Inequality Sign Switch

Date: 01/13/2003 at 00:48:52
From: Devin
Subject: Inequalities

How do you solve the inequality

   -1/8w < 3 ?

         -1/8w < 3
  -8/1 x -1/8w < 3
             w < -24

Date: 01/13/2003 at 09:22:58
From: Doctor Rick
Subject: Re: Inequalities

Hi, Devin.

I believe you mean (-1/8)w < 3, rather than -1/(8w) < 3. It's hard to 
be sure, so I prefer to use parentheses so no one could misinterpret 
what I write.

In your thoughts, you didn't write out that you were multiplying the 
right side by -8/1 as well, but I can see that this is what you're 
doing:

      (-1/8)w < 3
  (-8)(-1/8)w < (-8)3
            w < -24

You multiplied by the reciprocal of -1/8, which is -8/1, or -8. 
That's correct - but when we multiply an inequality through by a 
negative quantity, we have to reverse the direction of the inequality.

      (-1/8)w < 3
  (-8)(-1/8)w > (-8)3
            w > -24

Let me show you that this is needed by doing the problem another way. 
First I add (1/8)w to each side, then I subtract 3 from each side:

           (-1/8)w < 3
  (1/8)w + (-1/8)w < (1/8)w + 3
                 0 < (1/8)w + 3

             0 - 3 < (1/8)w + 3 - 3
                -3 < (1/8)w

Now I can multiply both sides by 8:

  -3*8 < 8(1/8)w
   -24 < w

This is the same as

     w > -24

Do you see how the sign got switched around? When I solved the problem 
so that I never multiplied by a negative number, I got the quantities 
on opposite sides of the inequality. When I switch them back to the 
sides I wanted them on, I had to reverse the inequality sign too.

You can check the results by picking a number greater than -24, and a 
number less than -24, and seeing which gives you a true inequality. 
For instance, if w = -23 (which is greater than -24, right?), I get

      (-1/8)w < 3
  (-1/8)(-23) < 3
         23/8 < 3
        2 7/8 < 3

This is a true statement. If I tried a value of w less than -24, I'd 
find that it results in a false statement.

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

Date: 02/01/2003 at 17:57:20
From: Devin
Subject: Thank you (Inequalities)

Thank you Dr. Rick. My teacher showed us how to work the problem, but 
the way you explained "why" the turn around of the inequality sign 
happens makes it clear to me now. Thanks a lot!
Devin