Feasible and Infeasible Regions: Bounded and Unbounded

Feasible region, precisely. The feasible region of an LPP is the set of all points $(x,y)$ in the plane that satisfy every constraint of the problem simultaneously -- both the non-trivial constraints and the non-negative restrictions $x\ge0,\ y\ge0$. Graphically (Section 3) it is exactly the shaded polygon or shaded area obtained by intersecting all the required half-planes. A point lying outside this region -- violating at least one constraint -- lies in the infeasible region.

Bounded regions. A feasible region is called bounded if it can be entirely enclosed within some sufficiently large circle (equivalently, if both the $x$-coordinate and the $y$-coordinate of every point in the region are less than some fixed finite number). This typically happens when EVERY non-trivial constraint is of $\le$ type (an upper limit on a resource), since each such constraint, together with $x,y\ge0$, confines the region to a finite triangle or polygon near the origin. A bounded feasible region has finitely many corner points, and -- as the Corner Point Theorem in Section 6 guarantees -- an optimal value of the objective function is ALWAYS attained somewhere on this region, at one of those corners.

Unbounded regions. A feasible region is called unbounded if it extends without limit in at least one direction -- there is no finite circle large enough to contain it. This typically happens when at least one non-trivial constraint is of $\ge$ type (a minimum requirement rather than a cap), since such a constraint, together with $x,y\ge0$, permits $x$ or $y$ to grow arbitrarily large while still satisfying the inequality. An unbounded region still has finitely many corner points along its "inner" boundary, but the region also contains points arbitrarily far from the origin in one or more directions.

Why an unbounded region needs extra care. For a BOUNDED region, the largest (or smallest) value of $Z$ among the finitely many corner points is automatically the true maximum (or minimum) of $Z$ over the whole region -- there is nowhere else in a bounded region for $Z$ to be larger, precisely because the region contains no points "further out" than its corners in any relevant direction. For an UNBOUNDED region, this guarantee can fail: since the region keeps extending outward, it is possible for $Z$ to keep growing (or shrinking) forever along some unbounded direction, so that no maximum (or minimum) value exists at all, even though every corner point gives a perfectly finite number. …