Answer
$x=\frac{4\pi}{3}$
Work Step by Step
$2~cos\frac{x}{2}+1=0$
$cos\frac{x}{2}=-\frac{1}{2}$
$\frac{x}{2}=\frac{2\pi}{3}$ or $\frac{x}{2}=\frac{4\pi}{3}$
(with $0\lt\frac{x}{2}\lt2\pi$)
If $x$ is in the interval $[0,2\pi)$, then $\frac{x}{2}$ is in the interval $[0,\pi)$
The solution for $x$ is in the interval $[0,2\pi)$:
$\frac{x}{2}=\frac{2\pi}{3}$
$x=\frac{4\pi}{3}$
