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How Absolute Value Works
Absolute value measures distance from zero — it is always non-negative, regardless of sign.
Definition
|x| = x if x ≥ 0; |x| = −x if x < 0
Always returns a non-negative value. Distance from zero.
Equation |ax+b| = c
c < 0 → no solution; c = 0 → one; c > 0 → two
Split: ax+b = c or ax+b = −c, then solve each.
Inequality |ax+b| < c
−c < ax+b < c → bounded interval
Solution is a single connected region between two bounds.
Inequality |ax+b| > c
ax+b < −c or ax+b > c → two rays
Solution is two separate intervals extending to ±∞.
What Is Absolute Value?
The absolute value of a number is its distance from zero on the number line, regardless of direction. Written with vertical bars — |x| — it is always non-negative. Formally: |x| = x when x ≥ 0, and |x| = −x when x < 0. So |−7| = 7 and |7| = 7, because both −7 and 7 are exactly 7 units away from zero.
Absolute value appears throughout mathematics and engineering: tolerance bands in manufacturing (a dimension must be within ±0.5 mm of spec), error margins in statistics, distance calculations, signal processing, and optimization. Any time you care about magnitude but not sign, you're using the concept of absolute value.
How to Solve an Absolute Value Equation
To solve |ax + b| = c, first check the sign of c. If c < 0, there is no solution — the absolute value can never produce a negative number. If c = 0, there is exactly one solution: ax + b = 0, so x = −b/a.
When c > 0, split into two separate linear equations:
- Case 1: ax + b = c → solve for x
- Case 2: ax + b = −c → solve for x
This gives two solutions. For example, to solve |x + 3| = 5: Case 1 gives x + 3 = 5 → x = 2; Case 2 gives x + 3 = −5 → x = −8. Check both by substituting back: |2 + 3| = |5| = 5 ✓ and |−8 + 3| = |−5| = 5 ✓.
How to Solve Absolute Value Inequalities
The approach differs based on whether the operator is "less than" or "greater than":
|ax + b| < c (or ≤ c) gives a bounded interval. Rewrite as −c < ax + b < c, then divide through by a (flip the direction if a < 0). The solution is a single segment between two finite bounds, written as L < x < R or [L, R] in interval notation.
|ax + b| > c (or ≥ c) gives a union of two rays. Rewrite as ax + b < −c OR ax + b > c, then solve each separately. The solution is two disconnected intervals: (−∞, L) ∪ (R, +∞).
Special cases worth remembering: if c < 0 with a "<" operator, there is no solution (absolute value is always ≥ 0, so it can't be less than a negative). If c < 0 with a ">" operator, all real numbers are solutions (absolute value always exceeds any negative). These shortcuts save time on exams.
When the Absolute Value Has No Solution
A common exam trap is |expression| = −5 or |expression| < −2. Since |·| ≥ 0 always, these can never be true. No amount of solving for x will make an absolute value negative. Recognise these immediately and write "no solution" or ∅ without further work.
Similarly, |expression| ≥ 0 is always true — every real number satisfies it, so the solution is all of ℝ. The calculator detects and labels these fast cases explicitly.
Absolute Value on the Number Line
Every absolute value result corresponds to a region on the number line. A single computed value |x| is a dot. An equation |ax+b| = c produces up to two dots. An inequality with "<" produces a shaded segment (between the two boundary dots); an inequality with ">" produces two shaded rays extending to ±∞.
Open circles (○) mark strict boundaries (< or >); closed circles (●) mark inclusive boundaries (≤ or ≥). This matches standard interval notation: parentheses ( ) exclude endpoints, brackets [ ] include them.
Absolute Value FAQ
What does absolute value mean?
Absolute value is the distance of a number from zero on the number line, always expressed as a non-negative value. |x| = x when x is positive or zero, and |x| = −x when x is negative. So |-7| = 7 and |7| = 7 — both are 7 units away from zero.
How do you solve |ax + b| = c?
First check c: if c < 0, there is no solution. If c = 0, one solution: x = -b/a. If c > 0, split into two equations: ax+b = c and ax+b = -c. Solve each separately to get two solutions: x = (c-b)/a and x = (-c-b)/a. Always verify by substituting back into the original equation.
Why does |x| = c have two solutions when c > 0?
Because two distinct numbers are exactly c units from zero: c itself (to the right) and -c (to the left). The equation |x+3| = 5 means 'x+3 is 5 units from zero' — which is satisfied by both x+3 = 5 and x+3 = -5, giving x = 2 and x = -8.
How do you solve |x| < c?
Rewrite as a compound inequality: -c < x < c. The solution is all x between -c and c, written as the interval (-c, c). Use square brackets [-c, c] for ≤. If c ≤ 0, there is no solution because |x| is always ≥ 0.
How do you solve |x| > c?
Split into two cases: x < -c OR x > c. The solution is a union of two rays, written (-∞, -c) ∪ (c, +∞). Use ≤ and ≥ for the corresponding inclusive version. If c < 0, the answer is all real numbers because an absolute value always exceeds any negative number.
What happens when |ax + b| = 0?
There is exactly one solution: ax + b must equal zero, so x = -b/a. Zero is the only number whose absolute value is zero. This is a special case where the two cases of the equation merge into one (since 0 and -0 are the same point).
Can an absolute value equation have no solution?
Yes — when the right-hand side is negative. |ax+b| = -3 has no solution because an absolute value can never be negative. Similarly, |ax+b| < -1 has no solution (nothing can be less than -1 if it's already ≥ 0). These are quick no-algebra answers on exams.
What is absolute value used for in real life?
Absolute value appears in: engineering tolerances (a part must be within ±0.5 mm of the target size), statistics (mean absolute deviation measures average error regardless of direction), computer graphics (distance calculations), signal processing (magnitude of a signal), economics (price change magnitude), and navigation (how far off-course regardless of direction).
Related Calculators
Coordinate Geometry Calculator
Distance formula uses absolute differences — |x₂−x₁| and |y₂−y₁|.
Quadratic Formula Calculator
Solve ax²+bx+c=0 with two solutions — a related equation-solving pattern.
Percentage Calculator
Percentage change uses |original| internally to handle signed values.
Circle Calculator
Radius is always positive — an absolute distance from centre to edge.
Surface Area Calculator
Dimensions are always positive — rooted in the same non-negative constraint.