Adding and Subtracting Inequalities

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

but we can't add them:

    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 
subtracted by -1 and add:

    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/