Infinity and Inequalities

Date: 06/05/2002 at 20:56:00
From: Latasha Carter
Subject: negative and positive infinity

Dear Dr. Math,

Hi, my name is Latasha Carter and I am having a problem 
understanding negative and positive infinity. Can you help me 
understand this problem with infinity?

We have just begun this lesson and I already have homework on it. Can
you break it down in simple terms so that I can memorize it and learn
it better?

Here's an example of the problem: 

  2x - 3 < 7

In this problem we are looking for the solution set using positive and
negative infinity. Please e-mail me as soon as possible.

                         Thank you, 
                    Troubled Math Student
                        Latasha Carter

Date: 06/05/2002 at 23:32:23
From: Doctor Peterson
Subject: Re: negative and positive infinity

Hi, Latasha.

Infinity, whether positive or negative, is not a number, but just 
sort of a signpost saying "keep going that way and never stop". (The 
word "infinite" actually means "no finish".) We indicate it on a 
number line by an arrow:

    <-----------------------+-------------------------->
   -oo                      0                         +oo

We don't actually label the arrows that way, or think of them as 
actually representing infinity; we just know that the arrow means the 
line goes farther than any number in each direction.

Now, we can solve your inequality by doing the same manipulations you 
do to solve an equation, as long as we don't multiply or divide by a 
negative number. (If you do, you have to reverse the direction of the 
inequality; the real no-no is to multiply or divide by something 
whose sign you don't know!) In this example, the left side says we 
multiply by 2 and then subtract 3, so we undo those in the opposite 
order, adding 3 and then dividing by 2. We do the same thing to both 
sides:

    2x - 3 < 7

    2x < 7 + 3

    2x < 10

    x < 10 / 2

    x < 5

That's the solution, and it doesn't involve infinity. If we graph it, 
we see infinity start to come into play:

    <=======================+====o--------------------->
   -oo                      0    5                    +oo

Any x less than 5 is a solution; and we can say in a sense that any x 
between negative infinity and 5 is a solution.

I suspect that you are supposed to use infinity because you are using 
interval notation, where for example the numbers between 1 and 2 
would be indicated as (1,2), and the round parentheses mean that 1 
and 2 are not included.

Using that notation, we can write our solution set as

    (-oo, 5)

meaning that anything between negative infinity and 5 is a solution, 
NOT INCLUDING either -oo or 5. Note that infinity can NEVER be 
included in a solution, since it is not a number. You can think of 
negative infinity here as if it were something less than all numbers, 
that is put here in place of a number just to give us something to 
mark the left end of the interval. It really means, as I suggested at 
the start, "go all the way to the left without stopping", or "there 
is no left end to this interval". And that's all it means!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/