1. **State the problem:** Simplify the expression $\frac{1}{2} (8x - 10) + 9x$. 2. **Recall the distributive property:** $a(b + c) = ab + ac$. We will distribute $\frac{1}{2}$ over the terms inside the parentheses. 3. **Distribute:** $$\frac{1}{2} (8x - 10) = \frac{1}{2} \times 8x - \frac{1}{2} \times 10 = 4x - 5$$ 4. **Rewrite the expression:** $$4x - 5 + 9x$$ 5. **Combine like terms:** $$4x + 9x = 13x$$ 6. **Final simplified expression:** $$13x - 5$$ This means the original expression simplifies to $13x - 5$.