Solving Rational Equations
Clear the denominators, then check for solutions that break the original.
The explanation
To solve an equation with fractions, multiply every term by the LCD to clear them, then solve normally.
The extra step that matters: check your answers.
Multiplying by an expression containing x can create solutions that do not actually work. These are extraneous solutions, and they are not arithmetic mistakes — they are a genuine side effect of the method.
If your answer makes any original denominator zero, discard it. If both answers do, the equation has no solution.
Write down the restrictions *before* you solve. Then checking is instant.
Multiplying both sides by the LCD is only an equivalence transformation where the LCD is nonzero. At values making it zero, the step is multiplication by zero, which can introduce roots that satisfy the cleared equation but not the original — the definition of an extraneous solution.
The reliable procedure: factor all denominators, state the excluded values, multiply through by the LCD, solve the resulting polynomial equation, then discard any root that is excluded.
Proportions a/b = c/d may be cross multiplied, which is the LCD method specialised to two fractions.
The same extraneous-root phenomenon appears with radical equations, where squaring both sides is likewise non-reversible, and for the same structural reason: applying a non-injective operation to both sides can enlarge the solution set. In both cases checking is part of the method rather than an optional verification.
Worked example
Solve 1/(x − 3) + 1 = 6/(x² − 9)... first note x ≠ ±3.
- Factor: x² − 9 = (x − 3)(x + 3); LCD = (x − 3)(x + 3).
- Multiply through: (x + 3) + (x² − 9) = 6.
- x² + x − 6 = 6 → x² + x − 12 = 0 → (x + 4)(x − 3) = 0.
- Candidates x = −4 and x = 3, but x = 3 is excluded.
Answer: x = −4 only (x = 3 is extraneous).
Common mistakes
- Reporting an extraneous root because the algebra produced it.
- Multiplying only the fraction terms by the LCD and skipping the whole numbers.