Q1: Suppose that the following constraints have been provided for a linear programming model.

$\displaystyle -x_{1}+2x_{2}\leq\\50$ (1)

$\displaystyle -2x_{1}+x_{2}\leq\\50$ (2)

and

$\displaystyle x_{1}\geq\\0, x_{2}\geq\\0$.

a) Demonstrate that the feasible region is unbounded.

b) If the objective is to maximize $\displaystyle Z=-x_{1}+x_{2}$, does that model have an optimal soultion? If so, find it. If now, explain why not.

c) Repeat part (b) when the objective is to maximize $\displaystyle Z=x_{1}-x_{2}$.

Does "unbounded" mean something specific when dealing with LP problems? When, I graph the above lines, I am not sure where the feasible region is, since the inequality tells me to shade below both lines in the first quadrant. Furthermore, since line (2) is above line (1) I have to shade over line (1) down to the horizontal axis, which doesn't seem right, as that would mean the constraints are not being satisfied. Does this mean the the feasible region is unbouned? Also,

Moreover, if the feasible region is unbouned and the gradiants of both objective functions from parts (b) and (c) are in directions away from the first quadrant, how can there be optimal solutions?

Q2: Minimize $\displaystyle Z=15x_{1}+20x_{2}$

subject to:

$\displaystyle x_{1}+2x_{2}\geq\\10$ (1)

$\displaystyle 2x_{1}-3x_{2}\leq\\6$ (2)

$\displaystyle x_{1}+x_{2}\leq\\6$ (3)

and $\displaystyle x_{1}\geq\\0,x_{2}\geq\\0$.

For this question, I am also not sure where my feasible region is (or if one even exists). I have graphed all the lines and drawn arrows in the direction of the inequality signs, but I am not sure where to shade. I am assuming this means there is no feasible region.

Should I be checking to makie sure the inequalities hold or do I just assume they do and and shade above or below the lines accordingly?

Thanks in advanced for your time

disclaimer: My entire second week of class has been cancled due to snow days, so I only have one example of how to solve an LP in my notes and have yet to recieve my text book. Thats my excuse for these elementary questions....