ctg(3x)≥-1 inequation

Given the inequality:
$$\cot{\left(3 x \right)} \geq -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(3 x \right)} = -1$$
Solve:
Given the equation
$$\cot{\left(3 x \right)} = -1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = \pi n + \operatorname{acot}{\left(-1 \right)}$$
Or
$$3 x = \pi n - \frac{\pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
get the intermediate answer:
$$x = \frac{\pi n}{3} - \frac{\pi}{12}$$
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{12}$$
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{12}$$
This roots
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{12}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{3} - \frac{\pi}{12}\right) - \frac{1}{10}$$
=
$$\frac{\pi n}{3} - \frac{\pi}{12} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(3 x \right)} \geq -1$$
$$\cot{\left(3 \left(\frac{\pi n}{3} - \frac{\pi}{12} - \frac{1}{10}\right) \right)} \geq -1$$

    /3    pi\      
-cot|-- + --| >= -1
    \10   4 /      

the solution of our inequality is:
$$x \leq \frac{\pi n}{3} - \frac{\pi}{12}$$

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