1. **State the problem:** Simplify the rational expression $$\frac{x^2 - 7x - 18}{2x} \div (x - 9)$$. 2. **Rewrite the division as multiplication:** Dividing by a quantity is the same as multiplying by its reciprocal. $$\frac{x^2 - 7x - 18}{2x} \div (x - 9) = \frac{x^2 - 7x - 18}{2x} \times \frac{1}{x - 9}$$ 3. **Factor the numerator:** Factor the quadratic expression in the numerator. $$x^2 - 7x - 18 = (x - 9)(x + 2)$$ 4. **Substitute the factorization:** $$\frac{(x - 9)(x + 2)}{2x} \times \frac{1}{x - 9}$$ 5. **Cancel common factors:** The factor $(x - 9)$ appears in numerator and denominator. $$\frac{\cancel{(x - 9)}(x + 2)}{2x} \times \frac{1}{\cancel{(x - 9)}} = \frac{x + 2}{2x}$$ 6. **Final simplified expression:** $$\boxed{\frac{x + 2}{2x}}$$ This is the simplified form of the original rational expression.