Question

The common region of the solutions of the inequations $x + 2y \ge 4$, $2x - y \le 6$ and $x, y > 0$ is

• A. bounded and origin side
• B. unbounded and non-origin side
• C. unbounded and origin side
• D. bounded and non-origin side

Hint:

Whenever a linear programming question has constraints with opposing signs like $\ge$ on positive variables mixed with a subtraction line like $2x - y \le 6$, the bounding lines often do not form a closed polygon in the first quadrant. This lack of a closed boundary immediately indicates an unbounded feasible layout!

Answer 1

The Correct Option is B

Step 1: List the constraints.
We have $x + 2y \ge 4$, $2x - y \le 6$, with $x, y > 0$, so we work in the first quadrant.
Step 2: Test the origin in the first constraint.
Put $(0,0)$ into $x+2y\ge 4$: $0 \ge 4$ is false, so the feasible side is away from the origin (non-origin side).
Step 3: Test the origin in the second constraint.
Put $(0,0)$ into $2x - y \le 6$: $0 \le 6$ is true, so this half-plane contains the origin.
Step 4: Intersect within the first quadrant.
The region lies above the line $x+2y=4$ and below the line $2x-y=6$, restricted to $x,y>0$.
Step 5: Check for closure.
As $y$ grows large the region keeps widening with no upper line to cap it, so it never closes off.
Step 6: Classify.
The region extends to infinity, so it is unbounded, and because the first constraint excludes the origin it lies on the non-origin side.
\[ \boxed{\text{unbounded and non-origin side}} \]