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tan(3*x+pi/6)<0 inequation

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tan|3*x + --| < 0
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\tan{\left(3 x + \frac{\pi}{6} \right)} < 0

tan(3*x + pi/6) < 0

Detail solution

Given the inequality:

\tan{\left(3 x + \frac{\pi}{6} \right)} < 0

To solve this inequality, we must first solve the corresponding equation:

\tan{\left(3 x + \frac{\pi}{6} \right)} = 0

Solve:
Given the equation

\tan{\left(3 x + \frac{\pi}{6} \right)} = 0

- this is the simplest trigonometric equation

with the change of sign in 0

We get:

\tan{\left(3 x + \frac{\pi}{6} \right)} = 0

This equation is transformed to

3 x + \frac{\pi}{6} = \pi n + \operatorname{atan}{\left(0 \right)}

3 x + \frac{\pi}{6} = \pi n

, where n - is a integer
Move

\frac{\pi}{6}

to right part of the equation
with the opposite sign, in total:

3 x = \pi n - \frac{\pi}{6}

Divide both parts of the equation by

3

x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}

x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}

This roots

x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}

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(\frac{\pi n}{3} - \frac{\pi}{18}\right) + - \frac{1}{10}

\frac{\pi n}{3} - \frac{\pi}{18} - \frac{1}{10}

substitute to the expression

\tan{\left(3 x + \frac{\pi}{6} \right)} < 0

\tan{\left(3 \left(\frac{\pi n}{3} - \frac{\pi}{18} - \frac{1}{10}\right) + \frac{\pi}{6} \right)} < 0

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

the solution of our inequality is:

x < \frac{\pi n}{3} - \frac{\pi}{18}

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