Question

In the xy-plane, region R consists of all the points (x,y) such that $2x+3y\le 6$. Is the point (r,s) in region R?
(1) $3r+2s=6$
(2) $r\le 3$ and $s\le 2$

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is E Statements (1) and (2) together are not sufficient.

1. Both (r,s) = (2,0) and (r,s) = (0,3) satisfy the equation 3r + 2s = 6, since 3(2) + 2(0) = 6 and 3(0) + 2(3) = 6. However, 2(2) + 3(0) = 4, so (2,0) is in region R, while 2(0) + 3(3) = 9,, so (0,3) is not in region R; NOT sufficient.
2. Both (r,s) = (0,0) and (r,s) = (3,2) satisfy the inequalities and . However, 2(0) + 3(0) = 0, so (0,0) is in region R, while 2(3) + 3(2) = 12, so (3,2) is not in region R; NOT sufficient.

• Taking (1) and (2) together, it can be seen that both (r,s) = (2,0) and (r,s) = (1,1.5) satisfy 3r + 2s = 6, and . However, 2(2) + 3(0) = 4 , so (2,0) is in region R, while 2(1) + 3(1.5) = 6.5 , so (1, 1.5) is not in region R. Therefore, (1) and (2) together are not sufficient.
• The correct answer is E; both statements together are still not sufficient.