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Question 107952: graph the linear equalities
3x + 4y <12
x + 3y < 6
x > 0
y > 0
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Start with the given system of inequalities
In order to graph this system of inequalities, we need to graph each inequality one at a time.
First lets graph the first inequality
In order to graph , we need to graph the equation  (just replace the inequality sign with an equal sign).
So lets graph the line (note: if you need help with graphing, check out this solver)
 graph of
Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point
Substitute (0,0) into the inequality
 Plug in  and 
 Simplify
(note: for some reason, some of the following images do not display correctly in Internet Explorer. So I would recommend the use of Firefox to see these images.)
Since this inequality is true, we simply shade the entire region that contains (0,0)
 Graph of with the boundary (which is the line in red) and the shaded region (in green)
(note: since the inequality contains a less-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)
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Now lets graph the second inequality
In order to graph , we need to graph the equation  (just replace the inequality sign with an equal sign).
So lets graph the line (note: if you need help with graphing, check out this solver)
 graph of
Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point
Substitute (0,0) into the inequality
 Plug in and
 Simplify
Since this inequality is true, we simply shade the entire region that contains (0,0)
 Graph of with the boundary (which is the line in red) and the shaded region (in green)
(note: since the inequality contains a less-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)
---------------------------------------------------------------
Now lets graph the third inequality
In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign).
So lets graph the line (simply draw a vertical line through )
 graph of (note:the graph is the line that is overlapping the y-axis. So it may be hard to see)
Now lets pick a test point, say (1,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point
Substitute (1,0) into the inequality
 Plug in  and
 Simplify
Since this inequality is true, we simply shade the entire region that contains (1,0)
 Graph of with the boundary (which is the line in red) and the shaded region (in green)
(note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)
---------------------------------------------------------------
Now lets graph the fourth inequality
In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign).
So lets graph the line (simply draw a horizontal line through )
 graph of (note:the graph is the line that is overlapping the x-axis. So it may be hard to see)
Now lets pick a test point, say (0,1). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point
Substitute (0,1) into the inequality
Plug in and 
Simplify
Since this inequality is true, we simply shade the entire region that contains (0,1)
 Graph of with the boundary (which is the line in red) and the shaded region (in green)
(note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)
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So we essentially have these 4 regions:
Region #1
 Graph of
Region #2
 Graph of
Region #3
 Graph of
Region #4
 Graph of
When these inequalities are graphed on the same coordinate system, the regions overlap to produce this region. It's a little hard to see, but after evenly shading each region, the intersecting region will be the most shaded in.
Here is a cleaner look at the intersection of regions
 Here is the intersection of the 4 regions represented by the series of dots
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