The linear constraints for which the shaded region in the figure is the solution set, are
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 2x+y\ge 2,\ \ x+2y\le 8,\ \ x-y\le 1,\ \ x,\ \ y\ge 0 \)
The shaded region lies in the first quadrant, so \( x,\ y\ge 0 \). Each boundary line is tested with the origin O(0, 0), which lies clearly on one side of every line.
2x + y = 2: the shaded region is on the side away from the origin. At O, 0 ≥ 2 is false, so the region is
\( \displaystyle 2x+y\ge 2 \)
x + 2y = 8: the shaded region is on the same side as the origin. At O, 0 ≤ 8 is true, so
\( \displaystyle x+2y\le 8 \)
x − y = 1: the shaded region is on the same side as the origin (above and to the left of the line). At O, 0 ≤ 1 is true, so
\( \displaystyle x-y\le 1 \)
So the constraints are \( 2x+y\ge 2,\ x+2y\le 8,\ x-y\le 1,\ x,y\ge 0 \).
Shortcut: Take a point inside the shaded region, for example (1, 2), and substitute: 2 + 2 = 4 ≥ 2, 1 + 4 = 5 ≤ 8 and 1 − 2 = −1 ≤ 1. Only Option (3) is satisfied.