1. **State the problem:** Simplify the expression $$\frac{x^2 - 15x + 54}{x^2 - 14x + 48} \div \frac{1}{x - 8}$$. 2. **Rewrite the division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal, so $$\frac{x^2 - 15x + 54}{x^2 - 14x + 48} \times \frac{x - 8}{1}$$ 3. **Factor the quadratics:** - Factor numerator of first fraction: $$x^2 - 15x + 54 = (x - 6)(x - 9)$$ - Factor denominator of first fraction: $$x^2 - 14x + 48 = (x - 6)(x - 8)$$ 4. **Substitute the factored forms:** $$\frac{(x - 6)(x - 9)}{(x - 6)(x - 8)} \times \frac{x - 8}{1}$$ 5. **Cancel common factors:** Cancel $(x - 6)$ from numerator and denominator: $$\frac{\cancel{(x - 6)}(x - 9)}{\cancel{(x - 6)}(x - 8)} \times \frac{x - 8}{1} = \frac{x - 9}{x - 8} \times \frac{x - 8}{1}$$ Cancel $(x - 8)$ from numerator and denominator: $$\frac{x - 9}{\cancel{(x - 8)}} \times \frac{\cancel{(x - 8)}}{1} = x - 9$$ 6. **Final answer:** $$\boxed{x - 9}$$ This means the original expression simplifies to $x - 9$, with the restrictions that $x \neq 6$ and $x \neq 8$ to avoid division by zero.