Let $ f(x)=|x-\frac{\pi}{6}|+|x+\frac{\pi}{3}|,g(x)=arcsin\frac{x^3-x+2}{2}.$ $ g(x)$ dtfined, when $ -2\le x^3-x+2\le 2$. If $ x>\frac{\pi}{6}$, then $ f(x)>\frac{\pi}{2}\ge g(x)$ - no solution. If $ -\frac{\pi}{3}\le x\le \frac{\pi}{6}$, then ${ f(x)=\pi}{2}$, so $ x^3-x=0\to x=0 \ or \ x=-1$. If $ x<\frac{-\pi}{3}$, then $ f(x)=-\frac{\pi}{6}-2x>\frac{\pi}{2}$ - no solution. Only $ x=0,x=-1$ are solutions.