Why does the inequality sign change when both sides are multilpied or divided by a negative number?

Answer 1

I bet you're thinking of > and <, the "greater than" and the "less than" signs, and not the "inequality" thing-- the = sign with a / through it. Am I right? Think about the positive integers 3 and 5. It should be easy to see that 3 is smaller than 5. Now here's a trick. If you think about the number line, even if you go back and forth over the zero point, "smaller" is always to the left, and "greater" is always to the right.

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-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9

<-------------- smaller larger --------------->

The weird thing is that when you're moving to the left and you cross zero, the numbers look like they're getting bigger-- except for the negative sign. You have to use your imagination with these negative integers. Say that you have 10 cd's. If I take two of them, I'm really adding "negative 2" to the total. This leaves you with 8 cd's. The "negative 2" made you total smaller, but not by much. I give back the two I took, and now I take 8 of them. I'm really adding "negative 8" to the 10 cd's. This leaves you with only two! So the "negative 8" had the effect of making your collection MUCH smaller than the "negative 2" did. So in a sense, the -8 is "smaller" than the -2. It fits with the number line, and the fact that numbers get smaller as you move to the left.

Now back to 3 and 5. Look at the number line. +3 < +5. three is smaller than 5. If I multiply them both by -1, I now have -3 and -5. If you look at the number line, you will see that -5 is to the left of -3! This means that

-3 > -5.

So to put it simply... when you multiply or divide a number with a negative number, the number becomes a negative number, therefore causing it to be smaller than the original number. Therefore the sign changes accordingly.