rational equation

A rational equation is an equation in which at least one rational expression appears — a variable sits in a denominator somewhere. Examples are 1/x = 5, (x + 1)/(x - 2) = 3, and 1/x + 1/(x - 1) = 1. Solving them is where rational expressions stop being objects to simplify and become things to solve for.

The standard strategy is to clear the denominators: multiply both sides by the least common denominator so that every fraction disappears, leaving an ordinary polynomial equation you already know how to solve — often linear or quadratic. Then solve that simpler equation.

But there is a trap that makes the last step non-negotiable. Multiplying by the LCD can introduce values that satisfy the cleared equation yet make an original denominator zero. Such values are extraneous solutions, and they are not real solutions. You must check every candidate against the original equation and throw out any that hits an excluded value.