Lesson 49: Rational Equations and Extraneous Solutions
A rational equation is an equation containing one or more rational expressions. When solving rational equations, some solutions may be extraneous (they make the original denominator zero) and must be excluded.
Steps to Solve Rational Equations
- Factor all denominators completely.
- Identify restrictions: values that make any denominator zero.
- Multiply both sides of the equation by the least common denominator (LCD) to eliminate fractions.
- Solve the resulting equation.
- Check each solution in the original equation and exclude any extraneous solutions.
Examples
Example 1: Solve 1/(x − 2) + 2/(x + 3) = 1
- LCD = (x − 2)(x + 3)
- Multiply both sides: (x + 3) + 2(x − 2) = (x − 2)(x + 3)
- Simplify: x + 3 + 2x − 4 = x² + x − 6 → 3x − 1 = x² + x − 6
- Rearrange: 0 = x² − 2x − 5
- Solve quadratic: x = 1 ± √6
- Check restrictions: x ≠2, −3 → both solutions valid
Example 2: Solve 1/(x − 1) + 1/(x + 2) = 0
- LCD = (x − 1)(x + 2)
- Multiply both sides: (x + 2) + (x − 1) = 0 → 2x + 1 = 0 → x = −1/2
- Check restrictions: x ≠1, −2 → solution valid
Practice Problems
- Solve: 2/(x − 3) − 1/(x + 1) = 1
- Solve: 1/(x + 2) + 3/(x − 1) = 2
- Solve: 4/(x² − 4) = 1/(x − 2)
- Solve: (x + 1)/(x − 3) = 2/(x − 3)
- Solve: 1/(x − 2) + 2/(x − 5) = 3/(x − 2)