log[1/2,2*x-11]<-1 inequation

Given the inequality:
$$\log{\left(\frac{x}{\frac{11}{5}} - 11 \right)} < -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{x}{\frac{11}{5}} - 11 \right)} = -1$$
Solve:
Given the equation
$$\log{\left(\frac{x}{\frac{11}{5}} - 11 \right)} = -1$$
$$\log{\left(\frac{5 x}{11} - 11 \right)} = -1$$
This equation is of the form:

log(v)=p

By definition log

v=e^p

then
$$\frac{5 x}{11} - 11 = e^{- 1^{-1}}$$
simplify
$$\frac{5 x}{11} - 11 = e^{-1}$$
$$\frac{5 x}{11} = e^{-1} + 11$$
$$x = \frac{11}{5 e} + \frac{121}{5}$$
$$x_{1} = \frac{11 \left(1 + 11 e\right)}{5 e}$$
$$x_{1} = \frac{11 \left(1 + 11 e\right)}{5 e}$$
This roots
$$x_{1} = \frac{11 \left(1 + 11 e\right)}{5 e}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{11 \left(1 + 11 e\right)}{5 e^{1}}$$
=
$$- \frac{1}{10} + \frac{11 \left(1 + 11 e\right)}{5 e}$$
substitute to the expression
$$\log{\left(\frac{x}{\frac{11}{5}} - 11 \right)} < -1$$
$$\log{\left(-11 + \frac{- \frac{1}{10} + \frac{11 \left(1 + 11 e\right)}{5 e^{1}}}{\frac{11}{5}} \right)} < -1$$

   /  243               -1\     
log|- --- + (1 + 11*E)*e  | < -1
   \   22                 /     

the solution of our inequality is:
$$x < \frac{11 \left(1 + 11 e\right)}{5 e}$$

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