1. **State the problem:** Solve the inequality $-x - \frac{5}{2} \geq -8 + \frac{1}{2}(-x + 1)$.\n\n2. **Write down the inequality:** $$-x - \frac{5}{2} \geq -8 + \frac{1}{2}(-x + 1)$$\n\n3. **Distribute the $\frac{1}{2}$ on the right side:** $$-x - \frac{5}{2} \geq -8 + \frac{1}{2} \cdot (-x) + \frac{1}{2} \cdot 1$$ $$-x - \frac{5}{2} \geq -8 - \frac{x}{2} + \frac{1}{2}$$\n\n4. **Combine constants on the right side:** $$-8 + \frac{1}{2} = -\frac{16}{2} + \frac{1}{2} = -\frac{15}{2}$$ So the inequality becomes: $$-x - \frac{5}{2} \geq -\frac{15}{2} - \frac{x}{2}$$\n\n5. **Add $\frac{x}{2}$ to both sides to get all $x$ terms on the left:** $$-x + \frac{x}{2} - \frac{5}{2} \geq -\frac{15}{2}$$ $$-\frac{2x}{2} + \frac{x}{2} - \frac{5}{2} \geq -\frac{15}{2}$$ $$-\frac{x}{2} - \frac{5}{2} \geq -\frac{15}{2}$$\n\n6. **Add $\frac{5}{2}$ to both sides to isolate the $x$ term:** $$-\frac{x}{2} \geq -\frac{15}{2} + \frac{5}{2}$$ $$-\frac{x}{2} \geq -\frac{10}{2}$$ $$-\frac{x}{2} \geq -5$$\n\n7. **Multiply both sides by $-2$ to solve for $x$.** Remember to reverse the inequality sign when multiplying by a negative number: $$x \leq (-5) \times (-2)$$ $$x \leq 10$$\n\n**Final answer:** $$\boxed{x \leq 10}$$