Inequalities on a Number Line

Date: 05/07/2003 at 21:46:02
From: Gayle 
Subject: How do you graph inequalities on a number line?

Will you please give me an example how to graph an inequality on a 
number line?

  all real numbers less then 2
  all real numbers greater then -3
  all real numbers less then or equal to -1
  all real numbers greater then or equal to 0

Date: 05/08/2003 at 12:30:56
From: Doctor Peterson
Subject: Re: How do you graph inequalities on a number line?

Hi, Gayle.

The graph of an inequality like these amounts to shading in the part 
of the number line that it describes. We use an open circle to 
indicate a value that is not included in the graph but is an endpoint 
of it, and a solid circle for an endpoint that is included.

For example, the inequality

  x < 3

describes all real numbers less than 3. On the number line, this 
means everything to the left of 3, and not including 3. So we draw a 
number line

  <--+---+---+---+---+---+---+---+---+---+---+---+---+-->
    -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6

and then we make a heavy line over the part of it to the left of 5, 
and make an open circle at 3 to show it is not included:

  <==+===+===+===+===+===+===+===+===+===o---+---+---+-->
    -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6

The opposite of this is

  x >= 3

which means all real numbers greater than or equal to 3. So everything 
to the right of 3 is included, and 3 is also included:

  <--+---+---+---+---+---+---+---+---+---*===+===+===+==>
    -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6

Does that help?

Now, can you tell what inequality this represents?

  <--+---+---+---+---o===+===+===+===+===*---+---+---+-->
    -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6

It will take two comparisons to describe it: all real numbers greater 
than -2 AND less than or equal to 3. We can write this as

  x > -2 and x <= 3

or combine it as

  -2 < x <= 3

You will get to this kind of problem eventually; I give it to you 
just so you won't be surprised by it.

If you have any further questions, feel free to write back.

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