1. **State the problem:** Solve the equation $$4 - \frac{x-9}{8} = \frac{x}{22} - \frac{1}{2}$$ for $x$. 2. **Rewrite the equation:** $$4 - \frac{x-9}{8} = \frac{x}{22} - \frac{1}{2}$$ 3. **Isolate fractions and constants:** Move all terms involving $x$ to one side and constants to the other. 4. **Multiply both sides by the least common denominator (LCD):** The denominators are 8, 22, and 2. The LCD is $$\text{lcm}(8,22,2) = 88$$. Multiply every term by 88: $$88 \times 4 - 88 \times \frac{x-9}{8} = 88 \times \frac{x}{22} - 88 \times \frac{1}{2}$$ 5. **Simplify each term:** $$352 - 11(x-9) = 4x - 44$$ 6. **Distribute and simplify:** $$352 - 11x + 99 = 4x - 44$$ Combine like terms on the left: $$451 - 11x = 4x - 44$$ 7. **Bring all $x$ terms to one side and constants to the other:** $$451 + 44 = 4x + 11x$$ $$495 = 15x$$ 8. **Solve for $x$:** $$x = \frac{495}{15}$$ 9. **Simplify the fraction:** $$x = \frac{\cancel{495}^{33}}{\cancel{15}^{1}} = 33$$ **Final answer:** $$\boxed{33}$$