1. **State the problem:** Simplify the expression $-\frac{1}{2}[x - 2(x-1)]$. 2. **Apply the distributive property inside the brackets:** $$x - 2(x-1) = x - 2x + 2$$ 3. **Simplify inside the brackets:** $$x - 2x + 2 = -x + 2$$ 4. **Rewrite the original expression with the simplified bracket:** $$-\frac{1}{2}[-x + 2]$$ 5. **Distribute $-\frac{1}{2}$ to each term inside the bracket:** $$-\frac{1}{2} \times (-x) + -\frac{1}{2} \times 2$$ 6. **Calculate each term:** $$\frac{1}{2}x - 1$$ 7. **Final simplified expression:** $$\frac{1}{2}x - 1$$