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Rational Equation Solver

Solve rational equations (fractions with variables) step-by-step. This solver finds domain restrictions, clears denominators using the LCD idea, solves the resulting equation, and removes extraneous solutions. Includes quick picks and a number-line visual.

Background

A rational equation contains one or more rational expressions like 1/(x−2). You can multiply both sides by a common denominator to eliminate fractions — but you must exclude any value that makes a denominator 0. That’s where extraneous solutions come from.

Enter equation

Allowed: + − * / ( ) ^. Examples: (x+1)/(x-2)=3, 1/x=2/(x+3). Exponents must be integers (0–12).

Options:

Chips fill the equation and solve immediately.

Result:

No results yet — enter an equation and click Solve.

How to use this solver

  • Type your equation using x, fractions, and parentheses.
  • Click Solve to see domain restrictions, clearing denominators, and solutions.
  • Any value that makes a denominator 0 is excluded.
  • Use “extraneous check” to see which candidates fail in the original equation.

How this solver works

  • Step 1: Find domain restrictions (denominators can’t be 0).
  • Step 2: Multiply both sides by a common denominator (LCD idea) to clear fractions.
  • Step 3: Solve the resulting equation to get candidate solutions.
  • Step 4: Substitute back into the original equation to remove extraneous solutions.

Formula & Equation Used

A rational equation often looks like: A(x)/B(x) = C(x)/D(x)

Domain restrictions: B(x) ≠ 0, D(x) ≠ 0

Clearing denominators (LCD idea): multiply both sides by a common denominator so the fractions disappear.

Example Problems & Step-by-Step Solutions

Example 1 — Two fractions (LCD)

Solve 1/(x-1) + 2/(x+1) = 1.

  1. Restrictions: x ≠ 1, x ≠ −1.
  2. Multiply by (x−1)(x+1) to clear fractions.
  3. Solve the resulting equation → candidates.
  4. Check candidates in the original equation to remove extraneous.

Example 2 — Proportion-style

Solve (x+2)/(x-1) = (x-3)/(x+1).

  1. Restrictions: x ≠ 1, x ≠ −1.
  2. Multiply by (x−1)(x+1) (or cross-multiply carefully).
  3. Solve for candidates, then verify in the original.

Example 3 — Extraneous solution appears

Solve x/(x-1) = 2/(x-1).

  1. Restriction: x ≠ 1 (denominator can’t be 0).
  2. Multiply both sides by (x−1) to clear denominators.
  3. You get x = 2 as a candidate solution.
  4. Check in the original equation: 2/(2−1) = 2 and 2/(2−1) = 2, so it works.
  5. Final answer: x = 2, with the domain restriction x ≠ 1.

Frequently Asked Questions

Q: What is an extraneous solution?

A value that appears after clearing denominators but fails in the original equation (often because it makes a denominator 0 or changes the equation when multiplied by 0).

Q: Why do we need domain restrictions?

Because rational expressions are undefined where the denominator equals 0.

Q: Do I always need the “LCD”?

You need a common denominator that clears every fraction. The LCD is the smallest choice, but multiplying by a larger common denominator still works (as long as you track restrictions).

Q: Does this solve complex solutions?

This version focuses on real solutions and real-domain restrictions.