tg^2x-3tgx+2<0 inequation

Given the inequality:
$$\left(\tan^{2}{\left(x \right)} - 3 \tan{\left(x \right)}\right) + 2 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\tan^{2}{\left(x \right)} - 3 \tan{\left(x \right)}\right) + 2 = 0$$
Solve:
Given the equation
$$\left(\tan^{2}{\left(x \right)} - 3 \tan{\left(x \right)}\right) + 2 = 0$$
transform
$$\tan^{2}{\left(x \right)} - 3 \tan{\left(x \right)} + 2 = 0$$
$$\left(\tan^{2}{\left(x \right)} - 3 \tan{\left(x \right)}\right) + 2 = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
This equation is of the form

a*w^2 + b*w + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -3$$
$$c = 2$$
, then

D = b^2 - 4 * a * c = 

(-3)^2 - 4 * (1) * (2) = 1

Because D > 0, then the equation has two roots.

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

or
$$w_{1} = 2$$
$$w_{2} = 1$$
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(1 \right)}$$
$$x_{2} = \pi n + \frac{\pi}{4}$$
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(2 \right)}$$
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(2 \right)}$$
This roots
$$x_{1} = \frac{\pi}{4}$$
$$x_{2} = \operatorname{atan}{\left(2 \right)}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{4}$$
=
$$- \frac{1}{10} + \frac{\pi}{4}$$
substitute to the expression
$$\left(\tan^{2}{\left(x \right)} - 3 \tan{\left(x \right)}\right) + 2 < 0$$
$$\left(- 3 \tan{\left(- \frac{1}{10} + \frac{\pi}{4} \right)} + \tan^{2}{\left(- \frac{1}{10} + \frac{\pi}{4} \right)}\right) + 2 < 0$$

       2/1    pi\        /1    pi\    
2 + cot |-- + --| - 3*cot|-- + --| < 0
        \10   4 /        \10   4 /    

but

       2/1    pi\        /1    pi\    
2 + cot |-- + --| - 3*cot|-- + --| > 0
        \10   4 /        \10   4 /    

Then
$$x < \frac{\pi}{4}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{\pi}{4} \wedge x < \operatorname{atan}{\left(2 \right)}$$

         _____  
        /     \  
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       x1      x2