TITLE: Why do we reverse inequality sign when dividing by negative number?
QUESTION [20 upvotes]: We all learned in our early years that when dividing both sides by a negative number, we reverse the inequality sign.
Take
$-3x < 9$
To solve for $x$, we divide both sides by $-3$ and get
$$x > -3.$$
Why is the reversal of inequality? What is going in terms of number line that will help me understand the concept better?

REPLY [2 votes]: Let $-x>b$
Add $x$ to both sides
$0>b+x$
Subtract $b$ from both sides
$-b>x$
$x<-b$