Why Flip the Inequality Sign?

Date: 10/26/2001 at 11:30:05
From: Sean
Subject: Flipping the Inequality sign 

I have a question about an inequality problem:  

Solve and graph 5 - 3x => 17. (=> is greater than or equal to). 

Please tell me why you flip the inequality sign when dividing by a 
negative number.

Thanks.
Sean

Date: 10/26/2001 at 12:09:56
From: Doctor Peterson
Subject: Re: Flipping the Inequality sign 

Hi, Sean.

Let's try solving your inequality in two ways, one dividing by a 
negative number and the other avoiding that.

First,

    5 - 3x >= 17

       -3x >= 12

         x <= -4

Here I subtracted 5, then divided by -3 and reversed the inequality. 

Next,

    5 - 3x >= 17

         5 >= 17 + 3x

       -12 >= 3x

        -4 >= x

This time I added 3x to both sides, eliminating the negative 
coefficient; then I subtracted 17 and divided by 3. Notice that this 
gives the same result, though it's written backwards. We know there's 
nothing tricky in this method; so the reversal of the inequality must 
be right when we do it the other way.

We can prove the general rule the same way:

    if a >= b

    then a - b >= 0  (subtracting b from both sides)

    and -b >= -a     (subtracting a from both sides)

    so that -a <= -b (rewriting the inequality in reverse)

This shows that when you multiply an inequality by -1 (which is part 
of what you do when you divide by a negative number), you reverse the 
inequality.

Here are some other discussions of this from our archives, which I 
found by searching for the words "reverse inequality negative":

   Negatives and Inequalities
   http://mathforum.org/library/drmath/view/53142.html   

   Reversing the Inequality
   http://mathforum.org/library/drmath/view/57463.html   

   Inequalities and Negative Numbers
   http://mathforum.org/library/drmath/view/53177.html   

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