ctg(x-П)<√21 inequation

Given the inequality:
$$\cot{\left(x - \pi \right)} < \sqrt{21}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(x - \pi \right)} = \sqrt{21}$$
Solve:
Given the equation
$$\cot{\left(x - \pi \right)} = \sqrt{21}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}$$
Or
$$x = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}$$
, where n - is a integer
$$x_{1} = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}$$
$$x_{1} = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}$$
This roots
$$x_{1} = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \operatorname{acot}{\left(\sqrt{21} \right)}\right) - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \operatorname{acot}{\left(\sqrt{21} \right)}$$
substitute to the expression
$$\cot{\left(x - \pi \right)} < \sqrt{21}$$
$$\cot{\left(\left(\pi n - \frac{1}{10} + \operatorname{acot}{\left(\sqrt{21} \right)}\right) - \pi \right)} < \sqrt{21}$$

    /1        /  ____\\     ____
-cot|-- - acot\\/ 21 /| < \/ 21 
    \10               /   

but

    /1        /  ____\\     ____
-cot|-- - acot\\/ 21 /| > \/ 21 
    \10               /   

Then
$$x < \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}$$
no execute
the solution of our inequality is:
$$x > \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}$$

         _____  
        /
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       x_1