TITLE: When solving inequalities, does $(x-9)$ count as a negative number?
QUESTION [6 upvotes]: I know that when solving inequalities, you reverse the sign when dividing or multiplying by a negative number, but right now I need to solve $\frac{x+9}{x-9}≤2$. Do I need to reverse the sign when I multiply $x-9$ across?
I'm sorry if this is outrageously obvious...

REPLY [2 votes]: 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 $x\geq27$