SOLUTION: I need help with this last problem. Graph the system of inequalities: y_>x and x<3

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Question 87681: I need help with this last problem.
Graph the system of inequalities: y_>x and x<3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given system of inequalities
y%3E=x
x%3C3

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


First lets graph the first inequality y%3E=x
In order to graph y%3E=x, we need to graph the equation y=x (just replace the inequality sign with an equal sign).
So lets graph the line y=x (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+x%29+ graph of y=x
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 y%3E=x

Substitute (0,0) into the inequality
1%3E=0 Plug in x=0 and y=1
1%3E=0 Simplify
Since this inequality is true, we simply shade the entire region that contains (0,1)
Graph of y%3E=x with the boundary (which is the line y=x in red) and the shaded region (in green)

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Now lets graph the second inequality x%3C3
In order to graph x%3C3, we need to graph the equation x=3 (just replace the inequality sign with an equal sign).
So lets graph the line x=3 (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-3%29%29+ graph of x=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%3C3

Substitute (0,0) into the inequality
%280%29%3C3 Plug in x=0 and y=0
0%3C3 Simplify
Since this inequality is true, we simply shade the entire region that contains (0,0)
Graph of x%3C3 with the boundary (which is the line x=3 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 y%3E=x


Region #2
Graph of x%3C3




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.







Here is a cleaner look at the intersection of regions




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