Why does the inequality symbol need to be reversed when multiplying or by a negative number?

Answer 1

Use two arbitrary numbers represented by the letters A and B. Now divide the question into a few cases.

Case 1. A < B and both are positive. Take any A and B and put them on your number line. Now look at the opposite of B which is -B and notice it will be farther to the left than -A which is the opposite of A. So we have -B<-A which we can written as -A>-B.

So for the first case, we see why we reverse the inequality symbol.

Case 2. Now look at A<B where both are negative. The exact same thing that happens in case 1 above, happens in this case. Try two number, say -2 and -1.

Case 3. Now say A is negative and B is positive. The other way around is the same thing. So Since A is negative, its opposite is positive and the opposite of B is negative. Any negative number is smaller than a positive number. So once again, we reverse the inequality symbol.

Try all three of these cases with some positive and negative number and a number line and you will see how this works, then generalize to A and B where they are arbitrary numbers. This is the way mathematician often prove things. They first use numbers and then generalize.