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

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

Divide both parts of the equation by log(2)

The equation is transformed to
$$\sin{\left(\frac{x}{2} \right)} = - \frac{1}{\log{\left(2 \right)}}$$
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.
$$x_{1} = 2 \pi + 2 \operatorname{asin}{\left(\frac{1}{\log{\left(2 \right)}} \right)}$$
$$x_{2} = - 2 \operatorname{asin}{\left(\frac{1}{\log{\left(2 \right)}} \right)}$$
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

$$\log{\left(2 \right)} \sin{\left(\frac{0}{2} \right)} < -1$$

0 < -1

but

0 > -1

so the inequality has no solutions