Question

The common region of the solution of the inequations $x+y \ge 5$, $y \le 4$, $x \ge 2$, $x, y \ge 0$ is

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

Hint:

Whenever a linear programming question contains a constraint specifying a variable is greater than a positive value (like $x \ge 2$) without a corresponding upper bound line (like $x \le k$), the region will almost always extend infinitely along that axis, making it unbounded!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a system of linear inequalities defining a feasible region on a 2D Cartesian coordinate plane. We need to analyze the boundary constraints to determine whether the enclosed region is bounded or unbounded, and check if it lies on the origin side or the non-origin side of the main boundary line.

Step 2: Key Formula or Approach:
To check the side of a line $Ax + By \ge C$ relative to the origin, substitute $(0,0)$ into the inequality: If $0 \ge C$ is true, the region includes the origin (origin side). If $0 \ge C$ is false, the region excludes the origin (non-origin side). We will analyze the intersection geometry of all constraints to check if the area is locked within a finite perimeter.

Step 3: Detailed Explanation:
Let's analyze each inequality constraint independently: 1. $x, y \ge 0$: Limits the feasible region entirely to the first quadrant. 2. $x \ge 2$: A vertical boundary line line passing through $(2,0)$ extending to the right. 3. $y \le 4$: A horizontal boundary line passing through $(0,4)$ extending downwards. 4. $x + y \ge 5$: A diagonal boundary line passing through $(5,0)$ and $(0,5)$. Let's test the reference point $(0,0)$ in the diagonal inequality: $$0 + 0 \ge 5 \implies 0 \ge 5 \quad (\text{False})$$ Because this statement is false, the solution set lies on the non-origin side of the line $x+y=5$. Now let's examine the boundaries together in the first quadrant: As $x$ increases towards positive infinity ($x \to \infty$), the constraints $x \ge 2$ and $x+y \ge 5$ remain satisfied while keeping $y$ fixed at a value like $y=1$ (since $1 \le 4$). There is no upper vertical or rightward boundary pinning $x$ to a finite limit. Therefore, the region extends infinitely to the right, making it completely unbounded.

Step 4: Final Answer:
The common feasible region is unbounded and lies on the non-origin side, matching option (A).