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Absolute Value Equation and Inequality Calculator

Enter or paste a complete problem such as |2x − 3| = 5 or |3x + 1| ≤ 7. Solve for x and get exact answers, every algebra step, interval and set-builder notation, a number line, and an optional function graph.

Also solves common multi-step forms such as −2|3x + 1| + 4 ≥ −10, including isolation and both inequality reversals.

Enter an absolute value statement

Type one absolute-value group and one relation. Supported: |ax+b| ? c, p|ax+b|+q ? r, or the same with the absolute-value side on the right. Use integers, finite decimals, or fractions.

Current scope: one linear expression inside one absolute-value group. Multiple groups such as |x|+|x−1|=3, variables outside the group, and nonlinear interiors such as |x²−1|=3 are identified but not solved.

|2x − 3| = 5
Try an exact statement

Solution, notation, and visuals

Solution summary
x = −1 or x = 4
Solution set
{−1, 4}
Exact root(s)−1 and 4
Isolated form|2x − 3| = 5
Endpoints−1 and 4 (included)
Check|2(−1) − 3| = 5 and |2(4) − 3| = 5 ✓

Number line

Step-by-step solution

The algebra steps will appear here.

Formulas and decision rules

Absolute value is distance from zero. Its piecewise definition is:

|u|={uif u≥0−uif u<0

Direct equation roots

For a ≠ 0 and c > 0:

x=c−ba or x=−c−ba

Center and radius

For |ax+b| < c, the solution lies within radius c/|a| of center −b/a. Greater-than relations select the outside rays.

Inside: AND

|u|<c⇔−c<u<c

Outside: OR

|u|>c⇔u<−c or u>c

Threshold decision table

Relationc < 0c = 0c > 0
=No solutionOne root: u = 0Two cases: u = ±c
<No solutionNo solution−c < u < c (AND)
≤No solutionu = 0−c ≤ u ≤ c (AND)
>All real xu ≠ 0u < −c or u > c (OR)
≥All real xAll real xu ≤ −c or u ≥ c (OR)

Method reference: OpenStax College Algebra 2e, §2.7 explains interval notation, inequality reversal, and absolute-value inequalities.

Worked examples for every case

=: two exact roots — |2x−3|=5
  1. Split: 2x−3=−5 or 2x−3=5.
  2. Add 3: 2x=−2 or 2x=8.
  3. Divide by 2: x=−1 or x=4.
  4. Check: both substitutions give absolute value 5.
<: inside, open endpoints — |3x+1|<7
  1. −7<3x+1<7.
  2. Subtract 1: −8<3x<6.
  3. Divide by 3: −8/3<x<2.
  4. Interval: (−8/3,2).
≤: inside, closed endpoints — |x−4|≤2
  1. −2≤x−4≤2.
  2. Add 4: 2≤x≤6.
  3. Interval: [2,6]; both boundaries are included.
>: outside, open endpoints — |x+2|>3
  1. x+2<−3 or x+2>3.
  2. Subtract 2: x<−5 or x>1.
  3. Interval: (−∞,−5) ∪ (1,∞).
≥ and negative a — |−2x+4|≥6
  1. −2x+4≤−6 or −2x+4≥6.
  2. −2x≤−10 or −2x≥2.
  3. Divide by −2 and reverse both signs: x≥5 or x≤−1.
Fractional coefficient — |(3/4)x−1|=5/2
  1. (3/4)x−1=−5/2 or (3/4)x−1=5/2.
  2. (3/4)x=−3/2 or (3/4)x=7/2.
  3. Multiply by 4/3: x=−2 or x=14/3.
One solution and zero threshold — |x−3|=0
  1. An absolute value is zero only when its interior is zero.
  2. x−3=0, so x=3.
No solution — |x|=−2
  1. |x|≥0 for every real x.
  2. It cannot equal −2, so the solution set is ∅.
All real numbers — |x|≥0
  1. Absolute value is nonnegative by definition.
  2. The statement is true for every real x: (−∞,∞).
Constant expression (a=0) — |0x+3|≥2
  1. |0x+3|=3 for every x.
  2. 3≥2 is true, so every real number works.

Common mistakes and quick corrections

Forgetting the negative case

|x|=4 gives x=4 and x=−4.

Using OR for an inside interval

|x|<3 means −3<x<3 (AND), not two outside rays.

Not reversing a sign

−2x>6 becomes x<−3 after division by −2.

Including strict endpoints

|x|<2 uses open endpoints: (−2,2).

Ignoring a negative threshold

|x|<−1 has no solution because |x|≥0.

Mixing brackets and parentheses

x≥2 is [2,∞): include 2 with a bracket, never bracket infinity.

Distance and tolerance applications

Comfortable temperature

Within 4°F of 72°F is |T−72|≤4, so 68≤T≤76.

Manufacturing tolerance

A part within 0.5 mm of 10 mm is |x−10|<0.5, so 9.5<x<10.5.

Distance from a landmark

At least 3 km from position 8 is |x−8|≥3, so x≤5 or x≥11.

Methodology, verification, privacy, and limits

Authorship: Starlight Tools. No independent credentialed mathematics reviewer is currently listed. The transparent calculation methodology follows the absolute-value and inequality rules cited above. Last methodology review: 31 August 2026.

Test evidence: the page’s automated self-checks cover two-root, one-root, no-solution, all-real-number, all five relations, negative outer and inner coefficients, fractional coefficients, zero and negative thresholds, constant expressions, and a right-side absolute-value group. A failed self-check is reported in the browser console rather than hidden.

Numbers are converted to reduced fractions and processed with BigInt arithmetic. Each numeric part is limited to 30 digits. Scientific and repeating-decimal notation are unsupported; use an exact fraction. Multiple absolute-value groups, nonlinear interiors, and variables outside the absolute-value group are outside this calculator’s one-group linear scope.

All solving happens locally in your browser. The calculator does not transmit or store your equation or inequality.

Absolute value solver FAQs

How do you solve an absolute value equation?

For positive c, split |u| = c into u = -c or u = c, then solve both linear equations. A zero threshold gives one case, and a negative threshold gives no real solution.

How do you solve an absolute value inequality?

For positive c, less-than relations become one compound inequality between -c and c joined by AND. Greater-than relations become two outside inequalities joined by OR. Inclusive symbols include the endpoints.

Why can an absolute value problem have no solution?

An absolute value cannot be negative. A constant absolute-value expression can also make the comparison false for every real x.

Can the solution be all real numbers?

Yes. For example, |x| ≥ 0 is always true. If a = 0, the expression is constant and the original comparison is either true for every real x or false for all of them.

Can I enter fractions and decimals?

Yes. Use integers, finite decimals, or fractions such as -3/4. Exact fractions are reduced automatically.

Is this absolute value solver private?

Yes. It solves locally in your browser and does not send or store the values you enter.

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