Is xy

Question

Is xy < 0?

(1) |x + y| < |x| + |y|
(2) |x| - |y| < |x – y|

gmatophobia · Answer

This question is based on a property

|x + y| < |x| + |y| → x and y have opposite positive or negative signs 
 |x + y| = |x| + |y| → x and y have same the positive or negative signs

|x| - |y| < |x – y| → x and y have opposite positive or negative signs or when x and y have the same positive or negative signs and |x| < |y| 
 |x| - |y| = |x – y| → x and y have the same positive or negative signs and |x|  \geq  |y|

We can apply this concept to the below question -

Statement 1

(1) |x + y| < |x| + |y|

Using the property above, we can conclude that x and y have opposite positive - negative signs. Hence, we can conclude that x*y is less than zero.

The statement is sufficient and we can eliminate A, and D.

Statement 2

(2) |x| - |y| < |x – y|

The inequality can hold true when x and y have opposite positive negative signs or when x and y have the same positive negative sign and |x| < |y|.

Hence, in one case x*y will be less than zero and in the other case x*y will be greater than zero. The statement is not sufficient to answer the question.

Option A

Legionnaire246 · Answer

Could you please explain how the property highlighted above works?

jamiedimonn · Answer

The inequality can hold true when x and y have opposite positive negative signs or when x and y have the same positive negative sign and |x| < |y|.

Hence, in one case x*y will be less than zero and in the other case x*y will be greater than zero. The statement is not sufficient to answer the question.

The 2nd statement holds good only when x and y have opposite signs when both x and y have same sign the statement fails ans should be D.

Bunuel  could you confirm?

Bunuel · Answer

The answer is, and should be, A, as indicated in the original post under the spoiler.

|x + y| < |x| + |y| holds true only when x and y have opposite signs. You can test this by plugging in numbers. When x and y have the same sign (so when xy ≥ 0), |x + y| = |x| + |y|, and when they have opposite signs, |x + y| < |x| + |y|. Therefore, the first statement is sufficient to conclude that xy < 0.

|x| - |y| < |x – y|, however, can be true when x and y have the same sign (e.g., x = 1 and y = 2; x = -1 and y = -2) as well as when they have opposite signs (e.g., x = 1 and y = -1; x = -1 and y = 1). Thus, the second statement is not sufficient to conclude whether xy < 0.

P.S. Pure algebraic questions are no longer a part of the  DS syllabus  of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:
 
 Word Problems 
 Work Problems 
 Distance Problems 
 Mixture Problems 
 Percent and Interest Problems 
 Overlapping Sets Problems 
 Statistics Problems 
 Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check  GMAT Syllabus for Focus Edition

You can also visit the  Data Sufficiency forum  and filter questions by  OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources  to see the types of questions currently tested on the GMAT.

So, you can ignore this and similar questions.

Hope it helps.

Is xy < 0? (1) |x + y| < |x| + |y| (2) |x| - |y| < |x y|

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