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.
There is no analytical solution to the problem. 

Numerical solution suggests $x\ge 1.9345632$
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.