(√14-4)x˂30-8√14 inequation

Given the inequality:
$$x \left(-4 + \sqrt{14}\right) < 30 - 8 \sqrt{14}$$
To solve this inequality, we must first solve the corresponding equation:
$$x \left(-4 + \sqrt{14}\right) = 30 - 8 \sqrt{14}$$
Solve:
Given the linear equation:

(sqrt(14)-4)*x = 30-8*sqrt(14)

Expand brackets in the left part

sqrt+14-4)*x = 30-8*sqrt(14)

Expand brackets in the right part

sqrt+14-4)*x = 30-8*sqrt14

Move free summands (without x)
from left part to right part, we given:
$$x \left(-4 + \sqrt{14}\right) + 4 = 34 - 8 \sqrt{14}$$
Divide both parts of the equation by (4 + x*(-4 + sqrt(14)))/x

x = 34 - 8*sqrt(14) / ((4 + x*(-4 + sqrt(14)))/x)

$$x_{1} = -4 + \sqrt{14}$$
$$x_{1} = -4 + \sqrt{14}$$
This roots
$$x_{1} = -4 + \sqrt{14}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(-4 + \sqrt{14}\right) + - \frac{1}{10}$$
=
$$- \frac{41}{10} + \sqrt{14}$$
substitute to the expression
$$x \left(-4 + \sqrt{14}\right) < 30 - 8 \sqrt{14}$$
$$\left(- \frac{41}{10} + \sqrt{14}\right) \left(-4 + \sqrt{14}\right) < 30 - 8 \sqrt{14}$$

/       ____\ /  41     ____\            ____
\-4 + \/ 14 /*|- -- + \/ 14 | < 30 - 8*\/ 14 
              \  10         /   

but

/       ____\ /  41     ____\            ____
\-4 + \/ 14 /*|- -- + \/ 14 | > 30 - 8*\/ 14 
              \  10         /   

Then
$$x < -4 + \sqrt{14}$$
no execute
the solution of our inequality is:
$$x > -4 + \sqrt{14}$$

         _____  
        /
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       x1