Multiplying or Dividing by a Negative Number

Date: 04/08/99 at 11:37:10
From: Ling-Seang Yap
Subject: When x(x-4) < 0

When x(x-4) < 0 , if we divide (x-4) on both sides of this inequality, 
we get x < 0, but a negative x does not give a true answer for it. 

Example:

 x (x-4) < 0
       x < 0

But if we substitute x with, say -4

 -1 (-1 - 4) < 0
   -1 ( -5 ) < 0
         5   < 0 (false?)

Hope you can shed light on this question. 
Thanks.

Date: 04/08/99 at 12:28:22
From: Doctor Peterson
Subject: Re: when x(x-4) < 0

The problem in what you did is that when you multiply or divide an 
inequality by a negative number, it reverses the inequality. So if 
x - 4 is negative, when you divide by it it changes to x > 0. Since 
your example case of x = -4 makes x - 4 negative, you find that 
x(x-4) > 0.

The best way to handle this kind of problem is to notice that there 
are two ways for the product x(x-4) to be negative: either x is 
negative and x-4 is positive, or x is positive and x-4 is negative. 
Let's take those cases one at a time:

If x < 0 and x-4 > 0, then we have x > 4, which is impossible if 
x < 0. So this case gives no solutions.

If x > 0 and x-4 < 0, then x < 4, and our answer is

    0 < x < 4

That is, x(x-4) is negative when x is between 0 and 4.

If we graph y = x(x-4), it is a parabola:

          *    +                    *
               |
               +
               |
           *   +                   *
               |
               +
            *  |                  *
               +
               |
             * +                 *
               |
              *+                *
               |
     --+---+---*---+---+---+---*---+---+--
               |
               +*             *
               |
               +
               |  *         *
               +   *       *
               |    *     *
               +       *
               |

As you can see, it is negative just where we said.

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