sin(1-2x)<(-sqrt(2))*1/2 inequation

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

Divide both parts of the equation by -1

The equation is transformed to
$$\sin{\left(2 x - 1 \right)} = \frac{\sqrt{2}}{2}$$
This equation is transformed to
$$2 x - 1 = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$2 x - 1 = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$2 x - 1 = 2 \pi n + \frac{\pi}{4}$$
$$2 x - 1 = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Move
$$-1$$
to right part of the equation
with the opposite sign, in total:
$$2 x = 2 \pi n + \frac{\pi}{4} + 1$$
$$2 x = 2 \pi n + 1 + \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$2$$
$$x_{1} = \pi n + \frac{\pi}{8} + \frac{1}{2}$$
$$x_{2} = \pi n + \frac{1}{2} + \frac{3 \pi}{8}$$
$$x_{1} = \pi n + \frac{\pi}{8} + \frac{1}{2}$$
$$x_{2} = \pi n + \frac{1}{2} + \frac{3 \pi}{8}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{8} + \frac{1}{2}$$
$$x_{2} = \pi n + \frac{1}{2} + \frac{3 \pi}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{8} + \frac{1}{2}\right) + - \frac{1}{10}$$
=
$$\pi n + \frac{\pi}{8} + \frac{2}{5}$$
substitute to the expression
$$\sin{\left(1 - 2 x \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\sin{\left(1 - 2 \left(\pi n + \frac{\pi}{8} + \frac{2}{5}\right) \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$

                             ___ 
    /  1   pi         \   -\/ 2  
-sin|- - + -- + 2*pi*n| < -------
    \  5   4          /      2   
                          

but

                             ___ 
    /  1   pi         \   -\/ 2  
-sin|- - + -- + 2*pi*n| > -------
    \  5   4          /      2   
                          

Then
$$x < \pi n + \frac{\pi}{8} + \frac{1}{2}$$
no execute
one of the solutions of our inequality is:
$$x > \pi n + \frac{\pi}{8} + \frac{1}{2} \wedge x < \pi n + \frac{1}{2} + \frac{3 \pi}{8}$$

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