Q.The feasible region for an LPP is always a _________ polygon.

The key idea is that the feasible region of a Linear Programming Problem (LPP) is defined by a set of linear inequalities. Each inequality represents a half-plane, and the intersection of these half-planes is a convex set bounded by straight-line segments. …

The key idea here is that every linear inequality in an LPP — like $2x + 3y \leq 6$ or $x \geq 0$ — defines a half-plane in the coordinate plane. The feasible region is the intersection of all these half-planes. A fundamental property of half-planes is that they are convex sets, and the intersection of convex sets is always convex. So the feasible region, being the common area where all constraints hold, must be convex.

Why does convexity matter? It means that if you pick any two points inside the feasible region, the entire straight line segment joining them also lies inside the region. This property is what guarantees that the optimal solution (if it exists) occurs at a corner point — a vertex of the polygon. Without convexity, that neat result wouldn't hold.

Now, is the feasible region always a polygon? Yes, because each linear constraint cuts the plane with a straight line, and the intersection of finitely many half-planes produces a shape bounded by straight line segments — that's a polygon. But it's not just any polygon; it's specifically a convex polygon.

1. Each constraint gives a half-plane. For example, $x + y \leq 4$ means all points on or below the line $x + y = 4$. This half-plane is convex.

2. Intersection preserves convexity. If you take two convex sets, their overlap is also convex. Since the feasible region is the intersection of many convex half-planes, it inherits convexity. …

Method: Reasoning about the shape of a feasible region

For a blank describing the feasible region of an LPP, argue from how the region is constructed out of the constraints.

The reasoning pattern

Step 1: See each constraint as a half-plane.

Every linear inequality (such as $x + y \le 4$ or $x \ge 0$) defines a half-plane, which is a convex set.

Step 2: Use the intersection property. …

Common Mistakes

Mistake 1: Writing "bounded" instead of "convex."

Why it's wrong: the blank asks what is always true, and an LPP region is not always bounded (e.g. $x\ge 0,\ y\ge 0$ alone is unbounded) -- but it is always convex, being an intersection of half-planes …