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Date: 09/29/98 at 22:23:03
From: Ariela Fernandez
Subject: Inequalities

Help! I have forgotten how to add or subtract inequalities. Can you add
inequalities in which the sign is facing the same way (for example "x
is greater than or equal to 4 and 4x is greater than or equal to 7")?
And can you add/subtract inequalities in which the signs face opposite
ways? Help me.

Ariela
```

```
Date: 09/30/98 at 12:29:35
From: Doctor Peterson
Subject: Re: Inequalities

Hi, Ariela. Like many things in math, if you forget the rule you can
just apply some common sense to figure it out.

Suppose you and a friend are sitting on a seesaw, and your side is down
because you are heavier. Two friends come along, and you ask the
heavier one to join you on your side, and the lighter one to join your
friend. Can you be sure your side will still be heavier, or is there
some chance that it might reverse? Now suppose instead that you know
that you and a friend together weigh more than two other friends
together. If the friend on your side gets off, and someone who is
lighter gets off the other side, might the relation change?

The answer is that you CAN add inequalities with the same direction,
and you can subtract opposite inequalities, but not the other way
around. For example, since we know that:

5 > 4  and
3 > 1

we can add them and know that:

8 > 5

but we can't subtract them and get:

2 > 3  (WRONG!)

However, we can write it as:

5 > 4  and
1 < 3

and then we can subtract them:

4 > 1

6 > 7  (WRONG!)

I prefer only to add inequalities that go in the same direction, and
never subtract inequalities; instead, I multiply the one to be

5 > 4  and  -->  5 >  4
1 < 3       --> -1 > -3
---------
4 >  1

That way there's only one rule to remember: the sum of the two greater
quantities is greater than the sum of the two lesser quantities. That
makes sense, doesn't it?

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
Middle School Algebra

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