6.4.1. (12 points) Solve the inequality $\sqrt{x-4}+\sqrt{x+1}+\sqrt{2 x}-\sqrt{33-x}>4$. In your answer, write the sum of all its integer solutions.
Answer: 525.

Solution. Rewrite the inequality as

$$
\sqrt{x-4}+\sqrt{x+1}+\sqrt{2 x}>\sqrt{33-x}+4
$$

Find the domain of the variable $x$:

$$
\left\{\begin{array}{l}
x-4 \geqslant 0 \\
x+1 \geqslant 0 \\
2 x \geqslant 0 \\
33-x \geqslant 0
\end{array}\right.
$$

from which $x \in[4,33]$. We are interested in the integer values from this interval that satisfy the inequality. Note that the left side of the obtained inequality is increasing, while the right side is strictly decreasing. They coincide at $x=8$. For $x \in\{4,5,6,7\}$, the left side is less than the right side, and for $x \in\{9,10, \ldots, 33\}$, the left side is greater than the right side. We calculate the sum. $9+10+\ldots+33=\frac{9+33}{2} \cdot 25=525$.