I have tried many times to identify my error, but I can't recognize it. I think it may have to do with my properties. This is the inequality that I tried to solve by having the same denominator and using a sign diagram. I didn't have x=2 as part of the sign diagram.
$$ 2x^{-1/3}(x-3)^{1/3}+x^{2/3}(x-3)^{-2/3}\ge 0 $$
The easiest way to see this is to let $z = x^{-1/3}(x-3)^{1/3}$, then your inequality says $$2z+z^2 \ge 0\\z(2+z) \ge 0$$ which implies that either $z \ge 0$ and $z+2\ge 0$ or $z \le 0$ and $z+2 \le 0$.
Combine 2 inequalities in each case into the stronger one, and solve both cases for $x$, plugging in the original definition of $z$.