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Negatives and Inequalities

Date: 09/10/98 at 19:27:52
From: Prerana Jain
Subject: Inequalities - why switch the sign?

Could you please explain to me why when we multiply or divide an 
inequality by a negative number, we have to reverse the inequality 

Thank you.

Date: 09/10/98 at 20:08:58
From: Doctor Sam
Subject: Re: Inequalities - why switch the sign?


Good question. It is puzzling why the sign should switch. Here is one 
way to think about it. Multiplying a quantity by -1 changes it into 
its opposite. For example, 3 becomes the opposite of 3 or -3, and 12 
becomes the opposite of 12, which is -12.

Now think about how these numbers fall on a number line:


Since 3 < 12, three is closer to zero and twelve is farther away.  
That's true in general. Numbers with larger magnitudes are farther 
away from zero.

Now what happens if we take opposites?  That is, if we multiply or 
divide these numbers by -1? Opposites are the same distance from zero 
so -12 will be farther away from 0 than -3:

   --- -12 --------------- -3 ----- 0 ---------------------------

This is true in general. If  A < B and both are positive, then the 
opposite of B will be farther to the left than the opposite of A, that 
is, -B < -A which means the same thing as -A > -B. If you just look at 
the symbols it seems as if we put minus signs in front of each letter 
and switch the inequality symbol. But if you look at the meaning of 
the symbols on the number line it is much clearer:

     -B            -A          0          A              B

If you look at a number line picture for A < B where both numbers are 
negative instead of positive (for example -4 < -1) you will see the 
same pattern occurs when you take the opposite of each.

The final case is A < B where A is negative and B is positive. You can 
draw a number line picture for this case too, but it is almost simpler 
to just think about the meaning of the opposites. If A is negative, 
then its opposite, -A, is a positive number. If B is positive, then its 
opposite, -B, is a negative number. Of course, any negative number is 
smaller than any positive number so -B < -A.

I hope that helps.

- Doctor Sam, The Math Forum
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Associated Topics:
High School Basic Algebra
High School Negative Numbers
Middle School Algebra
Middle School Negative Numbers

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