Official Guide Explanation:
Data Sufficiency #121




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Solution and Metadata

Question: 121
Page: 283
Difficulty: 6 (Moderately Difficult)
Category 1: Algebra > Inequalities > other
Category 2: Geometry > Coordinate Geometry > Other

Explanation: An inequality represents a region on the coordinate plane. While the equation 2x + 3y = 6 is a line, the inequality is a region bounded by that line. If the point (r,s) is in the region, that means that given the values of r (the x - coordinate) and s (the y - coordinate), it is true that 2r + 3s ≤ 6. We're trying to determine whether that's the case.

Statement (1) is insufficient. If 3r + 2s = 6, we don't know whether 2r + 3s ≤ 6. If r = 2 and s = 0, the answer is "yes." If r = 0 and s = 3, the answer is "no."

Statement (2) is also insufficient. If r = 1 and s = 1, the answer is "yes." If r = 2 and s = 1, the answer is "no."

Taken together, the statements are still insufficient. The first example I mentioned, when r = 2 and s = 0, follows the rules given in both statements, so we know that the answer could be "yes." To establish that the statements are insufficient, we need to find a counterexample. If s = 2 and r = (2/3), then it is true that 3r + 2s = 6, but 2r + 3s = (4/3) + 6 , which is greater than 6, so the answer is "no." Choice (E) is correct.

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