Simple Rational Equations

Any fraction with x downstairs has a forbidden zone: whatever value of x makes that denominator zero isn't allowed, because division by zero has no answer. So before you even start solving, scan each denominator and note which x-values would zero it out. Those are excluded from the start, no matter what the algebra later suggests.

To actually solve, multiply both sides by whatever clears every denominator at once. For two single fractions set equal, like a/(x-b) = c/(x+d), that means cross multiplying: multiply each numerator by the other side's denominator. This works because multiplying both sides of an equation by the same nonzero quantity keeps it balanced — you're just rewriting the equation in a form without fractions. Once cleared, you're left with a linear or quadratic equation you already know how to solve.

The last step is the one people skip: check every solution against the excluded values you found at the start. If a solution matches one of them, it's rejected — it's called extraneous, a value produced by the algebra that doesn't actually satisfy the original equation. This matters more when clearing denominators turns the equation into a quadratic, since now you might get two candidate solutions, and only checking tells you if both survive, one does, or — in rarer cases — neither does. That link to quadratics is why this topic sits right next to quadratic solving: the fraction-clearing step is just the doorway into algebra you already have.