HomeCalculatorsMathAlgebra & EquationsRational Inequality Calculator
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Rational Inequality Calculator

Solve rational inequalities with exact coefficient decisions and excluded poles. Inspect open or closed solution intervals, share inputs, and export the set.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
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 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 · 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.

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 1.5.1 · 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
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.
  • University of Oxford Mathematical Institute — Algebra — Linear/quadratic inequalities and sign of factors
    Supports: Factoring polynomials and determining sign intervals, with explicit quadratic-inequality examples. For a rational inequality the calculator applies the same factor-sign rules to numerator and denominator separately and excludes denominator zeros; that rational extension is a derived application, not a claim that the page tests this solver. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version
1.5.1

Background

Interpretation and common distinctions.

What does this rational inequality calculator compute?

(ax + b)/(cx + d) ○ rhs with ○ in {<,≤,>,≥} is an interval, a union of rays, ℝ, punctured ℝ, or ∅.

The published formula or method is (ax + b)/(cx + d) ○ rhs. The following references support the stated mathematics or method:

  • University of Oxford Mathematical Institute — Algebra: Factoring polynomials and determining sign intervals, with explicit quadratic-inequality examples. For a rational inequality the calculator applies the same factor-sign rules to numerator and denominator separately and excludes denominator zeros; that rational extension is a derived application, not a claim that the page tests this solver. 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)/(x + 1) < 1 with −2 < x < −1. 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 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 absolute value equations in one variable. Find zero, one, two, or infinitely many solutions, preserve small roots, and share or export the full result. › → 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. ›

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.