Basic operations with rational expressions

A rational expression is just a fraction whose parts contain variables. Adding or subtracting them works exactly like numeric fractions: rewrite each over a common denominator, then combine the numerators. When the denominators already match, you simply add the numerators, as in $\frac{3}{x} + \frac{5}{x} = \frac{8}{x}$.

When the denominators differ, factor them first. The factored forms reveal the least common denominator and let you rewrite each fraction over it before combining. For just two fractions you can also cross-multiply: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$.

To simplify or multiply rational expressions, factor every numerator and denominator and cancel the factors they share, for example $\frac{2x^2 - 18}{x - 3} = \frac{2(x-3)(x+3)}{x-3} = 2(x+3)$, which is valid when $x \ne 3$.

To solve an equation containing rational expressions, clear the denominators by multiplying every term by the common denominator. This turns it into an ordinary polynomial equation you can solve normally.

Combine $\frac{2x^2}{x + 2} - \frac{1}{x + 2}$. The denominators already match, so subtract the numerators: $\frac{2x^2 - 1}{x + 2}$.

Simplify $\frac{x^2 - 4}{x} \times \frac{3}{x + 2}$. Factor the numerator: $\frac{(x+2)(x-2)}{x} \times \frac{3}{x+2}$. Cancel the common $(x+2)$: $\frac{3(x-2)}{x}$.

Solve $\frac{4}{x} = \frac{x}{2x - 4}$ by clearing denominators (cross-multiplying): $4(2x - 4) = x \cdot x$, so $8x - 16 = x^2$, giving $x^2 - 8x + 16 = 0$.