1. **State the problem:** Solve the inequality $-2\ln(x-1) \geq 0$. 2. **Rewrite the inequality:** Divide both sides by $-2$. Since dividing by a negative number reverses the inequality, we get: $$\ln(x-1) \leq 0$$ 3. **Recall the property of logarithms:** $\ln(a) \leq 0$ means $a \leq 1$ because $\ln(1) = 0$ and $\ln(x)$ is increasing. 4. **Apply the property:** $$x - 1 \leq 1$$ 5. **Solve for $x$:** $$x \leq 2$$ 6. **Domain restriction:** Since $\ln(x-1)$ is defined only for $x-1 > 0$, we have: $$x > 1$$ 7. **Combine domain and solution:** $$1 < x \leq 2$$ **Final answer:** The solution set is $\boxed{(1, 2]}$.