1Cademy - Dividing $$\frac{x+9}{6-x} \div \frac{x^2-81}{x-6}$$

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Rational Expression

Example

Dividing $\frac{x+9}{6-x} \div \frac{x^2-81}{x-6}$

Divide $\frac{x+9}{6-x} \div \frac{x^2-81}{x-6}$ .

Step 1 — Rewrite as multiplication by the reciprocal. Flip the second fraction and change the division sign to multiplication:

$\frac{x+9}{6-x} \cdot \frac{x-6}{x^2-81}$

Step 2 — Factor the numerators and denominators completely. Recognize that $x^2 - 81$ is a difference of squares: $x^2 - 81 = (x-9)(x+9)$ . The expression becomes:

$\frac{x+9}{6-x} \cdot \frac{x-6}{(x-9)(x+9)}$

Step 3 — Multiply the numerators and denominators:

$\frac{(x+9)(x-6)}{(6-x)(x-9)(x+9)}$

Step 4 — Simplify by dividing out common factors. Cancel the common factor $(x+9)$ from the numerator and denominator. Notice that $(x-6)$ and $(6-x)$ are opposites — they differ only by a factor of $-1$ — so $\frac{x-6}{6-x} = -1$ . After canceling:

$-\frac{1}{x-9}$

This example illustrates a key technique: when a factor in the numerator and a factor in the denominator are opposites (such as $(x-6)$ and $(6-x)$ ), they divide to $-1$ rather than $1$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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