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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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 7/7 declared partitions (solve, inequality, abs, abs-inequality, rational, rational-inequality, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
The solution set of ax + b ○ rhs on ℝ for ○ in {<,≤,>,≥}, with an engine-linked number line.
Scope
Degree 1 only
Verification
Engine tested · Source checked · v1.5.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.5.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.discrete-algebra-cvp
Verification revision
2026-09-22.discrete-algebra-cvp · 3/3 property · digest b48961b2e1e8
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. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
9/9 golden · 3/3 CVP boundary · 6/6 invalid · 3/3 property · 5/5 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
7 legacy golden · 3 legacy boundary · legacy regression suite · 9/9 oracle-backed golden · 6/6 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.5.0 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.

a > 0x ○ (rhs − b) / a
a < 0flip ○
a = 0ℝ if b ○ rhs, else ∅
iSame engine as /linear-equation. Not ax²+bx+c. Not a system. Not CAS. Default op is <.

How to use

1

Enter a, b, rhs, and ○

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

2

Read the half-line

x < 4, or ℝ / ∅ when a = 0. The number line shows the solution set. Quadratic inequalities are out of this engine.

Example calculations

Common configurations with formula and result.

ϟ

a positive

2x + 3 < 11

x < 4
(-∞, 4)
ϟ

a negative

−2x + 3 < 11

x > −4
(-4, ∞)

Linear Inequality calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
ax + b ○ rhs with ○ in {<,≤,>,≥} is a half-line on ℝ when a ≠ 0. This page is mode=inequality on math.algebra.linear_equation. a < 0 flips the inequality. 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 {<,≤,>,≥}, with an engine-linked number line.
Inputs
  • a
  • b?
  • rhs
  • op
Outputs
  • interval
  • bound
  • status
  • sense
Formula
x ○ (rhs − b) / a, flipped when a < 0
Assumptions
  • Degree 1 only
  • Real coefficients
  • The Result Card includes an engine-linked number line of the published interval (open/closed endpoints follow ○). Same interval as REST/MCP.
  • Not quadratic
  • Not a system
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • missing a or rhs → MISSING_REQUIRED_INPUT
  • unknown op → INVALID_MODE
  • unknown mode (including quadratic, cas) → INVALID_MODE
  • a = 0 → ℝ or ∅
Example
a=2 b=3 rhs=11 op=lt → x<4
Validation cases

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

  • a=2 b=3 rhs=11 op=lt → x<4
  • mode=quadratic a=1 b=0 rhs=0 op=lt → 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
1.5.0

Background

Interpretation and common distinctions.

What does this linear inequality calculator compute?

ax + b ○ rhs with ○ in {<,≤,>,≥} is a half-line on ℝ when a ≠ 0.

The published formula or method is x ○ (rhs − b) / a, flipped when a < 0. 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 x < 4. 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 v1.5.0 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.

Other calculators in this family: Absolute Value Calculator, Absolute Value Inequality Calculator, Cubic Equation Calculator, Linear Equation Calculator, Polynomial Evaluate Calculator, Polynomial Roots Calculator, Quadratic Formula Calculator, Quadratic 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. › → 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. › → 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. › → 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. › → Polynomial Evaluate Calculator Horner nested multiplication for a real polynomial at a real x. Same engine as Polynomial Roots. Not factoring. Not a second engine. Not CAS. Runs locally. › → Polynomial Roots Calculator Find all roots of a real polynomial of degree 1–8. Closed form for linear, quadratic, and cubic (Cardano); Durand–Kerner otherwise. Cubic discovery /cubic-equation. Horner evaluation /polynomial-evaluate. Not the quadratic formula page. Runs locally. › → Quadratic Formula Calculator Solve ax² + bx + c = 0 with the quadratic formula. Inequality discovery /quadratic-inequality. Free online tool — no sign-up. Clear formulas and worked examples. Runs locally in your browser. › → Quadratic Inequality Calculator Solve ax² + bx + c ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Quadratic. Sign of a is the outside of the roots. Not cubic. Not a second engine. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second linear engine?

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

What if a is negative?

The inequality flips. −2x + 3 < 11 becomes x > −4.

Do you solve quadratic inequalities?

Yes. /calc/math/quadratic-inequality is mode=inequality on math.quadratic, not this page.