1. **State the problem:** Solve the equation $$4 - x - \frac{9}{8} = \frac{x}{22} - \frac{1}{2}$$ for $x$. 2. **Rewrite the equation:** Combine like terms on the left side: $$4 - \frac{9}{8} - x = \frac{x}{22} - \frac{1}{2}$$ 3. **Simplify constants on the left:** Convert 4 to eighths: $$4 = \frac{32}{8}$$ So, $$\frac{32}{8} - \frac{9}{8} - x = \frac{x}{22} - \frac{1}{2}$$ 4. **Subtract fractions:** $$\frac{32 - 9}{8} - x = \frac{x}{22} - \frac{1}{2}$$ $$\frac{23}{8} - x = \frac{x}{22} - \frac{1}{2}$$ 5. **Add $x$ to both sides:** $$\frac{23}{8} = x + \frac{x}{22} - \frac{1}{2}$$ 6. **Add $\frac{1}{2}$ to both sides:** $$\frac{23}{8} + \frac{1}{2} = x + \frac{x}{22}$$ 7. **Find common denominator for left side:** $$\frac{23}{8} + \frac{4}{8} = \frac{27}{8}$$ 8. **Combine $x$ terms on right side:** $$x + \frac{x}{22} = x\left(1 + \frac{1}{22}\right) = x \cdot \frac{23}{22}$$ 9. **Rewrite equation:** $$\frac{27}{8} = x \cdot \frac{23}{22}$$ 10. **Solve for $x$ by dividing both sides:** $$x = \frac{27}{8} \div \frac{23}{22} = \frac{27}{8} \times \frac{22}{23}$$ 11. **Multiply fractions:** $$x = \frac{27 \times 22}{8 \times 23} = \frac{594}{184}$$ 12. **Simplify fraction by dividing numerator and denominator by 2:** $$x = \frac{\cancel{594}^\text{297}}{\cancel{184}^\text{92}}$$ 13. **Final answer:** $$x = \frac{297}{92}$$