SOLUTION: Solve the following system of linear inequalities by graphing. x – 2y less than or equal to 4 x greater than or equal to 1

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Question 85010: Solve the following system of linear inequalities by graphing.
x – 2y less than or equal to 4
x greater than or equal to 1
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-2y%3C=4

x%3E=1



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-2y%3C=4 we need to graph the equation x-2y=4 (just replace the inequality sign with an equal sign). So lets graph the line x-2y=4 (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%284-1%2Ax%29%2F-2%29 graph of x-2y=4
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-2y%3C=4
%280%29-2%280%29%3C=4 Plug in x=0, y=0

0%3C=4 Simplify


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



Here is the graph of x-2y%3C=4 with the graph of the line(x-2y=4) 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 x%3E=1 we need to graph the equation x=1 (just replace the inequality sign with an equal sign). So lets graph the line x=1 (note: if you need help with graphing, check out this solver)
graph%28+400%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C1000%28x-1000%2F1000%29%29 graph of x=1
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%3E=1
%280%29%3E=1 Plug in x=0, y=0

0%3E=1 Simplify


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



Here is the graph of x%3E=1 with the graph of the line(x=1) 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-2y%3C=4
Region #2 which is the graph of x%3E=1


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)