SOLUTION: solve the following system of linear inequalities by graphing. x-2y greater than 4 x less than 4 can you show me the graph too. Thank you

Algebra ->  Linear-equations -> SOLUTION: solve the following system of linear inequalities by graphing. x-2y greater than 4 x less than 4 can you show me the graph too. Thank you      Log On


   

Question 89191: solve the following system of linear inequalities by graphing.
x-2y greater than 4
x less than 4
can you show me the graph too.
Thank you
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%3E4
x%3C4

In order to graph this system of inequalities, we need to graph each inequality one at a time.


First lets graph the first inequality x-2y%3E4
In order to graph x-2y%3E4, 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+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C+%281%2F2%29x-2%29+ graph of x-2y=4
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 x-2y%3E4 with the test point

Substitute (0,0) into the inequality
%280%29-2%280%29%3E4 Plug in x=0 and y=0
0%3E4 Simplify

Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line
Graph of x-2y%3E4 with the boundary (which is the line x-2y=4 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 second inequality x%3C4
In order to graph x%3C4, we need to graph the equation x=4 (just replace the inequality sign with an equal sign).
So lets graph the line x=4 (note: if you need help with graphing, check out this solver)

+graph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C+1000%28x-4%29%29+ graph of x=4
Now lets pick a test point, say (-3,1). 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 x%3C4 with the test point

Substitute (-3,1) into the inequality
%28-3%29%3C4 Plug in x=-3 and y=1
-3%3C4 Simplify

Since this inequality is true, we simply shade the entire region that contains (-3,1)
Graph of x%3C4 with the boundary (which is the line x=4 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)
---------------------------------------------------------------


So we essentially have these 2 regions:

Region #1
Graph of x-2y%3E4


Region #2
Graph of x%3C4




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. (note: for some reason, this image and the following image does not display in Internet Explorer. So I would recommend the use of Firefox to see these images.)







Here is a cleaner look at the intersection of regions




Here is the intersection of the 2 regions represented by the series of dots