5. Solve the inequality $x+6-\sqrt{(x+6)|x+5|+|x+4|} \geq 0$.

#
# Solution:

$$
\sqrt{(x+6)|x+5|+|x+4|} \leq x+6 \Leftrightarrow\left\{\begin{array}{l}
x+6 \geq 0 \\
(x+6)|x+5|+|x+4| \leq(x+6)^{2} \\
(x+6)|x+5|+|x+4| \geq 0
\end{array}\right.
$$

Given $x+6 \geq 0$, the inequality $(x+6)|x+5|+|x+4| \geq 0$ holds, and the system is equivalent to $\quad$ the following $\quad\left\{\begin{array}{l}x \geq-6 ; \\ (x+6)|x+5|+|x+4| \leq x^{2}+12 x+36 .\end{array} \Leftrightarrow\right.$

![](https://cdn.mathpix.com/cropped/2024_05_06_797a639488ab43e0ad60g-13.jpg?height=368&width=1487&top_left_y=1752&top_left_x=296)
$x \in[-5,+\infty)$.

Answer: $x \in[-5,+\infty)$.