9.014. Find the set of integer values of $x$ that satisfy the system of inequalities

$$
\left\{\begin{array}{l}
\frac{x+8}{x+2}>2 \\
\lg (x-1)<1
\end{array}\right.
$$
Solution.

Considering the domain of definition, we solve the second inequality of the system:

$$
\left\{\begin{array} { l } 
{ \frac { x + 8 } { x + 2 } - 2 > 0 , } \\
{ 0  0 } \\
{ 1 < x < 1 1 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
(x-4)(x+2)<0 \\
1<x<11
\end{array}\right.\right.\right.
$$

Using a number line, we find the integer solutions of the system $x_{1}=2$; $x_{2}=3$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-521.jpg?height=132&width=610&top_left_y=982&top_left_x=383)

Answer: $\quad x_{1}=2 ; x_{2}=3$.