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## Official Guide Explanation:Data Sufficiency #121

Background

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

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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