Answer
The corresponding values of $x$ are $$\{\frac{\pi}{3}, \pi, \frac{4\pi}{3}\}$$
Work Step by Step
1) The question asks for solutions over the interval $[0,2\pi)$, meaning that
$$x\in[0,2\pi)$$
or $$0\le x\lt2\pi$$
2) Now your work leads to $$2x=\frac{2\pi}{3}, 2\pi, \frac{8\pi}{3}$$
To find corresponding values of $x$ from $2x$, we divide both sides by $2$
$$x=\frac{\pi}{3}, \pi, \frac{4\pi}{3}$$
3) Finally, it is crucial to compare the values of $x$ in 2) with the range found in 1).
We see that all three values of $x=\frac{\pi}{3}, \pi, \frac{4\pi}{3}$ fit in the range of $x\in[0,2\pi)$. None would be eliminated as a result.
The corresponding values of $x$ are $$\{\frac{\pi}{3}, \pi, \frac{4\pi}{3}\}$$
