Reversing the Inequality Symbol

Date: 03/09/99 at 20:43:20
From: Rob DeMoe
Subject: Reversal of the Inequality Symbol 

I have a question on solving inequalities. Why do we reverse the 
inequality symbol when dividing/multiplying by a negative number?  
     
I realize that I could simply show students that if we check our 
solution without reversing the symbol we would notice an error. Is 
there some theorm or proof? I would feel a lot more comfortable if I 
knew the reasoning behind this. I hope you can help.

Date: 03/10/99 at 09:05:09
From: Doctor Peterson
Subject: Re: Reversal of the Inequality Symbol 

Working with inqualities and negative numbers can be tricky! Here is 
how I like to think of it.

Equations can be thought of as balanced scales, where the total weight 
on the left balances what is on the right. Inequalities are like 
unbalanced scales, where all you know is which side is "down" 
(heavier). So for example

    3x > 6

can be thought of as 3 unknown weights labeled "X" on the left, 
heavier than a 6-gram weight on the right.

Negative numbers complicate it a bit. A negative constant can be 
thought of as a helium balloon (barely) able to lift a certain weight. 
A negative number times a variable might mean there is an antigravity 
machine under it so it pulls up as hard as it would normally push 
down!

So the equation

    -4y < -36

would be 4 Y's on the left with an antigravity machine, and a "-36" 
gram balloon on the right. The right side is "heavier," which in this 
case means not that it is pushing down more, but that it is pulling up 
less!

To solve it, let us first replace the -36 gram balloon with a 36-gram 
weight and an antigravity machine:

    -(4y) < -(36)

Now turn off the antigravity machines:

    4y > 36

Why did I reverse the direction of the "<"? That is the key to this 
whole thing: antigravity machines are like turning the whole world 
upside down, so the side that was down is now up:

    -4y      -36
     ^        ^
  \  |        |
     \        |
        \     |
           \  |
              \
                 \

becomes

     4y       36
     |        |
     |        v  /
     |        /
     |     /
     v  /
     /
  /

because 4y, which pulled up harder before, now pushes down harder.

Now we can work with positive numbers, and divide both weights by 4 to 
get

    y > 9

So any number BIGGER than 9 will work. For example, for y = 10,

    -4y = -40 < -36

Do you see how the larger number, 10 > 9, becomes the smaller number 
(-40 < -36) when it is multiplied by a negative number? That is the 
key. The rule is that when you multiply an inequality by a negative 
number, you have to reverse the direction. Or if you prefer, you can 
do this:

   -4y < -36       Add 4y
     0 < 4y - 36   Add 36
    36 < 4y        Divide by 4
     9 < y         Reverse the whole inequality
     y > 9

By avoiding multiplication by a negative number, I avoided the need to 
reverse signs until the end.

Now that we have seen with our imagination what is going on, let us try 
to prove the rule that if

    a > b

then

    -a < -b.

Start with the original inequality and subtract a from both sides:

    0 > b - a

Now subtract b from both sides:

    -b > -a

But that is the same as

    -a < -b

which we were looking for. It is really pretty simple - so simple it 
does not grab your attention the way helium balloons and antigravity 
machines do! That is why I like to start the way I did.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/