jmc

Squaring both sides of the left-hand inequality yields $n<4n-6⟹6\le 3n⟹2\le n$.

Squaring both sides of the right-hand inequality yields $4n-6<2n+5⟹2n<11⟹n<5.5$. Thus, $n$ must be one of $\{2,3,4,5\}$, of which we can check all work. As such, the answer is $\boxed{4}$.