An equation that contains at least one rational expression.
NoteRational equations often require finding a common denominator for all the terms involved. The process for solving a rational equation often produces extra solutions that need to be rejected.
Simple Rational Equations
The simplest rational equations are ones that already have a common denominator.
$$\frac{2x+5}{(x+2)(x-3)} = \frac{13}{(x+2)(x-3)}$$
Since these have the same denominator already, the denominators can be ignored to create the equation with just the numerators.
$$2x+5 = 13$$
NoteThis is only possible because the denominators are the same. If the denominators are different, this process will not work.
NoteBe careful when finding a common denominator. Make sure to multiply each term by the correct factor to get the common denominator.
Extraneous Solutions
When solving rational equations, it's important to check for extraneous solutions. These are solutions that satisfy the equation after clearing the denominators, but do not satisfy the original equation.
NoteAlways check your solutions in the original equation to make sure they are not extraneous.
Self review1. Solve the equation $\frac{3}{x} = \frac{2}{x+1} + \frac{1}{x(x+1)}$. 2. Solve the equation $\frac{1}{x-2} + \frac{1}{x+2} = \frac{2}{x^2-4}$. 3. Solve the equation $\frac{2x}{x+1} = \frac{3}{x} + \frac{1}{x(x+1)}$.
