Method guideCalculus I · Algebra IIIntermediate

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.

LaTeX article Updated July 13, 2026

\frac{p(x)}{q(x)}=\frac{r(x)}{s(x)}

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.

q(x)s(x)\ne0

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.

d\left(\frac a p+q\right)=\frac{da}p+dq

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.

\frac ab=\frac cd\Rightarrow ad=bc

Worked example

Common mistakes

Clearing some denominators but not every term.
Finding restrictions after solving.
Keeping an excluded candidate.

Keep these ideas

State restrictions first.
Multiply every term by the LCD.
Check candidates in the original equation.