Mister Exam
tan(3*x-pi/4)<=sqrt(3) inequation
The solution
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/ pi\ ___ tan|3*x - --| <= \/ 3 \ 4 /
$$\tan{\left(3 x - \frac{\pi}{4} \right)} \leq \sqrt{3}$$
tan(3*x - pi/4) <= sqrt(3)
Detail solution
Given the inequality:
$$\tan{\left(3 x - \frac{\pi}{4} \right)} \leq \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(3 x - \frac{\pi}{4} \right)} = \sqrt{3}$$
Solve:
$$x_{1} = - \frac{5 \pi}{36}$$
$$x_{1} = - \frac{5 \pi}{36}$$
This roots
$$x_{1} = - \frac{5 \pi}{36}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{5 \pi}{36} - \frac{1}{10}$$
=
$$- \frac{5 \pi}{36} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(3 x - \frac{\pi}{4} \right)} \leq \sqrt{3}$$
$$\tan{\left(3 \left(- \frac{5 \pi}{36} - \frac{1}{10}\right) - \frac{\pi}{4} \right)} \leq \sqrt{3}$$
the solution of our inequality is:
$$x \leq - \frac{5 \pi}{36}$$
$$\tan{\left(3 x - \frac{\pi}{4} \right)} \leq \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(3 x - \frac{\pi}{4} \right)} = \sqrt{3}$$
Solve:
$$x_{1} = - \frac{5 \pi}{36}$$
$$x_{1} = - \frac{5 \pi}{36}$$
This roots
$$x_{1} = - \frac{5 \pi}{36}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{5 \pi}{36} - \frac{1}{10}$$
=
$$- \frac{5 \pi}{36} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(3 x - \frac{\pi}{4} \right)} \leq \sqrt{3}$$
$$\tan{\left(3 \left(- \frac{5 \pi}{36} - \frac{1}{10}\right) - \frac{\pi}{4} \right)} \leq \sqrt{3}$$
/3 pi\ ___ cot|-- + --| <= \/ 3 \10 6 /
the solution of our inequality is:
$$x \leq - \frac{5 \pi}{36}$$
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