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Rational Inequality Calculator

Solve (ax + b)/(cx + d) ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Linear Equation. Sign chart of N·D; the pole is excluded. 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)/(cx + d) ○ rhs on ℝ for ○ in {<,≤,>,≥}, excluding the pole.
Scope
Degree 1 numerator and denominator
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.

Clear (D² > 0)N·D ○ 0, N = (a − rhs·c)x − (rhs·d − b)
Critical pointsroot and pole x = −d/c
Polex ≠ −d / c
iSame engine as /linear-equation. Equation page is /rational-equation. Not a quadratic rational. Not CAS.

How to use

1

Enter a, b, c, d, ○, and rhs

(2x + 3)/(x + 1) < 1 is a=2, b=3, c=1, d=1, op=<, rhs=1. The equation page stays at /rational-equation.

2

Read the solution set

(−2, −1), or a union of rays, or ℝ \ {pole}. The pole is never included.

Example calculations

Common configurations with formula and result.

ϟ

Between root and pole

(2x + 3)/(x + 1) < 1

−2 < x < −1
(-2, -1)
ϟ

Outside

(2x + 3)/(x + 1) > 1

opens up
(-∞, -2) ∪ (-1, ∞)
ϟ

Identity except pole

(2x + 2)/(x + 1) ≤ 2

ℝ \ {−1}
ℝ \ {-1}

Rational Inequality calculator specification

Version 1.5.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
(ax + b)/(cx + d) ○ rhs with ○ in {<,≤,>,≥} is an interval, a union of rays, ℝ, punctured ℝ, or ∅. This page is mode=rational_inequality on math.algebra.linear_equation. The pole x = −d/c is never a solution. Identically 0 denominator is invalid. Not a second engine. Not quadratic. Not CAS.
What it calculates
The solution set of (ax + b)/(cx + d) ○ rhs on ℝ for ○ in {<,≤,>,≥}, excluding the pole.
Inputs
  • a
  • b?
  • c?
  • d?
  • rhs
  • op
Outputs
  • interval
  • lo
  • hi
  • pole
  • status
Formula
(ax + b)/(cx + d) ○ rhs
Assumptions
  • Degree 1 numerator and denominator
  • Real coefficients
  • Not quadratic
  • Not an equation
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • missing a or rhs → MISSING_REQUIRED_INPUT
  • c = d = 0 → INVALID_INPUT
  • unknown op → INVALID_MODE
  • unknown mode (including cas, quadratic) → INVALID_MODE
  • root equals pole → ℝ \ {pole} or ∅
Example
a=2 b=3 c=1 d=1 rhs=1 op=lt → (-2, -1)
Validation cases

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

  • a=2 b=3 c=1 d=1 rhs=1 op=lt → (-2, -1)
  • mode=cas a=2 b=3 c=1 d=1 rhs=1 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, Absolute Value Inequality Calculator, Column Space Calculator, Condition Number Calculator, Cubic Equation Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear Equation Calculator . Explore all Algebra & Equations.

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Frequently asked questions

Key distinctions behind the calculation.

Is this a second linear engine?

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

Why is the pole excluded?

cx + d = 0 makes the original fraction undefined. Multiplying by (cx+d)² does not put the pole back.

Where is (ax + b)/(cx + d) = rhs?

/calc/math/rational-equation is mode=rational on this same engine.