Graphing Inequalities

Date: 06/05/2003 at 23:52:25
From: Dena
Subject: Inequalities

How do you graph inequalities?

Ex: y - 2x > 16

It always comes out a different answer.

Date: 06/07/2003 at 13:06:55
From: Doctor Dotty
Subject: Re: Inequalities

Hi Dena,

I will use a similar inequality, y + 4x < 20.

We will end up with a graph, some of which will be shaded. Somehow we 
need to find the line that is the border between the shaded area and 
the non-shaded. This is at y + 4x = 20. 

This is because one side, where y + 4x < 20, isn't shaded, whereas the
other side is.

So, we need to plot this line on a graph.

We know that the line is linear (a straight line), as there are only
two variables, each with an exponent of 1. This means that there
aren't any powers, roots, etc.

So, let's find a couple of points that the line goes through.

What about when x = 0?

        y + 4x = 20
       y + 4*0 = 20
             y = 20

So the line goes through (0, 20).

What about when y = 0?

        y + 4x = 20
        0 + 4x = 20
             x = 5

So the line goes through (5, 0).

Let's plot these points and join them.

                     |                
                  20 +                
                     |.               
                     |                
                     |                
                     |  .             
                     |                
                     |                
                     |    .           
    -----------------+-----+----------
                     |     5          
                     |      .         
                     |                
                     |                
                     |        .       
                     |                
                     |                
                     |          .     
                     |                

Now we only need to know which side of the line to shade in.

The way we do that is simple. Let's choose any point. Let's take 
(0,0)  as it is easy.

        y + 4x <> 20

       0 + 4*0 <> 20

             0 <  20

At (0, 0), y + 4x < 20. This is in the area we want, so we can shade 
it:

    xxxxxxxxxxxxxxxxx|                
    xxxxxxxxxxxxxx20 +                
    xxxxxxxxxxxxxxxxx|.               
    xxxxxxxxxxxxxxxxx|x               
    xxxxxxxxxxxxxxxxx|x               
    xxxxxxxxxxxxxxxxx|xx.             
    xxxxxxxxxxxxxxxxx|xxx             
    xxxxxxxxxxxxxxxxx|xxx             
    xxxxxxxxxxxxxxxxx|xxxx.           
    -----------------+-----+----------
    xxxxxxxxxxxxxxxxx|xxxxx5          
    xxxxxxxxxxxxxxxxx|xxxxxx.         
    xxxxxxxxxxxxxxxxx|xxxxxx          
    xxxxxxxxxxxxxxxxx|xxxxxxx         
    xxxxxxxxxxxxxxxxx|xxxxxxxx.       
    xxxxxxxxxxxxxxxxx|xxxxxxxx        
    xxxxxxxxxxxxxxxxx|xxxxxxxxx       
    xxxxxxxxxxxxxxxxx|xxxxxxxxxx.     
    xxxxxxxxxxxxxxxxx|xxxxxxxxxxx     

The last thing is the line style. If the graph includes the line (i.e.
it is a 'greater than or equal' to or 'less than or equal to' graph)
then the line should be solid. If the graph does not include the line
(i.e. it is a 'strictly less than' or 'strictly greater than' graph)
then it is a broken line.

In our case, it is a 'strictly less than' graph, so the line should be
broken (dashed).

Can you draw the graph for your inequality in a similar way?

Write back if I can be of any more help - on this or anything else.

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