Q.Solve graphically: Minimize $Z = 20x + 10y$ subject to $x + 2y \ge 40,\ 3x + y \ge 30,\ x, y \ge 0$. Does $Z$ have a maximum value on this feasible region? Justify your answer.

The feasible region is bounded if it can be enclosed within some finite circle (typically when every non-trivial constraint is $\le$ type, confining the region near the origin), and unbounded if it extends without limit in some direction (typically when at least one constraint is $\ge$ type). For a bounded region, the largest/smallest value of $Z$ among the corner points is automatically the true maximum/minimum. For an unbounded region, this must be confirmed with the half-plane test: if the open half-plane $\{ax+by>k\}$ (for a maximum) or $\{ax+by<k\}$ …