sin(x)>=(-sqrt(2))/2 inequation

Given the inequality:
$$\sin{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$x = 2 \pi n - \frac{\pi}{4}$$
$$x = 2 \pi n + \frac{5 \pi}{4}$$
, where n - is a integer
$$x_{1} = 2 \pi n - \frac{\pi}{4}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{4}$$
$$x_{1} = 2 \pi n - \frac{\pi}{4}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{4}$$
This roots
$$x_{1} = 2 \pi n - \frac{\pi}{4}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \frac{\pi}{4}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{\pi}{4} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\sin{\left(2 \pi n - \frac{\pi}{4} - \frac{1}{10} \right)} \geq \frac{\left(-1\right) \sqrt{2}}{2}$$

                             ___ 
    /1    pi         \    -\/ 2  
-sin|-- + -- - 2*pi*n| >= -------
    \10   4          /       2   
                          

but

                            ___ 
    /1    pi         \   -\/ 2  
-sin|-- + -- - 2*pi*n| < -------
    \10   4          /      2   
                         

Then
$$x \leq 2 \pi n - \frac{\pi}{4}$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \pi n - \frac{\pi}{4} \wedge x \leq 2 \pi n + \frac{5 \pi}{4}$$

         _____  
        /     \  
-------•-------•-------
       x1      x2