Solving Rational Equations & Inequalities
A rational equation has variables in denominators. The strategy is to clear the fractions by multiplying every term by the common denominator, leaving an ordinary equation.
One caution: multiplying can introduce extraneous solutions that make an original denominator zero. Always check your answers back in the original equation.
Clear the denominators
Multiply every term by the least common denominator. This cancels the fractions and leaves a polynomial equation you can solve normally.
Solve the resulting equation by the usual methods — combining like terms, factoring, or the quadratic formula if needed.
Check for extraneous solutions
Any solution that makes an original denominator zero is extraneous — it must be thrown out, because the original expression is undefined there.
So after solving, substitute each answer into the original denominators. Keep only the ones that keep every denominator nonzero.
Worked examples
Example 1: clear and solve
Solve .
Answer:
Example 2: spotting extraneous
Why would be rejected in an equation with ?
Answer: It is extraneous
Example 3: rejecting an extraneous solution
Solve .
Answer:
Try one yourself
Common questions
How do I clear the fractions?
Multiply every term by the least common denominator. That cancels all the denominators, leaving a polynomial equation.
What is an extraneous solution?
A value that satisfies the cleared equation but makes an original denominator zero, so it cannot be a real solution.
Do I always need to check?
Yes — checking against the original denominators is the only way to catch extraneous solutions.
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