log((x+7),2x+14)≥2 inequation

Given the inequality:
$$\log{\left(x + 7 \right)} \geq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(x + 7 \right)} = 2$$
Solve:
$$x_{1} = - \frac{27}{4}$$
$$x_{1} = - \frac{27}{4}$$
This roots
$$x_{1} = - \frac{27}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{27}{4} + - \frac{1}{10}$$
=
$$- \frac{137}{20}$$
substitute to the expression
$$\log{\left(x + 7 \right)} \geq 2$$
$$\log{\left(- \frac{137}{20} + 7 \right)} \geq 2$$

      log(2)      
1 - --------- >= 2
    log(3/10)     

but

      log(2)     
1 - --------- < 2
    log(3/10)    

Then
$$x \leq - \frac{27}{4}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{27}{4}$$

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