When solving inequalities, does $(x-9)$ count as a negative number?

I suppose it depends on what $x$ is an element of. If we assume that $x\in\mathbb{R}$, we have to break this inequality into separate cases.

There is the case where $x>9$, in which case the sign does not change, and there is the case of $x<9$, in which the sign does change.

Of course, this is undefined for $x=9$.

This is an inequality. You should not multiply across by $(x-9)$ at all. Instead you should bring the $2$ over to the left and combine the fraction through a common denominator. Verify you arrive at: $\frac{-x+27}{x-9}≤0$ Now one should make separate numberlines for numerator and denominator and verify the signs on the numberlines. NUM:$$___________plus_______27__minus______$$ DENOM: $$___minus____9____plus_____________$$ When you "divide" the numberlines you look for the negative interval, which is $x<9$ or $x\geq27$