x^2log625(6-x)<=log5(x^2-12x+36) inequation

Given the inequality:
$$\frac{x^{2} \log{\left(6 - x \right)}}{\log{\left(625 \right)}} \leq \frac{\log{\left(x^{2} - 12 x + 36 \right)}}{\log{\left(5 \right)}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x^{2} \log{\left(6 - x \right)}}{\log{\left(625 \right)}} = \frac{\log{\left(x^{2} - 12 x + 36 \right)}}{\log{\left(5 \right)}}$$
Solve:
Given the equation
$$\frac{x^{2} \log{\left(6 - x \right)}}{\log{\left(625 \right)}} = \frac{\log{\left(x^{2} - 12 x + 36 \right)}}{\log{\left(5 \right)}}$$
transform
$$\frac{\frac{x^{2} \log{\left(6 - x \right)}}{4} - \log{\left(x^{2} - 12 x + 36 \right)}}{\log{\left(5 \right)}} = 0$$
$$\frac{x^{2} \log{\left(5 \right)} \log{\left(6 - x \right)} - \log{\left(625 \right)} \log{\left(x^{2} - 12 x + 36 \right)}}{\log{\left(5 \right)} \log{\left(625 \right)}} = 0$$
Do replacement
$$w = \log{\left(625 \right)}$$
Given the equation:
$$\frac{- w \log{\left(x^{2} - 12 x + 36 \right)} + x^{2} \log{\left(5 \right)} \log{\left(6 - x \right)}}{w \log{\left(5 \right)}} = 0$$
Multiply the equation sides by the denominator w*log(5)
we get:
$$w \left(- \frac{\log{\left(x^{2} - 12 x + 36 \right)}}{\log{\left(5 \right)}} + \frac{x^{2} \log{\left(6 - x \right)}}{w}\right) \log{\left(5 \right)} = 0$$
Expand brackets in the left part

w-log+36+x+2+12*xlog5 + x^2*log6+xw)*log5 = 0

Looking for similar summands in the left part:

w*(-log(36 + x^2 - 12*x)/log(5) + x^2*log(6 - x)/w)*log(5) = 0

Divide both parts of the equation by (-log(36 + x^2 - 12*x)/log(5) + x^2*log(6 - x)/w)*log(5)

w = 0 / ((-log(36 + x^2 - 12*x)/log(5) + x^2*log(6 - x)/w)*log(5))

We get the answer: w = x^2*log(5)*log(6 - x)/log(36 + x^2 - 12*x)
do backward replacement
$$\log{\left(625 \right)} = w$$
substitute w:
$$x_{1} = 2.82842712474619$$
$$x_{2} = 5$$
$$x_{3} = -2.82842712474619$$
$$x_{1} = 2.82842712474619$$
$$x_{2} = 5$$
$$x_{3} = -2.82842712474619$$
This roots
$$x_{3} = -2.82842712474619$$
$$x_{1} = 2.82842712474619$$
$$x_{2} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{3}$$
For example, let's take the point
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$-2.82842712474619 - \frac{1}{10}$$
=
$$-2.92842712474619$$
substitute to the expression
$$\frac{x^{2} \log{\left(6 - x \right)}}{\log{\left(625 \right)}} \leq \frac{\log{\left(x^{2} - 12 x + 36 \right)}}{\log{\left(5 \right)}}$$
$$\frac{\left(-2.92842712474619\right)^{2} \log{\left(6 - -2.92842712474619 \right)}}{\log{\left(625 \right)}} \leq \frac{\log{\left(\left(-2.92842712474619\right)^{2} - 12 \left(-2.92842712474619\right) + 36 \right)}}{\log{\left(5 \right)}}$$

18.7742356652868    4.37848049105602
---------------- <= ----------------
    log(625)             log(5)     

but

18.7742356652868    4.37848049105602
---------------- >= ----------------
    log(625)             log(5)     

Then
$$x \leq -2.82842712474619$$
no execute
one of the solutions of our inequality is:
$$x \geq -2.82842712474619 \wedge x \leq 2.82842712474619$$

         _____           _____  
        /     \         /
-------•-------•-------•-------
       x_3      x_1      x_2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -2.82842712474619 \wedge x \leq 2.82842712474619$$
$$x \geq 5$$