(2,5-√6)(10-4x)>0 inequation

Given the inequality:
$$\left(\frac{5}{2} - \sqrt{6}\right) \left(10 - 4 x\right) > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{5}{2} - \sqrt{6}\right) \left(10 - 4 x\right) = 0$$
Solve:
Given the equation:

((5/2)-sqrt(6))*(10-4*x) = 0

Expand expressions:

25 - 10*x - 10*sqrt(6) + 4*x*sqrt(6) = 0

Reducing, you get:

25 - 10*x - 10*sqrt(6) + 4*x*sqrt(6) = 0

Expand brackets in the left part

25 - 10*x - 10*sqrt6 + 4*x*sqrt6 = 0

Move free summands (without x)
from left part to right part, we given:
$$- 10 x + 4 \sqrt{6} x - 10 \sqrt{6} = -25$$
Divide both parts of the equation by (-10*x - 10*sqrt(6) + 4*x*sqrt(6))/x

x = -25 / ((-10*x - 10*sqrt(6) + 4*x*sqrt(6))/x)

We get the answer: x = 5/2
$$x_{1} = \frac{5}{2}$$
$$x_{1} = \frac{5}{2}$$
This roots
$$x_{1} = \frac{5}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{5}{2}$$
=
$$\frac{12}{5}$$
substitute to the expression
$$\left(\frac{5}{2} - \sqrt{6}\right) \left(10 - 4 x\right) > 0$$
$$\left(\frac{5}{2} - \sqrt{6}\right) \left(10 - \frac{4 \cdot 12}{5}\right) > 0$$

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    2*\/ 6     
1 - ------- > 0
       5       
    

the solution of our inequality is:
$$x < \frac{5}{2}$$

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