### 9.303 Find integer values of $x$ that satisfy the inequality

$$
\log _{0.3}(\sqrt{x+5}-x+1)>0
$$
Solution. The inequality is equivalent to the system:

$\left\{\begin{array}{l}\sqrt{x+5}-x+10, \\ x-\text { integer }\end{array} \Leftrightarrow\left\{\begin{array}{l}\sqrt{x+5}x-1, \\ x-\text { integer. }\end{array}\right.\right.$

From this, $x>0$ and, consequently, $x \geq 1$ ( $x$ - integer). Therefore,

$\left\{\begin{array}{l}(x+5)(x-1)^{2}, \\ x-\text { integer, } \\ x \geq 1\end{array} \Leftrightarrow\left\{\begin{array}{l}\frac{1+\sqrt{21}}{2}<x<4, \\ x-\text { integer }\end{array} \Leftrightarrow x=3\right.\right.$.

Answer: $x=3$.