SOLUTION: Solve the following system of linear inequalities by graphing. x + 2y is less than or equal to 3 2x – 3y is less than or equal to 6

Algebra ->  Graphs -> SOLUTION: Solve the following system of linear inequalities by graphing. x + 2y is less than or equal to 3 2x – 3y is less than or equal to 6       Log On


   

Question 85019: Solve the following system of linear inequalities by graphing.
x + 2y is less than or equal to 3
2x – 3y is less than or equal to 6
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Start with the given system of inequalities
x%2B2y%3C=3

2x-3y%3C=6



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 x%2B2y%3C=3 we need to graph the equation x%2B2y=3 (just replace the inequality sign with an equal sign). So lets graph the line x%2B2y=3 (note: if you need help with graphing, check out this solver)
graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%283-1%2Ax%29%2F2%29 graph of x%2B2y=3
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 x%2B2y%3C=3
%280%29%2B2%280%29%3C=3 Plug in x=0, y=0

0%3C=10 Simplify


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



Here is the graph of x%2B2y%3C=3 with the graph of the line(x%2B2y=3) 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 2x-3y%3C=6 we need to graph the equation 2x-3y=6 (just replace the inequality sign with an equal sign). So lets graph the line 2x-3y=6 (note: if you need help with graphing, check out this solver)
graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%286-2%2Ax%29%2F-3%29 graph of 2x-3y=6
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 2x-3y%3C=6
2%280%29-3%280%29%3C=6 Plug in x=0, y=0

0%3C=10 Simplify


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



Here is the graph of 2x-3y%3C=6 with the graph of the line(2x-3y=6) 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 2 regions
Region #1 which is the graph of x%2B2y%3C=3
Region #2 which is the graph of 2x-3y%3C=6


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 2 regions represented by the dots (note: this region extends to infinity)