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### Negatives and Inequalities

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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
sign?

Thank you.
```

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

Prerana,

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:

----------------------0----1----2----3--------------------12-----

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
Check out our web site! http://mathforum.org/dr.math/
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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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