sin(3x)≥√2/2 inequation

Given the inequality:
$$\sin{\left(3 x \right)} \geq \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(3 x \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(3 x \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$3 x = 2 \pi n + \frac{\pi}{4}$$
$$3 x = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{2 \pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{\pi}{4}$$
$$x_{1} = \frac{2 \pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{\pi}{4}$$
This roots
$$x_{1} = \frac{2 \pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{\pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{2 \pi n}{3} + \frac{\pi}{12}\right) + - \frac{1}{10}$$
=
$$\frac{2 \pi n}{3} - \frac{1}{10} + \frac{\pi}{12}$$
substitute to the expression
$$\sin{\left(3 x \right)} \geq \frac{\sqrt{2}}{2}$$
$$\sin{\left(3 \left(\frac{2 \pi n}{3} - \frac{1}{10} + \frac{\pi}{12}\right) \right)} \geq \frac{\sqrt{2}}{2}$$

                             ___
   /  3    pi         \    \/ 2 
sin|- -- + -- + 2*pi*n| >= -----
   \  10   4          /      2  
                           

but

                            ___
   /  3    pi         \   \/ 2 
sin|- -- + -- + 2*pi*n| < -----
   \  10   4          /     2  
                          

Then
$$x \leq \frac{2 \pi n}{3} + \frac{\pi}{12}$$
no execute
one of the solutions of our inequality is:
$$x \geq \frac{2 \pi n}{3} + \frac{\pi}{12} \wedge x \leq \frac{2 \pi n}{3} + \frac{\pi}{4}$$

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