Solving Rational Equations

A rational equation looks scary because x sits underneath a fraction bar. The fix is one move: multiply both sides by the least common denominator (LCD) of every term. That multiplication cancels each denominator exactly, because each fraction's bottom divides evenly into the LCD. What's left is a plain equation with no fractions at all, something you already know how to solve.

Once the denominators are gone, solve the remaining equation like any other: combine like terms, isolate x, factor if it's quadratic. But the job isn't done. Go back to the original equation and check two things: does your answer actually satisfy it, and does it make any original denominator zero? If it does, throw it out. Multiplying by the LCD can introduce extraneous solutions, numbers that solve the cleared-up equation but were never legal in the fraction form because they'd require dividing by zero.

This sits next to solving quadratic and polynomial equations in the algebra group, since clearing denominators often leaves you with exactly that: a quadratic to factor or a linear equation to isolate. The skill that makes this whole family work is recognizing the LCD quickly and multiplying every single term by it, not just one side.