-2sin2x<(3) inequation

Given the inequality:
$$- 2 \sin{\left(2 x \right)} < 3$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \sin{\left(2 x \right)} = 3$$
Solve:
Given the equation
$$- 2 \sin{\left(2 x \right)} = 3$$
- this is the simplest trigonometric equation

Divide both parts of the equation by -2

The equation is transformed to
$$\sin{\left(2 x \right)} = - \frac{3}{2}$$
As right part of the equation
modulo =

True

but sin
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
$$x_{1} = \frac{\pi}{2} + \frac{\operatorname{asin}{\left(\frac{3}{2} \right)}}{2}$$
$$x_{2} = - \frac{\operatorname{asin}{\left(\frac{3}{2} \right)}}{2}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$- 2 \sin{\left(0 \cdot 2 \right)} < 3$$

0 < 3

so the inequality is always executed