4/(x-3)-x<0 inequation

Given the inequality:
$$- x + \frac{4}{x - 3} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- x + \frac{4}{x - 3} = 0$$
Solve:
Given the equation:
$$- x + \frac{4}{x - 3} = 0$$
Multiply the equation sides by the denominators:
and -3 + x
we get:
$$\left(- x + \frac{4}{x - 3}\right) \left(x - 3\right) = 0 \left(x - 3\right)$$
$$- x^{2} + 3 x + 4 = 0$$
This equation is of the form

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

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

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

(3)^2 - 4 * (-1) * (4) = 25

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

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = -1$$
$$x_{2} = 4$$
$$x_{1} = -1$$
$$x_{2} = 4$$
$$x_{1} = -1$$
$$x_{2} = 4$$
This roots
$$x_{1} = -1$$
$$x_{2} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1 + - \frac{1}{10}$$
=
$$- \frac{11}{10}$$
substitute to the expression
$$- x + \frac{4}{x - 3} < 0$$
$$\frac{4}{-3 + - \frac{11}{10}} - - \frac{11}{10} < 0$$

 51    
--- < 0
410    

but

 51    
--- > 0
410    

Then
$$x < -1$$
no execute
one of the solutions of our inequality is:
$$x > -1 \wedge x < 4$$

         _____  
        /     \  
-------ο-------ο-------
       x1      x2