log((1/2))^(2)*(x-1)+5*log((1/2))*(x-1)>-6 inequation

Given the inequality:
$$\left(x - 1\right) 5 \log{\left(\frac{1}{2} \right)} + \left(x - 1\right) \log{\left(\frac{1}{2} \right)}^{2} > -6$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x - 1\right) 5 \log{\left(\frac{1}{2} \right)} + \left(x - 1\right) \log{\left(\frac{1}{2} \right)}^{2} = -6$$
Solve:
Given the equation:

log((1/2))^(2)*(x-1)+5*log((1/2))*(x-1) = -6

Expand expressions:

- log(2)^2 + x*log(2)^2 + (5*log(1/2))*(x - 1) = -6

- log(2)^2 + x*log(2)^2 + 5*log(2) - 5*x*log(2) = -6

Reducing, you get:

6 - log(2)^2 + 5*log(2) + x*log(2)^2 - 5*x*log(2) = 0

Expand brackets in the left part

6 - log2^2 + 5*log2 + x*log2^2 - 5*x*log2 = 0

Move free summands (without x)
from left part to right part, we given:
$$- 5 x \log{\left(2 \right)} + x \log{\left(2 \right)}^{2} - \log{\left(2 \right)}^{2} + 5 \log{\left(2 \right)} = -6$$
Divide both parts of the equation by (-log(2)^2 + 5*log(2) + x*log(2)^2 - 5*x*log(2))/x

x = -6 / ((-log(2)^2 + 5*log(2) + x*log(2)^2 - 5*x*log(2))/x)

We get the answer: x = (6 - log(2)^2 + log(32))/((5 - log(2))*log(2))
$$x_{1} = \frac{- \log{\left(2 \right)}^{2} + \log{\left(32 \right)} + 6}{\left(5 - \log{\left(2 \right)}\right) \log{\left(2 \right)}}$$
$$x_{1} = \frac{- \log{\left(2 \right)}^{2} + \log{\left(32 \right)} + 6}{\left(5 - \log{\left(2 \right)}\right) \log{\left(2 \right)}}$$
This roots
$$x_{1} = \frac{- \log{\left(2 \right)}^{2} + \log{\left(32 \right)} + 6}{\left(5 - \log{\left(2 \right)}\right) \log{\left(2 \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{- \log{\left(2 \right)}^{2} + \log{\left(32 \right)} + 6}{\left(5 - \log{\left(2 \right)}\right) \log{\left(2 \right)}}$$
=
$$- \frac{1}{10} + \frac{- \log{\left(2 \right)}^{2} + \log{\left(32 \right)} + 6}{\left(5 - \log{\left(2 \right)}\right) \log{\left(2 \right)}}$$
substitute to the expression
$$\left(x - 1\right) 5 \log{\left(\frac{1}{2} \right)} + \left(x - 1\right) \log{\left(\frac{1}{2} \right)}^{2} > -6$$
$$\left(-1 + \left(- \frac{1}{10} + \frac{- \log{\left(2 \right)}^{2} + \log{\left(32 \right)} + 6}{\left(5 - \log{\left(2 \right)}\right) \log{\left(2 \right)}}\right)\right) 5 \log{\left(\frac{1}{2} \right)} + \left(-1 + \left(- \frac{1}{10} + \frac{- \log{\left(2 \right)}^{2} + \log{\left(32 \right)} + 6}{\left(5 - \log{\left(2 \right)}\right) \log{\left(2 \right)}}\right)\right) \log{\left(\frac{1}{2} \right)}^{2} > -6$$

        /              2             \     /              2             \            
   2    |  11   6 - log (2) + log(32)|     |  11   6 - log (2) + log(32)|            
log (2)*|- -- + ---------------------| - 5*|- -- + ---------------------|*log(2) > -6
        \  10    (5 - log(2))*log(2) /     \  10    (5 - log(2))*log(2) /            
     

the solution of our inequality is:
$$x < \frac{- \log{\left(2 \right)}^{2} + \log{\left(32 \right)} + 6}{\left(5 - \log{\left(2 \right)}\right) \log{\left(2 \right)}}$$

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