-2cos(x/2)<1 inequation

Given the inequality:
$$- 2 \cos{\left(\frac{x}{2} \right)} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \cos{\left(\frac{x}{2} \right)} = 1$$
Solve:
Given the equation
$$- 2 \cos{\left(\frac{x}{2} \right)} = 1$$
- this is the simplest trigonometric equation

Divide both parts of the equation by -2

The equation is transformed to
$$\cos{\left(\frac{x}{2} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$\frac{x}{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$\frac{x}{2} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$\frac{x}{2} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
This roots
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{4 \pi}{3}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{4 \pi}{3}$$
substitute to the expression
$$- 2 \cos{\left(\frac{x}{2} \right)} < 1$$
$$- 2 \cos{\left(\frac{2 \pi n - \frac{1}{10} + \frac{4 \pi}{3}}{2} \right)} < 1$$

     /  1    pi       \    
2*sin|- -- + -- + pi*n| < 1
     \  20   6        /    

one of the solutions of our inequality is:
$$x < 2 \pi n + \frac{4 \pi}{3}$$

 _____           _____          
      \         /
-------ο-------ο-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 2 \pi n + \frac{4 \pi}{3}$$
$$x > 2 \pi n - \frac{2 \pi}{3}$$