SOLUTION: Solve the following system of linear inequalities by graphing. 3x + 4y is less than or equal to 12 x + 3y is less than or equal to 6 x is greater than or e

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Question 85018: Solve the following system of linear inequalities by graphing.
3x + 4y is less than or equal to 12
x + 3y is less than or equal to 6
x is greater than or equal to 0
y is greater than or equal to 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.

So 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, but this point is the easiest to work with), and evaluate the inequality
Plug in x=0, y=0

Simplify

Since this inequality is true, we shade the entire region containing (0,0)

Here is the graph of with the graph of the line() in red and the shaded region in green
(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)

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, but this point is the easiest to work with), and evaluate the inequality
Plug in x=0, y=0

Simplify

Since this inequality is true, we shade the entire region containing (0,0)

Here is the graph of with the graph of the line() in red and the shaded region in green
(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)

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 (note: if you need help with graphing, check out this solver )
graph of
Now lets pick a test point, say (1,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality
Plug in x=1

Simplify

Since this inequality is true, we shade the entire region that contains (1,0)

Here is the graph of with the graph of the line() in red and the shaded region in green
(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)

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 (note: if you need help with graphing, check out this solver )
graph of
Now lets pick a test point, say (0,1) (any point will work, but this point is the easiest to work with), and evaluate the inequality
Plug in y=1

Simplify

Since this inequality is true, we shade the entire region that contains (0,1)

Here is the graph of with the graph of the line() in red and the shaded region in green
(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)
So we essentially have these 4 regions
Region #1 which is the graph of
Region #2 which is the graph of
Region #3 which is the graph of
Region #4 which is the graph of

So these 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 dots