Solve $x-1 \ge \sin(x)$ I got as far as $-1 = \sin(x)-x$.
I don't know what to do next. Pretty sure I am forgetting some simplification rule.

Others have suggested there is no analytical solutions, however;
Let $x=\frac{5\pi }{2}$, then $\frac{5\pi }{2}-1\geq \sin(\frac{5\pi }{2})=1$. Just an example, you can go from here.
It seems there are infinite solutions.
