Rational Functions

Solving Rational Equations

Solving equations containing rational expressions, checking for extraneous solutions.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Asymptotes & Holes

Clearing Denominators Without Losing Track

An equation like 1/x + 1/(x−2) = 3/(x(x−2)) has variables buried in three different denominators. Before reading on, think about a strategy for eliminating every fraction at once — and reconsider, given what you already know about extraneous solutions from radical and logarithmic equations, whether clearing denominators might carry the same kind of risk.

Multiplying every term by the least common denominator (LCD) clears every fraction in one step, turning the equation into an ordinary polynomial equation. But this step carries the exact same danger as squaring a radical equation: multiplying by an expression that could equal 0 (at the excluded x-values that originally made a denominator 0) isn't fully reversible, and can smuggle in an extraneous solution.

Worked Example — Solving a Rational Equation

Solve 1/x + 1/(x − 2) = 3/(x(x − 2)). The LCD is x(x − 2). Multiply every term by it: (x − 2) + x = 3. Simplify: 2x − 2 = 3, so 2x = 5, giving x = 5/2. Check: x = 5/2 doesn't make either original denominator 0 (x ≠ 0 and x ≠ 2), so it's valid.

Worked Example — Identifying and Rejecting an Extraneous Solution

Solve (x)/(x − 3) = 3/(x − 3) + 2. Multiply every term by the LCD, x − 3: x = 3 + 2(x − 3). Simplify: x = 3 + 2x − 6 = 2x − 3. Solving: −x = −3, so x = 3. But x = 3 makes the original denominators, x − 3, equal to 0 — undefined. x = 3 is extraneous; this equation has no valid solution.

Worked Example — A Work-Rate Application

One pipe fills a pool in 6 hours alone; a second pipe fills it in 4 hours alone. Working together, how long does it take? Each pipe's rate is a fraction of the pool per hour: 1/6 and 1/4. Working together for t hours: t/6 + t/4 = 1. Multiply by the LCD, 12: 2t + 3t = 12, so 5t = 12, giving t = 12/5 = 2.4 hours.

Tip

Before multiplying by the LCD, note every x-value that would make any original denominator 0 — those values are automatically excluded from the solution set, no matter what the cleared equation seems to say.

Common Mistakes

Multiplying only some terms of the equation by the LCD, missing a term on either side.
Every single term in the equation, on both sides, must be multiplied by the same LCD — missing even one term breaks the equation's balance.
Forgetting to check a rational equation's solution against the original denominators, the same oversight that causes trouble with radical and logarithmic equations.
Always identify which x-values make any original denominator 0 before finalizing an answer, and reject any solution that matches one of them.

Key Takeaways

Multiplying by the least common denominator clears every fraction from a rational equation at once.
This step can introduce extraneous solutions at any x-value that made an original denominator 0.
Rational equations model real situations like combined work rates, using each contributor's rate as a fraction per unit time.

Summary

Rational equations complete a recurring theme across this course: radical, logarithmic, and rational equations all risk extraneous solutions whenever the solving process isn't fully reversible. The next unit shifts entirely, from continuous functions to sequences of individual terms.

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