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Absolute Value Calculator

Solve absolute value equations in one variable. Find zero, one, two, or infinitely many solutions, preserve small roots, and share or export the full result.

Instant result
Result
—

Enter values to calculate.

Inputs—
Solutions—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all six published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all six published modes
Assurance
Core
Declared partition coverage
PASS · 7/7 declared partitions (invalid-domain, mode-solve, mode-inequality, mode-abs, mode-abs_inequality, mode-rational, mode-rational_inequality) · Matrix
Numerical scope
Listed independent full-pack vectors only; classification uses exact comparisons, no absolute epsilon.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
The real solutions of |ax + b| = rhs.
Scope
Degree 1 only
Verification
Engine tested · Source checked · v1.5.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all six published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.5.1 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.meta-metrics-v2
Verification revision
2026-10-28.meta-metrics-v2 · 1/1 property · digest f9b7ec60bdd7
Legacy regression
18/18 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
178/178 golden · 1/1 CVP boundary · 12/12 invalid · 1/1 property · 2/2 metamorphic · 178/178 cross-interface · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
7 legacy golden · 3 legacy boundary · legacy regression suite · 178/178 oracle-backed golden · 12/12 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.5.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

rhs > 0, a ≠ 0ax + b = ± rhs
rhs = 0, a ≠ 0x = −b / a
rhs < 0∅
iSame engine as /linear-equation. If a = 0, the equation is |b| = rhs: all real x when it holds, otherwise no solution. Not |ax²+bx+c|. Inequalities are /absolute-value-inequality. Not CAS.

How to use

1

Enter a, b, and rhs

|2x + 3| = 11 is a=2, b=3, rhs=11. The equation page stays at /linear-equation.

2

Read one or two roots

x = −7 and x = 4. rhs < 0 is empty. Inequalities are /absolute-value-inequality.

Example calculations

Common configurations with formula and result.

ϟ

Two solutions

|2x + 3| = 11

2x + 3 = ± 11
x = −7, 4
ϟ

Zero

|2x + 3| = 0

x = −3/2
x = −1.5
ϟ

Negative rhs

|2x + 3| = −1

never
∅

Absolute Value calculator specification

Version 1.5.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
|ax + b| = rhs splits into ax + b = rhs and ax + b = −rhs when rhs > 0. This page is mode=abs on math.algebra.linear_equation. rhs < 0 is empty. a = 0 is ℝ or ∅. Not a second engine. Not quadratic. Not CAS.
What it calculates
The real solutions of |ax + b| = rhs.
Inputs
  • a
  • b?
  • rhs
Outputs
  • x
  • x2
  • n
  • status
  • solutions
Formula
ax + b = ± rhs
Assumptions
  • Degree 1 only
  • Real coefficients
  • Not quadratic
  • Not an inequality
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • missing a or rhs → MISSING_REQUIRED_INPUT
  • unknown mode (including cas, quadratic) → INVALID_MODE
  • rhs < 0 → empty
  • a = 0 → ℝ or ∅
Example
a=2 b=3 rhs=11 → x = −7, 4
Validation cases

2 published on this page · 18/18 tests · Production surface contract 6/6 · View evidence

  • a=2 b=3 rhs=11 → -7, 4
  • mode=cas a=2 b=3 rhs=11 → INVALID_MODE
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • Margaret Stevens — The Maths Workbook (University of Oxford) — Chapter 1, §5: Inequalities and Absolute Value
    Supports: The piecewise definition of |x|, reversal of inequalities under a negative factor, and worked absolute-value equation and inequality solutions. The calculator applies these rules to its stated linear arguments and zero/negative right-hand-side cases. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version
1.5.1

Background

Interpretation and common distinctions.

What does this absolute value calculator compute?

|ax + b| = rhs splits into ax + b = rhs and ax + b = −rhs when rhs > 0.

The published formula or method is ax + b = ± rhs. The following references support the stated mathematics or method:

  • Margaret Stevens — The Maths Workbook (University of Oxford): The piecewise definition of |x|, reversal of inequalities under a negative factor, and worked absolute-value equation and inequality solutions. The calculator applies these rules to its stated linear arguments and zero/negative right-hand-side cases. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses |2x + 3| = 11 with 2x + 3 = ± 11. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.algebra.linear_equation v1.5.1 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical and input contract

Finite binary64 coefficients or strict decimal/scientific strings up to512 characters. Nonzero text rounding to zero rejects NUMERIC_UNDERFLOW. Exact rational decisions use the accepted binary64 coefficients, not arbitrary-precision raw decimals; no epsilon merges roots, zero coefficients, or poles. b defaults to0. In rational modes c and d form the denominator and cannot both be0. In other modes c/d are right-hand-side aliases and must agree with rhs.

The complete result includes solution ranges with open/closed endpoints. Endpoints round once to binary64. Overflow, nonzero underflow, and distinct exact boundaries collapsing to one binary64 value reject. Exact arithmetic is bounded to16384 numerator/denominator bits.178 independent Fraction predicate/open-cell full-pack references cover listed cases, with a1ULP fixed-vector ceiling and strict serialized topology; this is not a whole-domain/runtime proof. The figure may be omitted when boundaries cannot be separated faithfully; the numerical interval remains authoritative.

Typing calculates locally. Shared URLs and REST/MCP transmit inputs and may enter logs. CSV/JSON retain raw provided inputs and the complete unrounded result pack.

Other calculators in this family: Absolute Value Inequality Calculator, Cubic Equation Calculator, Linear Equation Calculator, Linear Inequality Calculator, Polynomial Evaluate Calculator, Polynomial Roots Calculator, Quadratic Formula Calculator, Quadratic Inequality Calculator . Explore all Algebra & Equations.

→ Absolute Value Inequality Calculator Solve absolute value inequalities with open or closed bounds. Inspect bounded intervals, unions, and degenerate cases, then share or export the solution set. › → Cubic Equation Calculator Solve real cubic equations with Cardano and trigonometric forms. Inspect all three roots, repeated roots, numerical limits, share inputs, and export results. › → Linear Equation Calculator Solve linear equations in one variable, including identities and inconsistent cases. Compare coefficient decisions, share inputs, or export the full result. › → Linear Inequality Calculator Solve linear inequalities with strict or inclusive bounds. Reverse the relation for negative coefficients, inspect the number line, and export the solution set. › → Polynomial Evaluate Calculator Evaluate real polynomials of degree 1–8 with exact rational Horner arithmetic and one binary64 rounding. Inspect numerical limits, share inputs, and export. › → Polynomial Roots Calculator Find all roots of real polynomials of degree 1–8, including complex and repeated roots. Check numerical limits, share inputs, and export complete results. › → Quadratic Formula Calculator Solve real quadratic equations with two, repeated, or complex roots. Inspect the discriminant, vertex, numerical limits, worked steps, and complete exports. › → Quadratic Inequality Calculator Find solution intervals for real quadratic inequalities. Choose strict or inclusive endpoints, inspect numerical limits, share raw inputs, and export results. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second linear engine?

No. It is mode=abs on math.algebra.linear_equation, the same engine as /calc/math/linear-equation.

Why is a negative right-hand side empty?

Absolute value is never negative, so |ax + b| = −1 has no real x.

Do you solve |ax + b| < c?

/calc/math/absolute-value-inequality is mode=abs_inequality on this same engine.