7sin(x/2)<-1 inequation

Given the inequality:
$$7 \sin{\left(\frac{x}{2} \right)} < -1$$
To solve this inequality, we must first solve the corresponding equation:
$$7 \sin{\left(\frac{x}{2} \right)} = -1$$
Solve:
Given the equation
$$7 \sin{\left(\frac{x}{2} \right)} = -1$$
- this is the simplest trigonometric equation

Divide both parts of the equation by 7

The equation is transformed to
$$\sin{\left(\frac{x}{2} \right)} = - \frac{1}{7}$$
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{7} \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{7} \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{7} \right)}$$
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{7} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
$$x_{1} = 4 \pi n - 2 \operatorname{asin}{\left(\frac{1}{7} \right)}$$
$$x_{2} = 4 \pi n + 2 \operatorname{asin}{\left(\frac{1}{7} \right)} + 2 \pi$$
$$x_{1} = 4 \pi n - 2 \operatorname{asin}{\left(\frac{1}{7} \right)}$$
$$x_{2} = 4 \pi n + 2 \operatorname{asin}{\left(\frac{1}{7} \right)} + 2 \pi$$
This roots
$$x_{1} = 4 \pi n - 2 \operatorname{asin}{\left(\frac{1}{7} \right)}$$
$$x_{2} = 4 \pi n + 2 \operatorname{asin}{\left(\frac{1}{7} \right)} + 2 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(4 \pi n - 2 \operatorname{asin}{\left(\frac{1}{7} \right)}\right) + - \frac{1}{10}$$
=
$$4 \pi n - 2 \operatorname{asin}{\left(\frac{1}{7} \right)} - \frac{1}{10}$$
substitute to the expression
$$7 \sin{\left(\frac{x}{2} \right)} < -1$$
$$7 \sin{\left(\frac{4 \pi n - 2 \operatorname{asin}{\left(\frac{1}{7} \right)} - \frac{1}{10}}{2} \right)} < -1$$

-7*sin(1/20 - 2*pi*n + asin(1/7)) < -1

one of the solutions of our inequality is:
$$x < 4 \pi n - 2 \operatorname{asin}{\left(\frac{1}{7} \right)}$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 4 \pi n - 2 \operatorname{asin}{\left(\frac{1}{7} \right)}$$
$$x > 4 \pi n + 2 \operatorname{asin}{\left(\frac{1}{7} \right)} + 2 \pi$$