tan(x/4)<-1 inequation

Given the inequality:
$$\tan{\left(\frac{x}{4} \right)} < -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(\frac{x}{4} \right)} = -1$$
Solve:
Given the equation
$$\tan{\left(\frac{x}{4} \right)} = -1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{4} = \pi n + \operatorname{atan}{\left(-1 \right)}$$
Or
$$\frac{x}{4} = \pi n - \frac{\pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 4 \pi n - \pi$$
$$x_{1} = 4 \pi n - \pi$$
This roots
$$x_{1} = 4 \pi n - \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(4 \pi n - \pi\right) + - \frac{1}{10}$$
=
$$4 \pi n - \pi - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(\frac{x}{4} \right)} < -1$$
$$\tan{\left(\frac{4 \pi n - \pi - \frac{1}{10}}{4} \right)} < -1$$

    /1    pi       \     
-tan|-- + -- - pi*n| < -1
    \40   4        /     

the solution of our inequality is:
$$x < 4 \pi n - \pi$$

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