Question

Which ONE of the following options represents correctly ALL the constraints resulting in the feasible region for the given linear programming problem as shown in the figure?
Note: Figure not to scale

• A. \( x_1 \ge 0; x_2 \ge 0; x_2 \ge 3; x_1 - x_2 \le 1; x_1 - 0.5x_2 \ge 1 \)
• B. \( x_1 \ge 0; x_2 \ge 0; x_2 \le 3; x_1 - x_2 \le 1; x_1 + 0.5x_2 \ge 1 \)
• C. \( x_1 \ge 0; x_2 \ge 0; x_2 \le 4; x_1 - x_2 \le 1; x_1 + 0.5x_2 \ge 1 \)
• D. \( x_1 \ge 0; x_2 \ge 0; x_2 \le 3; x_1 + x_2 \le 1; x_1 + 0.5x_2 \ge 1 \)

Hint:

Find the line through each pair of boundary points, then check which side of each line the shaded triangle lies on.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The graph shows a shaded triangle bounded by three straight lines plus the two axis limits, and we need the set of inequalities that produces exactly this shaded region.
We will find each boundary line first, then decide which side of that line the shading sits on.

Step 2: Read the key points off the graph:
The shaded region has corner points near (0, 2), (1, 0) and (4, 3), with a flat top edge running along x2 = 3 up to the point (4, 3).
Since the region sits in the first quadrant, both \( x_1 \ge 0 \) and \( x_2 \ge 0 \) hold, matching the standard non-negativity rule for LP variables.

Step 3: Find the top boundary line:
The flat top of the triangle runs horizontally at height 3, so this line is \( x_2 = 3 \).
The shaded region lies below this line, so the constraint reads \( x_2 \le 3 \).

Step 4: Find the slanted line through (1,0) and (4,3):
The slope between these two points is \( m = \frac{3-0}{4-1} = 1 \).
\[ x_2 - 0 = 1 \times (x_1 - 1) \implies x_1 - x_2 = 1 \]
The shaded triangle sits on the upper-left side of this line, so the constraint is \( x_1 - x_2 \le 1 \).

Step 5: Find the slanted line through (0,2) and (1,0):
This line has intercepts \( x_1 = 1 \) and \( x_2 = 2 \), giving the intercept-form equation \( \frac{x_1}{1} + \frac{x_2}{2} = 1 \), or \( x_1 + 0.5x_2 = 1 \).
The shaded region sits on the outer side of this line, away from the origin, so the constraint is \( x_1 + 0.5x_2 \ge 1 \).

Step 6: Check option (A).
This option has \( x_2 \ge 3 \) instead of \( x_2 \le 3 \), which would shade above the flat top line, not the actual triangle, so this is wrong.

Step 7: Check option (B).
This matches every constraint we worked out: \( x_1 \ge 0, x_2 \ge 0, x_2 \le 3, x_1 - x_2 \le 1, x_1 + 0.5x_2 \ge 1 \).

Step 8: Check option (C).
This uses \( x_2 \le 4 \) instead of \( x_2 \le 3 \), which does not match the flat top edge seen at height 3 in the figure, so this is wrong.

Step 9: Check option (D).
This uses \( x_1 + x_2 \le 1 \) instead of \( x_1 - x_2 \le 1 \), a completely different line that does not pass through (1,0) and (4,3), so this is wrong.

Final Answer:
The full set of constraints for the feasible region is given in option (B). \[ \boxed{\text{Option (B)}} \]