Advertisement

Ok, now you're an expert at graphing lines like

x + y = 3

But, what if we stick an inequality in there?

x + y is less than or equal to 3

Don't worry.  If you use your head, it's not that bad.  It's just one more thing to worry about.

Let's graph the line:

the graph of x + y is less than or equal to 3, which passes through the points ( 0 , 3 ) and ( 3 , 0 )
This takes care of the
x + y = 3 part ...

So, what about the

<?

Remember that this rectangular coordinate grid is just made up of a bunch of ( x , y ) points.

All the points ON the line work in this part:

x + y = 3

Now, we just need to find all the points that work in this part:

x + y < 3

Here's how we find these points:

We just pick a point OFF the line and test it!

the graph of x + y < 3 ... test the point ( 0 , 0 ) The origin, ( 0 , 0 ), is really easy to work with, so let's try it:

x + y < 3 ... 0 + 0 < 3 ... 0 < 3

( 0 , 0 ) works in x + y < 3 !

Here's the deal: 

If ( 0 , 0 ) works, then ALL the points
on the same side of the line will work.

Don't believe me?  Let's try a couple more:

the graph of x + y < 3 ... test the points ( -1 , 2 ) or ( 2 , 3 ) ... try plugging the point ( -1 , 2 ) into the equation x + y < 3 ... -1 + 2 < 3 ... 1 < 3 ... yep!  ... try plugging the point ( 2 , 3 ) into the equation x + y < 3 ... 2 + 3 < 3 ... 5 < 3 ...nope!

So, our answer is this:

a graph of x + y is less than or equal to 3 ... the portion of the graph under the line is shaded All the ( x , y ) points on the line and in the shaded region work in

x + y is less than or equal to 3