Mister Exam
tan(-pi*1/3+5*x)>=-sqrt(3)/3 inequation
The solution
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___ /-pi \ -\/ 3 tan|---- + 5*x| >= ------- \ 3 / 3
$$\tan{\left(5 x + \frac{\left(-1\right) \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
tan(5*x + (-pi)/3) >= (-sqrt(3))/3
Detail solution
Given the inequality:
$$\tan{\left(5 x + \frac{\left(-1\right) \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(5 x + \frac{\left(-1\right) \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
$$x_{1} = \frac{\pi}{30}$$
$$x_{1} = \frac{\pi}{30}$$
This roots
$$x_{1} = \frac{\pi}{30}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{30}$$
=
$$- \frac{1}{10} + \frac{\pi}{30}$$
substitute to the expression
$$\tan{\left(5 x + \frac{\left(-1\right) \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\tan{\left(\frac{\left(-1\right) \pi}{3} + 5 \left(- \frac{1}{10} + \frac{\pi}{30}\right) \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
but
Then
$$x \leq \frac{\pi}{30}$$
no execute
the solution of our inequality is:
$$x \geq \frac{\pi}{30}$$
$$\tan{\left(5 x + \frac{\left(-1\right) \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(5 x + \frac{\left(-1\right) \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
$$x_{1} = \frac{\pi}{30}$$
$$x_{1} = \frac{\pi}{30}$$
This roots
$$x_{1} = \frac{\pi}{30}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{30}$$
=
$$- \frac{1}{10} + \frac{\pi}{30}$$
substitute to the expression
$$\tan{\left(5 x + \frac{\left(-1\right) \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\tan{\left(\frac{\left(-1\right) \pi}{3} + 5 \left(- \frac{1}{10} + \frac{\pi}{30}\right) \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
___
/1 pi\ -\/ 3
-tan|- + --| >= -------
\2 6 / 3
but
___
/1 pi\ -\/ 3
-tan|- + --| < -------
\2 6 / 3
Then
$$x \leq \frac{\pi}{30}$$
no execute
the solution of our inequality is:
$$x \geq \frac{\pi}{30}$$
_____
/
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x1
Solving inequality on a graph
Rapid solution
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/ / ___________ \ \ | | ___ ___ / ___ | | | | 1 + \/ 5 - \/ 6 *\/ 5 - \/ 5 | pi| And|-atan|-------------------------------------| <= x, x < --| | | ___________| 6 | | | ___ ____ ___ / ___ | | \ \\/ 3 + \/ 15 + \/ 2 *\/ 5 - \/ 5 / /
$$- \operatorname{atan}{\left(\frac{- \sqrt{6} \sqrt{5 - \sqrt{5}} + 1 + \sqrt{5}}{\sqrt{3} + \sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{15}} \right)} \leq x \wedge x < \frac{\pi}{6}$$
(x < pi/6)∧(-atan((1 + sqrt(5) - sqrt(6)*sqrt(5 - sqrt(5)))/(sqrt(3) + sqrt(15) + sqrt(2)*sqrt(5 - sqrt(5)))) <= x)
Rapid solution 2
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/ ___________ \
| ___ ___ / ___ |
| 1 + \/ 5 - \/ 6 *\/ 5 - \/ 5 | pi
[-atan|-------------------------------------|, --)
| ___________| 6
| ___ ____ ___ / ___ |
\\/ 3 + \/ 15 + \/ 2 *\/ 5 - \/ 5 /
$$x\ in\ \left[- \operatorname{atan}{\left(\frac{- \sqrt{6} \sqrt{5 - \sqrt{5}} + 1 + \sqrt{5}}{\sqrt{3} + \sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{15}} \right)}, \frac{\pi}{6}\right)$$
x in Interval.Ropen(-atan((-sqrt(6)*sqrt(5 - sqrt(5)) + 1 + sqrt(5))/(sqrt(3) + sqrt(2)*sqrt(5 - sqrt(5)) + sqrt(15))), pi/6)