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### 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/
```
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations
High School Linear Equations
High School Negative Numbers
Middle School Algebra
Middle School Equations
Middle School Graphing Equations
Middle School Negative Numbers

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