Solving rational equations without accepting forbidden answers
List restrictions, multiply every term by the least common denominator, solve the resulting equation, and reject candidates outside the original domain.
How to recognize the method, run it, and know when it is the wrong choice.
Updated July 13, 2026Restrictions first, denominator clearing second, solution check last. Clearing fractions does not erase the original domain.
Restrictions come before algebra
Set every original denominator factor unequal to zero. This exclusion list becomes the final filter for candidate solutions.
If a later operation produces an excluded value, report that it is extraneous rather than quietly keeping it.
The LCD must multiply every term
Multiplying both sides by the least common denominator clears fractions because each denominator divides it. Terms without visible denominators are multiplied too.
Write the multiplier beside each term before canceling to avoid skipping a constant or distributing incompletely.
Check in the original equation
A candidate may solve the cleared polynomial but fail the rational equation because a denominator becomes zero. Substitution into the original form catches that.
Cross-multiplication is a shortcut only for one fraction equal to one fraction; it does not replace the general LCD method.
Solve 2/(x − 1) = 3/(x + 2).
Record original restrictions.
Multiply by (x − 1)(x + 2).
Solve the linear equation.
Check in the original equation.
Common mistakes
- Clearing some denominators but not every term.
- Finding restrictions after solving.
- Keeping an excluded candidate.
Three takeaways
- State restrictions first.
- Multiply every term by the LCD.
- Check candidates in the original equation.