Rational equation solver

Rational Equation Calculator

Solve supported rational equations with x in one or two linear denominators, including mixed constants and sums of fractions, while keeping excluded values visible.

Try an example

What rational equations can I enter?

I use the rational solver when x appears in a denominator. It supports linear numerators, mixed constant terms, and up to two distinct linear variable denominators when clearing them produces an equation of degree two or less.

Examples include 1/x + 1 = 2, x/(x - 2) + 3 = 5/(x - 2), and 2/(x + 3) + 3/(x - 2) = 2. More complicated inputs that would create a cubic or require three different variable denominators are rejected instead of guessed.

Domain first, then a common denominator

I write down the values that make any denominator zero before changing the equation. The solver then multiplies through by the common denominator and solves the resulting linear or quadratic equation.

Every candidate is checked in the original rational equation. A value that makes a denominator zero or fails the original equation is removed even if it appears during the cleared-denominator algebra.

Try an example

Type or paste an equation, solve it, inspect the transformations, and use verification and the graph instead of relying on the final value alone.

1/x + 1 = 2
x/(x - 2) + 3 = 5/(x - 2)
2/(x + 3) + 3/(x - 2) = 2
x/(x - 1) = 1/(x - 1)

Worked examples

Follow the same steps the calculator uses for three equations in this category.

1/x + 1 = 2

x = 1

  1. Exclude values that make a denominator zero: x≠0

    1/x+1=2 → x≠0

  2. Cross-multiply to remove the rational denominators

    1/x+1=2 → x+1=2x

  3. Subtract 1 from both sides

    -x+1=0 → -1x=-1

  4. Divide both sides by -1

    -1x=-1 → x=1

Verified

x/(x - 2) + 3 = 5/(x - 2)

x = 11/4

  1. Exclude values that make a denominator zero: x≠2

    x/(x-2)+3=5/(x-2) → x≠2

  2. Cross-multiply to remove the rational denominators

    x/(x-2)+3=5/(x-2) → 4x-6=5

  3. Add 11 to both sides

    4x-11=0 → 4x=11

  4. Divide both sides by 4

    4x=11 → x=2.75

Verified

2/(x + 3) + 3/(x - 2) = 2

x = (3-√145)/4, x = (3+√145)/4

  1. Exclude values that make a denominator zero: x≠-3, x≠2

    2/(x+3)+3/(x-2)=2 → x≠-3, x≠2

  2. Cross-multiply to remove the rational denominators

    2/(x+3)+3/(x-2)=2 → 5x+5=2x^2+2x-12

  3. Calculate the discriminant: D = 145

    -2x^2+3x+17=0 → D=145

  4. Substitute the coefficients into the quadratic formula

    D=145 → x=(-3±√145)/-4

  5. Simplify the formula to get the real roots

    x=(-3±√145)/-4 → x=-2.260398644698 or x=3.760398644698

Verified

Common mistakes

  • This variable-denominator form is outside the supported rational-equation patterns.
  • The equation contains division by zero.
  • Substitute each answer into the original equation and compare both sides.

Frequently asked questions

What does this calculator support?
Handle supported linear numerators and denominators, keep domain restrictions visible, and verify candidates.
How does it verify an answer?
Substitute each answer into the original equation and compare both sides.
Which equations are outside the supported range?
Cubic and higher-degree polynomials, logarithmic, exponential, trigonometric, nested-radical, and multiple-absolute-value equations remain outside the supported solver scope.

Verification

Substitute each answer into the original equation and compare both sides.

Current solver scope

Cubic and higher-degree polynomials, logarithmic, exponential, trigonometric, nested-radical, and multiple-absolute-value equations remain outside the supported solver scope.

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