Prove that $\sqrt{x+2} + \sqrt{2x-2} + \sqrt{9-3x} < 3\sqrt{3}$

Prove that $\sqrt{x+2} + \sqrt{2x-2} + \sqrt{9-3x} < 3\sqrt{3}$ for x$\in\mathbb R$ and the square root exists. I observed that $1 \le x\le3$ and then i tried to raise the initial equation to the second power but nothing improved. Also, i am not sure if this is true, but if the equation holds for $x = 1$ and $x = 3$ then it must be true for all x in the interval.

Does anyone have any idea?

Use the root-mean square inequality:

$$\frac{a+b+c}{3} ≤ \sqrt{\frac{a^2+b^2+c^2}{3}}$$

for all real nonnegative $a,b,c$.

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