Solving Rational Equations

A rational equation is one with x in a denominator, such as 3 ÷ (2x + 5) − 1/8 = 3/8, or 1/x + 1/3 = 1 ÷ (3x) + 1. UK textbooks usually say “an equation with x in the denominator”; rational equation is the American name. You can tell one by looking underneath the fraction lines: if a letter is there, it is rational.

Five steps. Note what x cannot be: in 3 ÷ (2x + 5) − 1/8 = 3/8, x cannot be −2.5. Isolate the x terms: add 1/8 to both sides to get 3 ÷ (2x + 5) = 1/2. Clear the denominator: multiply both sides by (2x + 5), giving 3 = (2x + 5) ÷ 2. Repeat if needed: multiply by 2, so 6 = 2x + 5. Solve: 2x = 1, x = 1/2. Then check.

Multiply both sides of the equation by the denominator, so it cancels on the side with the fraction. If two terms have different denominators, multiply by something that clears both at once: with 1/x and 1 ÷ (3x), multiply everything by 3x. In the video’s second example that turns the equation into 3 − 1 = 2x, so x = 1.

Dividing by zero has no meaning, so any value of x that makes a denominator zero is not allowed. Find those values before you start. For 3 ÷ (2x + 5), the denominator is zero when 2x + 5 = 0, so x cannot be −2.5; for 1/x, x cannot be 0. If your solution turns out to be one of these, it is not a solution.

Because clearing denominators can hide a mistake, and because the answer must not be a forbidden value. Substitute it into the original equation and work out both sides. For x = 1 in 1/x + 1/3 = 1 ÷ (3x) + 1: the left side is 1 + 1/3 = 4/3 and the right side is 1/3 + 1 = 4/3, so x = 1 is correct.