x^2-6x-1<0 inequation

Given the inequality:
$$\left(x^{2} - 6 x\right) - 1 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{2} - 6 x\right) - 1 = 0$$
Solve:
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
$$a = 1$$
$$b = -6$$
$$c = -1$$
, then

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

(-6)^2 - 4 * (1) * (-1) = 40

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

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

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

or
$$x_{1} = 3 + \sqrt{10}$$
$$x_{2} = 3 - \sqrt{10}$$
$$x_{1} = 3 + \sqrt{10}$$
$$x_{2} = 3 - \sqrt{10}$$
$$x_{1} = 3 + \sqrt{10}$$
$$x_{2} = 3 - \sqrt{10}$$
This roots
$$x_{2} = 3 - \sqrt{10}$$
$$x_{1} = 3 + \sqrt{10}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(3 - \sqrt{10}\right) + - \frac{1}{10}$$
=
$$\frac{29}{10} - \sqrt{10}$$
substitute to the expression
$$\left(x^{2} - 6 x\right) - 1 < 0$$
$$-1 + \left(\left(\frac{29}{10} - \sqrt{10}\right)^{2} - 6 \left(\frac{29}{10} - \sqrt{10}\right)\right) < 0$$

                    2               
  92   /29     ____\        ____    
- -- + |-- - \/ 10 |  + 6*\/ 10  < 0
  5    \10         /                
    

but

                    2               
  92   /29     ____\        ____    
- -- + |-- - \/ 10 |  + 6*\/ 10  > 0
  5    \10         /                
    

Then
$$x < 3 - \sqrt{10}$$
no execute
one of the solutions of our inequality is:
$$x > 3 - \sqrt{10} \wedge x < 3 + \sqrt{10}$$

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