Inequalities and Negative Numbers

Date: Mon, 7 Nov 1994 12:40:55 -0500
From: Steve Barkin
Subject: Inequality

When multiplying both sides of an inequality by a negative number, 
why does the inequality sign change?

Chris Graham and Parks
Steve Barkin
Graham & Parks School
Cambridge, Ma 02139

From: Dr. Ken
Date: Mon, 7 Nov 1994 13:06:25 -0500 (EST)

Hello Steve!

Great question.  It's something that trips up a lot of people.  
Here are a couple of different ways to see why it's true:

1) Let's say you have an inequality, like 5 < 3 .  We can picture 
these on a number line like this:

<____________________________0__________3_______5__________>

But where are negative 3 and negative 5?  Over to the left:

<_______-5_______-3__________0__________3_______5__________>

So you can see that -3 > -5 .


2) The other good way to see why this is true (suggested to me 
by my Math Doctor friend Ethan) is to look at it algebraically. 
If x < y, then look at the following steps:

 x < y         subtract x from both sides
 0 < y - x     subtract y from both sides
-y < -x        flip it around (I'm not really changing anything)
-x > -y

That's the other way that I know of to see it.  Does this help?

-Ken "Dr." Math
__________

Date: Thu, 10 Nov 1994 21:01:15 -0500
From: Anonymous
Subject: Re: Inequality

Hi Steve, I assume you realized that Ken made a typo below when he
wrote 5 < 3 .

I also wondered whether it helped to ask the question back: "why 
shouldn't it be switched?"  Inequalities have "direction." Reversing 
"polarity" (positive to negative or vice versa) implies a change in 
direction, doesn't it? It would be strange if it didn't switch.  Maybe 
it would help if you could figure out why you didn't think it would 
switch.

This is the way electricity works.  Current goes from the positive side 
tot he negative.  If you switch the positive and negative ends then the
current goes the other direction.

Or another version: "if inequalities change, why don't equalities?"  
Because "equals" doesn't care what side you're on. a=b means b=a.  
Not so for inequalities.

-- steve ("chief of this wonderful staff")

>When multiplying both sides of an inequality by a negative number, why does
>the inequality sign change?

Ken writes:
>Great question.  It's something that trips up a lot of people.  Here are a
>couple of different ways to see why it's true:
>
>1) Let's say you have an inequality, like 5 < 3 .  We can picture these 
>on a number line like this: . . .

Date: Sat, 12 Nov 1994 11:07:00 -0500
From: steve barkin
Subject: Re: inequality

Hi Steve, 

Thank you for the correction to Ken's typo. I mentioned that there was 
an error with my class. Chris, one of my algebra students, wrote the 
letter. We are letting our students use the Internet through our addresses. 

One of the ideas in my 8th grade Algebra class is ownership of ideas. 
"If you cannot explain it, you don't own it. Often students come to me 
knowing formulas as A = pr2 yet cannot explain it. I must add that I 
have had MIT students who intern here also not being able to explain 
why, but know procedures. 

Next week they will get your reply and have to write a journal page of 
what you mean. Do they buy the expanation ?  I often ask them to prove 
it to me. My role has changed of not always being the expert in the 
classroom. 

Wait until they ask about the cones Volume  = 1/3 pr square times
height. They have derived it by pouring popcorn in cyliners, cans and 
liter cubes as they had other formulas. Yet they cannot explain the 1/3. 
The prism volume was clear. The 1/2 is understood.  The 1/3 has them 
stumped. I know of no physical model that is convincing to them at the 
beginning of Algebra 1. I hate saying to them to wait until calculus. 
I do say that some ideas in math will not be understood right away. 
Even if we are convinced they are true. 

My students often have to do a journal page with the heading "How
do you know this is true?"

Thanks for the reply.  
Steve Barkin
Graham & Parks School
Cambridge, Ma 02139