Solving |ax + b| = |cx + d| (Double Absolute Value)
When two absolute values are set equal, use the fact that |A| = |B| means A = B or A = −B:
- Case 1: ax + b = cx + d → (a − c)x = d − b
- Case 2: ax + b = −(cx + d) → (a + c)x = −d − b
Each case yields at most one solution. Special handling is needed when a = c (Case 1 becomes trivial) or a = −c (Case 2 becomes trivial).
Interval Notation Quick Reference
| Inequality | Meaning | Interval Notation | Number Line |
|---|---|---|---|
| |x| = 3 | x is 3 from 0 | {-3, 3} | Two filled dots |
| |x| < 3 | within 3 of 0 | (-3, 3) | Open-interval shade |
| |x| ≤ 3 | at most 3 from 0 | [-3, 3] | Closed-interval shade |
| |x| > 3 | more than 3 from 0 | (-∞,-3) ∪ (3,+∞) | Two outward rays |
| |x| ≥ 3 | at least 3 from 0 | (-∞,-3] ∪ [3,+∞) | Two outward closed rays |
Real-World Applications
- Engineering tolerances: A manufactured part is acceptable if |dimension − target| ≤ ε, directly an absolute value inequality.
- Error bounds: Measurement uncertainty is expressed as |measured − true| ≤ error margin.
- Temperature control: A thermostat keeps a room within ±2°C of the setpoint: |T − T₀| ≤ 2.
- Signal processing: Keeping a signal's deviation from a reference within a threshold.
- Quality control: Rejecting items whose weight differs from spec by more than a given amount: |w − w₀| > tol.
- Finance: Stop-loss rules: exit if |price − entry| > risk tolerance.