Overview
A rational expression is a quotient of polynomials and is defined only where its original denominator is nonzero. Factor before cancellation because only common factors cancel. To solve a rational equation, record excluded inputs, clear denominators on that domain, solve the resulting equation, and check candidates in the original equation.
Consider this example: For x ≠ −3, simplify (x² − 9)/(x + 3). Factor x² − 9 as (x − 3)(x + 3). Cancel the common nonzero factor x + 3 to obtain x − 3, while retaining x ≠ −3. Canceling the added 3 from individual terms would be invalid.
The method also handles a different case: Solve 1/(x − 1) + 1/(x + 1) = 3/4 over the real numbers. Exclude x = 1 and −1. Combine the fractions to 2x/(x² − 1) = 3/4. Cross-multiply: 8x = 3x² − 3, or 3x² − 8x − 3 = 0. Factoring gives (3x + 1)(x − 3) = 0, so x = −1/3 or 3. Neither is excluded, and substitution confirms both; 1 and −1 are denominator zeros, not solutions.
For a further application, consider: What is the real solution set of (x² − 4)/(x − 2) = 3x/(x + 1)? Exclude 2 and −1 before canceling x − 2. The equation becomes x + 2 = 3x/(x + 1). Multiply by x + 1: x² + 3x + 2 = 3x, so x² = −2. This has no real solutions.
Canceling terms in a sum is not canceling factors. Clearing denominators can create an apparent solution at a forbidden input; it must be rejected even if it solves the transformed polynomial equation. For a new quotient, list denominator zeros before simplifying. For a new equation, retain that list until every candidate is checked in the original fractions.