(1/25)^(-x-1)<=25^2 inequation

Given the inequality:
$$\left(\frac{1}{25}\right)^{- x - 1} \leq 625$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{1}{25}\right)^{- x - 1} = 625$$
Solve:
Given the equation:
$$\left(\frac{1}{25}\right)^{- x - 1} = 625$$
or
$$\left(\frac{1}{25}\right)^{- x - 1} - 625 = 0$$
or
$$25 \cdot 25^{x} = 625$$
or
$$25^{x} = 25$$
- this is the simplest exponential equation
Do replacement
$$v = 25^{x}$$
we get
$$v - 25 = 0$$
or
$$v - 25 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 25$$
do backward replacement
$$25^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(25 \right)}}$$
$$x_{1} = 25$$
$$x_{1} = 25$$
This roots
$$x_{1} = 25$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 25$$
=
$$\frac{249}{10}$$
substitute to the expression
$$\left(\frac{1}{25}\right)^{- x - 1} \leq 625$$
$$\left(\frac{1}{25}\right)^{- \frac{249}{10} - 1} \leq 625$$

                                      4/5       
444089209850062616169452667236328125*5    <= 625
       

but

                                      4/5       
444089209850062616169452667236328125*5    >= 625
       

Then
$$x \leq 25$$
no execute
the solution of our inequality is:
$$x \geq 25$$

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