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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 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)/(cx + d) ○ rhs on ℝ for ○ in {<,≤,>,≥}, excluding the pole, with an engine-linked number line.
Scope
Degree 1 numerator and denominator
Verification
Engine tested · Specification 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
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
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.

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. The number line marks the pole with an open ×.

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, with an engine-linked number line.
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
  • The Result Card includes an engine-linked number line of the published interval; the pole is an open mark and is never included. Same interval as REST/MCP.
  • 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 6/6 · 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, Cubic Equation Calculator, Linear Equation Calculator, Linear Inequality Calculator, Polynomial Evaluate Calculator, Polynomial Roots Calculator, Quadratic Formula 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. › → 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. › → 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. ›

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.