Solving a Rational Equation

Solving a rational equation means noting the excluded values, clearing denominators with the LCD, solving the cleared equation, and checking every answer in the original.

Solve 1/x + 1/2 = 3/4. Excluded value: x = 0. The LCD is 4x; multiplying every term gives 4 + 2x = 3x, so x = 4. Check in the original: 1/4 + 1/2 = 3/4 — true, and 4 is not excluded, so it stands. Now the cautionary run: x/(x − 2) = 2/(x − 2) clears to x = 2, but 2 is the excluded value itself — the cleared equation offered an answer the original forbids. Write no solution, not x = 2 — the audit outranks the algebra that produced it.