1. Solve the inequality $\frac{\sqrt{\frac{x}{\gamma}+(\alpha+2)}-\frac{x}{\gamma}-\alpha}{x^{2}+a x+b} \geqslant 0$.

In your answer, specify the number equal to the number of integer roots of this inequality. If there are no integer roots, or if there are infinitely many roots, enter the digit 0 in the answer sheet.

$$
\alpha=3, \gamma=1, a=-15, b=54 .
$$
# Answer: 7.

Solution. Given the numerical values, we solve the inequality:

$$
\frac{\sqrt{x+5}-x-3}{x^{2}-15 x+54} \geqslant 0 \Leftrightarrow \frac{\sqrt{x+5}-x-3}{(x-6)(x-9)} \geqslant 0
$$

Since $\sqrt{x+5} \geqslant x+3 \Leftrightarrow\left[\begin{array}{c}-5 \leqslant x \leqslant-3, \\ x+5 \geqslant(x+3)^{2}\end{array}\right.$, the expression in the numerator is non-negative for $x \in[-5 ;-1]$, and negative for $x>-1$.

It is also true that $x^{2}-15 x+54<0$ for $x \in(6 ; 9)$, otherwise this expression is non-negative.

Therefore, $x \in[-5 ;-1] \bigcup(6 ; 9)$, and the number of integer solutions is $5+2=7$.