HomeCalculatorsMathAlgebra & EquationsAbsolute Value Inequality Calculator
Math calculator

Absolute Value Inequality Calculator

Solve |ax + b| ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Linear Equation. Inside the branches is an interval; outside is a union. Not quadratic. Not a second engine. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 18/18 tests · Production surface contract 4/4 · v1.5.0
Input interpretation
Enter values to calculate.
Result
—
Model
The solution set of |ax + b| ○ rhs on ℝ for ○ in {<,≤,>,≥}.
Scope
Degree 1 only
Verification
Engine tested · 18/18 tests · Production surface contract 4/4 · Specification checked · v1.5.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
7 golden · 3 boundary · 8 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.5.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

inside, rhs > 0(lo, hi) or [lo, hi]
outside, rhs > 0(−∞, lo) ∪ (hi, ∞)
rhs < 0ℝ or ∅
iSame engine as /linear-equation. Equation page is /absolute-value. Not |ax²+bx+c|. Not CAS.

How to use

1

Enter a, b, ○, and rhs

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

2

Read the solution set

(−7, 4), or a union of rays, or ℝ / ∅. Linear inequalities without absolute value stay at /linear-inequality.

Example calculations

Common configurations with formula and result.

ϟ

Between the branches

|2x + 3| < 11

−7 < x < 4
(-7, 4)
ϟ

Outside the branches

|2x + 3| > 11

opens up
(-∞, -7) ∪ (4, ∞)
ϟ

Negative rhs

|2x + 3| < −1

never
∅

Absolute Value Inequality calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
|ax + b| ○ rhs with ○ in {<,≤,>,≥} is an interval, a union of rays, a point, ℝ, or ∅. This page is mode=abs_inequality on math.algebra.linear_equation. rhs < 0 is ℝ or ∅. a = 0 is ℝ or ∅. Not a second engine. Not quadratic. Not CAS.
What it calculates
The solution set of |ax + b| ○ rhs on ℝ for ○ in {<,≤,>,≥}.
Inputs
  • a
  • b?
  • rhs
  • op
Outputs
  • interval
  • lo
  • hi
  • status
Formula
|ax + b| ○ rhs
Assumptions
  • Degree 1 only
  • Real coefficients
  • Not quadratic
  • Not an equation
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • missing a or rhs → MISSING_REQUIRED_INPUT
  • unknown op → INVALID_MODE
  • unknown mode (including cas, quadratic) → INVALID_MODE
  • rhs < 0 → ℝ or ∅
  • a = 0 → ℝ or ∅
Example
a=2 b=3 rhs=11 op=lt → (-7, 4)
Validation cases

2 published on this page · 18/18 tests · Production surface contract 4/4 · View evidence

  • a=2 b=3 rhs=11 op=lt → (-7, 4)
  • mode=cas a=2 b=3 rhs=11 op=lt → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.5.0

Other calculators in this family: Absolute Value Calculator, Column Space Calculator, Condition Number Calculator, Cubic Equation Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear Equation Calculator, Linear Inequality Calculator . Explore all Algebra & Equations.

→ Absolute Value Calculator Solve |ax + b| = rhs. Same engine as Linear Equation. Two solutions when rhs > 0; empty when rhs < 0. Not quadratic. Not a second engine. Not CAS. Runs locally. › → Column Space Calculator Column space (image) of a small real square matrix from RREF pivot columns. Same engine as Linear System. Trivial image is {0}. Left-nullspace is a sibling. Runs locally. › → Condition Number Calculator κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix. Same engine as Linear System. Singular matrices return infinite. Not a 2-norm product. Runs locally. › → Cubic Equation Calculator Cardano / trigonometric closed form for a real cubic. Same engine as Polynomial Roots. Three real roots or one real plus a conjugate pair. Not Ferrari. Not a second cubic engine. Runs locally. › → Least Squares Calculator Linear least squares for a small real square system. Full rank uses QR; rank-deficient uses SVD. Same engine as Linear System. Not Statistics regression. Runs locally. › → Left Nullspace Calculator Left-nullspace (cokernel) of a small real square matrix as ker(Aᵀ). Same engine as Linear System. Trivial left kernel is {0}. Not Jordan form. Runs locally. › → Linear Equation Calculator Solve ax + b = rhs for x. Identity and inconsistent cases when a = 0. Inequality /linear-inequality. Absolute value /absolute-value. Not the quadratic formula. Runs locally. › → Linear Inequality Calculator Solve ax + b ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Linear Equation. Negative a flips the inequality. Not quadratic. Not a second engine. Runs locally. ›
}

Frequently asked questions

Key distinctions behind the calculation.

Is this a second linear engine?

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

Why is a negative right-hand side empty or all of ℝ?

Absolute value is never negative. |ax + b| < −1 is empty; |ax + b| > −1 is ℝ.

Where is |ax + b| = rhs?

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