SOLUTION: Solve the following system of linear inequalities by graphing. 3x – y < 2 x + y > 2

Algebra ->  Algebra  -> Inequalities -> SOLUTION: Solve the following system of linear inequalities by graphing. 3x – y < 2 x + y > 2       Log On

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Question 85713: Solve the following system of linear inequalities by graphing.
3x – y < 2
x + y > 2

Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
12.

Start with the given system of inequalities
3x-y%3C2

x%2By%3E2



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 3x-y%3C2 we need to graph the equation 3x-y=2 (just replace the inequality sign with an equal sign). So lets graph the line 3x-y=2 (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%282-3%2Ax%29%2F-1%29 graph of 3x-y=2
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 3x-y%3C2
3%280%29-%280%29%3C2 Plug in x=0, y=0

0%3C2 Simplify


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


drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B9.75%29%29
Here is the graph of 3x-y%3C2 with the graph of the line(3x-y=2) in red and the shaded region in green
(note: The red line should be a dashed line since it is not included in the region. Since the inequalityis a < sign, it tells us not to include the boundaries.)




Now lets graph the second inequality

In order to graph x%2By%3E2 we need to graph the equation x%2By=2 (just replace the inequality sign with an equal sign). So lets graph the line x%2By=2 (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%282-1%2Ax%29%2F1%29 graph of x%2By=2
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%2By%3E2
%280%29%2B%280%29%3E2 Plug in x=0, y=0

0%3E2 Simplify


Since this inequality is not true, we shade the entire region that doesn't contain (0,0)


drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B9.75%29%29
Here is the graph of x%2By%3E2 with the graph of the line(x%2By=2) in red and the shaded region in green
(note: The red line should be a dashed line since it is not included in the region. Since the inequalityis a > sign, it tells us not to include the boundaries.)
So we essentially have these 2 regions
drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B9.75%29%29 Region #1 which is the graph of 3x-y%3C2
drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2Cgraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B9.75%29%29 Region #2 which is the graph of x%2By%3E2


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.
%0D%0A%0D%0Adrawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C+%282-3%2Ax%29%2F-1%2B9.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B0%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B0.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B1.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B2.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B3%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B3.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B4.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B5.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B6%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B6.75%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B7.5%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B8.25%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B9%29%2C%0D%0Agraph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-1%2Ax%29%2F1%2C+%282-1%2Ax%29%2F1%2B9.75%29%29
Here is a cleaner look at the intersection of regions

drawing%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A++graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%282-3%2Ax%29%2F-1%2C%282-1%2Ax%29%2F1%29%2Ccircle%28-6.5%2C9.25%2C0.05%29%2C%0D%0A+circle%28-6.5%2C9.25%2C0.08%29%2Ccircle%28-4.75%2C7.5%2C0.05%29%2C%0D%0A+circle%28-4.75%2C7.5%2C0.08%29%2Ccircle%28-4.75%2C9.25%2C0.05%29%2C%0D%0A+circle%28-4.75%2C9.25%2C0.08%29%2Ccircle%28-3%2C5.75%2C0.05%29%2C%0D%0A+circle%28-3%2C5.75%2C0.08%29%2Ccircle%28-3%2C7.5%2C0.05%29%2C%0D%0A+circle%28-3%2C7.5%2C0.08%29%2Ccircle%28-3%2C9.25%2C0.05%29%2C%0D%0A+circle%28-3%2C9.25%2C0.08%29%2Ccircle%28-1.25%2C4%2C0.05%29%2C%0D%0A+circle%28-1.25%2C4%2C0.08%29%2Ccircle%28-1.25%2C5.75%2C0.05%29%2C%0D%0A+circle%28-1.25%2C5.75%2C0.08%29%2Ccircle%28-1.25%2C7.5%2C0.05%29%2C%0D%0A+circle%28-1.25%2C7.5%2C0.08%29%2Ccircle%28-1.25%2C9.25%2C0.05%29%2C%0D%0A+circle%28-1.25%2C9.25%2C0.08%29%2Ccircle%280.5%2C2.25%2C0.05%29%2C%0D%0A+circle%280.5%2C2.25%2C0.08%29%2Ccircle%280.5%2C4%2C0.05%29%2C%0D%0A+circle%280.5%2C4%2C0.08%29%2Ccircle%280.5%2C5.75%2C0.05%29%2C%0D%0A+circle%280.5%2C5.75%2C0.08%29%2Ccircle%280.5%2C7.5%2C0.05%29%2C%0D%0A+circle%280.5%2C7.5%2C0.08%29%2Ccircle%280.5%2C9.25%2C0.05%29%2C%0D%0A+circle%280.5%2C9.25%2C0.08%29%2Ccircle%282.25%2C5.75%2C0.05%29%2C%0D%0A+circle%282.25%2C5.75%2C0.08%29%2Ccircle%282.25%2C7.5%2C0.05%29%2C%0D%0A+circle%282.25%2C7.5%2C0.08%29%2Ccircle%282.25%2C9.25%2C0.05%29%2C%0D%0A+circle%282.25%2C9.25%2C0.08%29%29 Here is the intersection of the 2 regions represented by the dots