1. The problem is to simplify the expression $C - \frac{2}{3} \leq \sin(x)$.\n\n2. This is an inequality involving the sine function and a constant $C$. We want to understand the range of $C$ such that the inequality holds for some $x$.\n\n3. Recall that $\sin(x)$ ranges between $-1$ and $1$ for all real $x$.\n\n4. The inequality $C - \frac{2}{3} \leq \sin(x)$ means $\sin(x)$ is at least $C - \frac{2}{3}$.\n\n5. Since $\sin(x) \leq 1$, for the inequality to hold for some $x$, we must have $C - \frac{2}{3} \leq 1$.\n\n6. Solve for $C$: $$C - \frac{2}{3} \leq 1 \implies C \leq 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}.$$\n\n7. Therefore, the inequality holds for some $x$ if and only if $C \leq \frac{5}{3}$.\n\nFinal answer: $C \leq \frac{5}{3}$