Question

The region represented by the system of inequations \(3x + y \ge 3, 2x - y \ge - 5, x, y \ge 0\) is :

• A. unbounded in \(1^{st}\) quadrant
• B. bounded in \(1^{st}\) quadrant
• C. unbounded in \(2^{nd}\) quadrant
• D. bounded in \(2^{nd}\) quadrant

Hint:

To quickly see if a region is unbounded, check if a very large point like \((1000, 1000)\) satisfies the inequalities.
In LPP, if the constraints are mostly of the "\(\ge\)" type with positive coefficients, the region is likely unbounded in the positive direction.

Answer 1

The Correct Option is A

Concept:
• The constraints \(x \ge 0, y \ge 0\) restrict the feasible region to the first quadrant.
• For each inequation, plot the corresponding line and determine which side represents the inequality by testing a point (usually the origin \((0,0)\)).
• An unbounded region extends infinitely in at least one direction.

Step 1: Analyze the first inequality \(3x + y \ge 3\)
Plot the line \(3x + y = 3\).
Intercepts: \(x\)-intercept \(= (1, 0)\), \(y\)-intercept \(= (0, 3)\).
Test \((0,0)\): \(3(0) + 0 \ge 3 \implies 0 \ge 3\) (False).
The region is on the side away from the origin.

Step 2: Analyze the second inequality \(2x - y \ge -5\)
Plot the line \(2x - y = -5\).
Intercepts: \(x\)-intercept \(= (-2.5, 0)\), \(y\)-intercept \(= (0, 5)\).
Test \((0,0)\): \(2(0) - 0 \ge -5 \implies 0 \ge -5\) (True).
The region is on the side towards the origin.

Step 3: Determine if the intersection is bounded
In the first quadrant (\(x, y \ge 0\)):
The region is bounded below by the line \(y = 3 - 3x\).
The region is bounded on the other side by the line \(y = 2x + 5\).
Note that as \(x\) increases towards infinity, both \(3x+y \ge 3\) and \(2x-y \ge -5\) can be satisfied simultaneously for large values of \(y\) (specifically, \(y\) can range between \(0\) and \(2x+5\)).
For example, the point \((100, 100)\) satisfies both:
\(3(100) + 100 = 400 \ge 3\)
\(2(100) - 100 = 100 \ge -5\)
Since there is no upper limit on how large \(x\) or \(y\) can be while staying in the first quadrant and satisfying these constraints, the region is unbounded.