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?
Best Answer
Dividing by a negative number is the same as dividing by a positive number and then multiplying by $-1$. Dividing an inequality by a positive number retains the same inequality. But, multiplying by $-1$ is the same as switching the signs of the numbers on both sides of the inequality, which reverses the inequality: $$ \tag{1} a\lt b\quad\iff -a\gt -b. $$ You should be able to convince yourself why the above is true by looking at the number line and considering the various cases involved.
Seeing why (1) is true is not too hard.
Here is the hand waving approach I suggested above:
Consider, for example, in (1), the case when $a$ is negative and $b$ is positive. We have $a<b$. Then $-a$ is positive and $-b$ is negative. Thus, we have $-b<-a$.
As another case, suppose $a$ and $b$ are both negative with $a<b$. Switching the signs here makes the resulting numbers both positive with $-a>-b$ (you can see this by drawing the points on the number line and noting that with the given conditions, $b$ is closer to the origin than $a$):
The other cases can be handled similarly.
But, perhaps a bit of rigor is needed here.
Recall that
$$a<b\quad\text{ if and only if }\quad b-a\quad \text{ is positive.}$$ Now, $b-a$ is positive if and only if $(-a)-(-b) =-a+b=b-a$ is positive. So $a<b$ if and only if $-a> -b$.