Question

The profit on the sale of a small bag of popcorn is $1, while the profit on the sale of a medium bag of popcorn is $2. The Glee Club expects to sell fewer than 100 total bags of popcorn, but they hope to make more than $100 in profit. The system that represents this is s+m<100 and s+2m>100. The graph of the boundary lines is shown below. Which regions should be shaded (to form the solution)?

Answer 1

If we let s be the y axis, and m be the x axis.

s+m<100

s<100-m (0<m<100)

...

s+2m>100

s>100-2m (0<m<100)

....

Taken together we have:

100-2m<s<100-m

And 0<s<100

So you would end up with a region bounded by s=100-2m on the left, s=100-m on the right, (the lines themselves are not part of the solution set) a vertical line m=1 and a horizontal line s=1. Hopefully you can visualize that from your answer choices. :)

And as the other poster pointed out, if the axis are reversed you will of course have this area rotated about the 45° line y=x.

Answer 2

Assume the graph is drawn with m as the x-axis, and s as the y-axis. If it is drawn the other way, it would be just a reflection about the x=y line.
There are two unspecified but implicit conditions, namelym>=0 and s>=0, which correspond to the m- and s-axes (horiz. & vert.) when equality holds in either case.
s+m<100 is a solid line at 45 degrees intersecting the m- and s-axes at (0,100), and (100,0) respectively.
s+2m>100 is a solid line with slope -2 intersecting the axes at (0,50) and (100,0).
The feasible region (shaded) is between the two lines, but above the horizontal axis.
If the graph is drawn the other way, the feasible region is between the two lines, and to the right of the vertical axis.