Mister Exam
ctg(5x+(2pi)/(3))=<1 inequation
The solution
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/ 2*pi\ cot|5*x + ----| <= 1 \ 3 /
$$\cot{\left(5 x + \frac{2 \pi}{3} \right)} \leq 1$$
cot(5*x + (2*pi)/3) <= 1
Detail solution
Given the inequality:
$$\cot{\left(5 x + \frac{2 \pi}{3} \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(5 x + \frac{2 \pi}{3} \right)} = 1$$
Solve:
$$x_{1} = - \frac{\pi}{12}$$
$$x_{1} = - \frac{\pi}{12}$$
This roots
$$x_{1} = - \frac{\pi}{12}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\pi}{12} - \frac{1}{10}$$
=
$$- \frac{\pi}{12} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(5 x + \frac{2 \pi}{3} \right)} \leq 1$$
$$\cot{\left(5 \left(- \frac{\pi}{12} - \frac{1}{10}\right) + \frac{2 \pi}{3} \right)} \leq 1$$
but
Then
$$x \leq - \frac{\pi}{12}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{\pi}{12}$$
$$\cot{\left(5 x + \frac{2 \pi}{3} \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(5 x + \frac{2 \pi}{3} \right)} = 1$$
Solve:
$$x_{1} = - \frac{\pi}{12}$$
$$x_{1} = - \frac{\pi}{12}$$
This roots
$$x_{1} = - \frac{\pi}{12}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\pi}{12} - \frac{1}{10}$$
=
$$- \frac{\pi}{12} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(5 x + \frac{2 \pi}{3} \right)} \leq 1$$
$$\cot{\left(5 \left(- \frac{\pi}{12} - \frac{1}{10}\right) + \frac{2 \pi}{3} \right)} \leq 1$$
/1 pi\ tan|- + --| <= 1 \2 4 /
but
/1 pi\ tan|- + --| >= 1 \2 4 /
Then
$$x \leq - \frac{\pi}{12}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{\pi}{12}$$
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