tg3x<0 inequation

Given the inequality:
$$\tan{\left(3 x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(3 x \right)} = 0$$
Solve:
Given the equation
$$\tan{\left(3 x \right)} = 0$$
- this is the simplest trigonometric equation

with the change of sign in 0

We get:
$$\tan{\left(3 x \right)} = 0$$
This equation is transformed to
$$3 x = \pi n + \operatorname{atan}{\left(0 \right)}$$
Or
$$3 x = \pi n$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{\pi n}{3}$$
$$x_{1} = \frac{\pi n}{3}$$
This roots
$$x_{1} = \frac{\pi n}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\frac{\pi n}{3} + - \frac{1}{10}$$
=
$$\frac{\pi n}{3} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(3 x \right)} < 0$$
$$\tan{\left(3 \left(\frac{\pi n}{3} - \frac{1}{10}\right) \right)} < 0$$

tan(-3/10 + pi*n) < 0

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

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