2. Solve the inequality $\sqrt{\sqrt{x+1}-2}+\sqrt{x+82-18 \sqrt{x+1}}>5$.
Answer. $x \in[3 ; 35) \cup(120 ;+\infty)$.

Solution. Note that the second radicand can be written as $(\sqrt{x+1}-9)^{2}$. The inequality takes the form $\sqrt{\sqrt{x+1}-2}+|\sqrt{x+1}-9|>5$. Let $\sqrt{\sqrt{x+1}-2}=t$. Then we get

$$
t+\left|t^{2}-7\right|>5 \Leftrightarrow\left[\begin{array} { l } 
{ t ^ { 2 } - 7 > 5 - t } \\
{ t ^ { 2 } - 7  0 , } \\
{ t ^ { 2 } - t - 2 < 0 }
\end{array} \Leftrightarrow \left[\begin{array}{l}
t \in(-\infty ;-4) \cup(3 ;+\infty) \\
t \in(-1 ; 2)
\end{array}\right.\right.\right.
$$

Returning to the variable $x$, we get the system

![](https://cdn.mathpix.com/cropped/2024_05_06_6022a90abea870d9ae11g-01.jpg?height=290&width=1474&top_left_y=1594&top_left_x=291)