Mister Exam
sqrt3tg(3x+pi/4)<3 inequation
The solution
You have entered
[src]
_________________ / / pi\ / 3*tan|3*x + --| < 3 \/ \ 4 /
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} < 3$$
sqrt(3*tan(3*x + pi/4)) < 3
Detail solution
Given the inequality:
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} < 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} = 3$$
Solve:
Given the equation
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} = 3$$
transform
$$\sqrt{3} \sqrt{\tan{\left(3 x + \frac{\pi}{4} \right)}} - 3 = 0$$
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} - 3 = 0$$
Do replacement
$$w = \tan{\left(3 x + \frac{\pi}{4} \right)}$$
Given the equation
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} - 3 = 0$$
Because equation degree is equal to = 1/2 - does not contain even numbers in the numerator, then
the equation has single real root.
We raise the equation sides to 2-th degree:
We get:
$$\left(\sqrt{3}\right)^{2} \left(\sqrt{0 w + \tan{\left(3 x + \frac{\pi}{4} \right)}}\right)^{2} = 3^{2}$$
or
$$3 \tan{\left(3 x + \frac{\pi}{4} \right)} = 9$$
Expand brackets in the left part
This equation has no roots
do backward replacement
$$\tan{\left(3 x + \frac{\pi}{4} \right)} = w$$
Given the equation
$$\tan{\left(3 x + \frac{\pi}{4} \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x + \frac{\pi}{4} = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$3 x + \frac{\pi}{4} = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = \pi n + \operatorname{atan}{\left(w \right)} - \frac{\pi}{4}$$
Divide both parts of the equation by
$$3$$
substitute w:
$$x_{1} = - \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
$$x_{1} = - \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
This roots
$$x_{1} = - \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(- \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}\right)$$
=
$$- \frac{\pi}{12} - \frac{1}{10} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
substitute to the expression
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} < 3$$
$$\sqrt{3 \tan{\left(3 \left(- \frac{\pi}{12} - \frac{1}{10} + \frac{\operatorname{atan}{\left(3 \right)}}{3}\right) + \frac{\pi}{4} \right)}} < 3$$
the solution of our inequality is:
$$x < - \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} < 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} = 3$$
Solve:
Given the equation
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} = 3$$
transform
$$\sqrt{3} \sqrt{\tan{\left(3 x + \frac{\pi}{4} \right)}} - 3 = 0$$
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} - 3 = 0$$
Do replacement
$$w = \tan{\left(3 x + \frac{\pi}{4} \right)}$$
Given the equation
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} - 3 = 0$$
Because equation degree is equal to = 1/2 - does not contain even numbers in the numerator, then
the equation has single real root.
We raise the equation sides to 2-th degree:
We get:
$$\left(\sqrt{3}\right)^{2} \left(\sqrt{0 w + \tan{\left(3 x + \frac{\pi}{4} \right)}}\right)^{2} = 3^{2}$$
or
$$3 \tan{\left(3 x + \frac{\pi}{4} \right)} = 9$$
Expand brackets in the left part
3*tan3*x+pi/4 = 9
This equation has no roots
do backward replacement
$$\tan{\left(3 x + \frac{\pi}{4} \right)} = w$$
Given the equation
$$\tan{\left(3 x + \frac{\pi}{4} \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x + \frac{\pi}{4} = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$3 x + \frac{\pi}{4} = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = \pi n + \operatorname{atan}{\left(w \right)} - \frac{\pi}{4}$$
Divide both parts of the equation by
$$3$$
substitute w:
$$x_{1} = - \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
$$x_{1} = - \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
This roots
$$x_{1} = - \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(- \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}\right)$$
=
$$- \frac{\pi}{12} - \frac{1}{10} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
substitute to the expression
$$\sqrt{3 \tan{\left(3 x + \frac{\pi}{4} \right)}} < 3$$
$$\sqrt{3 \tan{\left(3 \left(- \frac{\pi}{12} - \frac{1}{10} + \frac{\operatorname{atan}{\left(3 \right)}}{3}\right) + \frac{\pi}{4} \right)}} < 3$$
___ ______________________
\/ 3 *\/ -tan(3/10 - atan(3)) < 3
the solution of our inequality is:
$$x < - \frac{\pi}{12} + \frac{\operatorname{atan}{\left(3 \right)}}{3}$$
_____
\
-------ο-------
x1
Solving inequality on a graph