sin(x)<=(-1/2) inequation

Given the inequality:
$$\sin{\left(x \right)} \leq - \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = - \frac{1}{2}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$x = 2 \pi n - \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
This roots
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \frac{\pi}{6}\right) - \frac{1}{10}$$
=
$$2 \pi n - \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} \leq - \frac{1}{2}$$
$$\sin{\left(2 \pi n - \frac{\pi}{6} - \frac{1}{10} \right)} \leq - \frac{1}{2}$$

    /1    pi\        
-sin|-- + --| <= -1/2
    \10   6 /        

one of the solutions of our inequality is:
$$x \leq 2 \pi n - \frac{\pi}{6}$$

 _____           _____          
      \         /
-------•-------•-------
       x_1      x_2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 2 \pi n - \frac{\pi}{6}$$
$$x \geq 2 \pi n + \frac{7 \pi}{6}$$