Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| Inequality | Solution Space | ||
| A. | \( 2x+3y\ge 6 \) | I. | half plane contains the origin, also points on the line \( 2x+3y=6 \) |
| B. | \( 2x+3y\le 6 \) | II. | half plane contains the origin but not points on the line \( 2x+3y=6 \) |
| C. | \( 2x+3y\gt 6 \) | III. | half plane not contains the origin but contains points on the line \( 2x+3y=6 \) |
| D. | \( 2x+3y\lt 6 \) | IV. | half plane neither contains the origin nor contains points on the line \( 2x+3y=6 \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A-III, B-I, C-IV, D-II
Two checks decide each half plane.
The origin: putting (0, 0) gives \( 2(0)+3(0)=0 \), and \( 0\lt 6 \), so the origin satisfies the “less than” inequalities only.
The boundary line: points on \( 2x+3y=6 \) are included only when the sign is \( \ge \) or \( \le \).
(A) \( 2x+3y\ge 6 \)
This matches (III): the origin is not included (0 ≥ 6 is false), but the line is included.
(B) \( 2x+3y\le 6 \)
This matches (I): the origin is included (0 ≤ 6), and so is the line.
(C) \( 2x+3y\gt 6 \)
This matches (IV): the origin is not included, and the strict sign excludes the line.
(D) \( 2x+3y\lt 6 \)
This matches (II): the origin is included (0 < 6), but the strict sign excludes the line.
So the matching is A-III, B-I, C-IV, D-II.