(11x-6)(x-7)<0 inequation

Given the inequality:
$$\left(x - 7\right) \left(11 x - 6\right) < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x - 7\right) \left(11 x - 6\right) = 0$$
Solve:
Expand the expression in the equation
$$\left(x - 7\right) \left(11 x - 6\right) = 0$$
We get the quadratic equation
$$11 x^{2} - 83 x + 42 = 0$$
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
$$a = 11$$
$$b = -83$$
$$c = 42$$
, then

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

(-83)^2 - 4 * (11) * (42) = 5041

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

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

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

or
$$x_{1} = 7$$
$$x_{2} = \frac{6}{11}$$
$$x_{1} = 7$$
$$x_{2} = \frac{6}{11}$$
$$x_{1} = 7$$
$$x_{2} = \frac{6}{11}$$
This roots
$$x_{2} = \frac{6}{11}$$
$$x_{1} = 7$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{6}{11}$$
=
$$\frac{49}{110}$$
substitute to the expression
$$\left(x - 7\right) \left(11 x - 6\right) < 0$$
$$\left(-7 + \frac{49}{110}\right) \left(-6 + \frac{11 \cdot 49}{110}\right) < 0$$

721    
--- < 0
100    

but

721    
--- > 0
100    

Then
$$x < \frac{6}{11}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{6}{11} \wedge x < 7$$

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       x2      x1