Graphing without the Y

Date: 01/28/2004 at 20:43:42
From: Stephanie
Subject: Algebra

I need help with graphing inequalities.  There is a certain one that 
doesn't make any sense to me.  It is:

  x > -5

There is no "y", so how do I do it?

Date: 01/29/2004 at 09:48:37
From: Doctor Ian
Subject: How do you deal with a missing y?

Hi Stephanie,

There are a couple of ways to look at this.  One is to forget about
the coordinate plane, and go back to the number line.  On a number
line, you want to emphasize (by making it bolder) the part of the line
that satisfies the inequality, e.g., to graph

  3 < x < 7
        -

I would do this:

   <--|--|--o===========*--|--|-->
      1  2  3  4  5  6  7  8  9 
 
Here the 'o' represents an open endpoint, meaning I can get almost
down to 3, but never quite there; while the '*' represents a closed
endpoint, meaning I can get all the way to 7. 

Does this make sense?

Now, suppose I really want to graph this on a coordinate plane.  Let's
think for a moment about the equation of a vertical line on the plane.
That might look like

  x = 4

right?  There's no y!  So what can we do?  Well, when no constraints
are placed on y, it means that y can take _any_ value at all.  In the
case of a vertical line, y can be 2, giving us the point

  (4,2)

or it can be 6.3, giving us the point

  (4,6.3)

and so on.  When we fill in _all_ the possible values of y, we get a
vertical line. 

In the case of an inequality, we get a vertical line through each
possible value of x.  That would include when x is 4,

               |
               |
               |
   <--|--|--o==|========*--|--|-->
      1  2  3  |  5  6  7  8  9 
               |
               |

and when x is 5.3, 

               |   |
               |   |
               |   |
   <--|--|--o==|===|====*--|--|-->
      1  2  3  |  5| 6  7  8  9 
               |   |
               |   |

and every other possible value of x in the specified range.  When we
fill all those lines in, what will we get?  We'll end up with a shaded
region of the plane, 

            o###########*
            o###########*
            o###########*
   <--|--|--o###########*--|--|-->
      1  2  o###########*  8  9 
            o###########*
            o###########*

Here, the 'o's would normally be drawn as a dashed line, showing that
we can't quite get over to x = 3; and the '*'s would be drawn as a
solid line, to show that we can get all the way to x = 7. 

Does this make sense? 

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