clearing denominators

Clearing denominators means wiping the fractions out of an equation by multiplying every term on both sides by a common denominator. The goal is to trade a messy equation full of fractions for a clean polynomial equation that has none — fractions are harder to manipulate than whole expressions, so you remove them up front.

Concretely: find the least common denominator of all the fractions, multiply both entire sides by it, and let each denominator cancel against the LCD. What remains is a polynomial equation you can solve by familiar methods. For 1/x + 1/2 = 3/4, multiplying by 4x clears everything and leaves 4 + 2x = 3x.

Two honest cautions. First, this move is legal only because you multiply both sides equally. Second — and this is the price of the technique — multiplying by something containing the variable can introduce extraneous solutions, so any answer you find must be checked back in the original equation before you trust it.