SOLUTION: Graph the inequality on a plane. 3x+4y≤12

Graph the boundary line which has its equation formed by replacing the
inequality symbol by =.  Draw the line solid since the inequality symbol is
 and not .

To graph the boundary line , get some
points on it:

 x | y
 0 |  3
 4 |  0
 8 | -3
-4 |  6
-8 |  9


Draw a solid line through those points, which is the boundary line.




Now we choose any point which is not on the green boundary line, as a 
test point to find out which side of the boundary line all the solutions
of the original inequality. 

,  

Let's choose, say the point (6,5) as a test point.  Any other point not on
the line would do just as well.  



So we substitute the test point (x,y) = (6,5) in the
inequality to see if it comes out true or false. 

,  

,  
   
,  

.

That is false, so the solutions to the original inequality are not on 
the side which our test point is on, and therefore its solutions are all
on the other side of the boundary line.

So we shade the side of the line that the test point was NOT on,
which was the lower side of the green line.  

I can't shade on here because this graphing program has no shading 
function, but you can shade the lower side of the green line on your 
paper.

Shortcut:

Since the origin )x,y) = (0,0) is the easiest possible point to test, and it
is not on the green boundary line, you can use it as a test point.  If we had
used the origin (0,0,) as a test point instead of (6,5), then substituting 
the origin

(x,y) = (0,0) 

in the original inequality:

,  

,  
   
,  

.

That is true, and so we would also have known to shade the side of the green
line which the origin (0,0) is on, and that would have also been the lower
side, the same side as we determined using the test point (6,5).  So 
you can use the origin (0,0) as a test point any time the boundary
line does not go through the origin.  Only then will you have to use
another point for a test point.  

Don't forget to shade the area below the green line on your paper, which
I can't do on here.  The line only will get marked wrong.  You must have the
region below the green line shaded.

Edwin