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I'm starting to learn about linear programming. On one of my HW questions I was able to write the LP, graph (shown below), and find the optimal solution (3,2) without a problem. When I checked my solution in the manual, I had everything correct expect the solution manual said that (0,0) is also in the feasible region and should be tested. Visually, (0,0) is not in the feasible region, so I must be missing something. This is the second question where (0,0) has been tested, even though it's not in the feasible region. I appreciate any insight you can provide.

Minimize $z=4p_1+p_2$

Q4 Graph

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    Since you have both $p_1 \geq 3$ and $p_1 \geq 0$, I suspect that there is a typo in either the question or the solution, because you are completely right that $(0,0)$ is not in the feasible region. – 2017-02-02

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I don't find any mistake in your solution. The feasible region is correctly determined. Of course, the origin is not a feasible solution here.

Sometimes the solution manuals are incorrect. So, don't worry.