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# M15-11

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Math Expert
Joined: 02 Sep 2009
Posts: 43828

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15 Sep 2014, 23:55
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Difficulty:

75% (hard)

Question Stats:

51% (01:05) correct 49% (02:04) wrong based on 105 sessions

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Is $$|x - y| \gt |x + y|$$?

(1) $$x^2 - y^2 = 9$$

(2) $$x - y = 2$$
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Sep 2009
Posts: 43828

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15 Sep 2014, 23:55
Official Solution:

Statement (1) by itself is insufficient. S1 gives us information about $$(x - y)(x + y)$$ but does not tell how $$(x - y)$$ and $$(x + y)$$ compare to each other.

Statement (2) by itself is insufficient. S2 gives no information about $$(x + y)$$.

Statements (1) and (2) combined are sufficient. From S1 and S2 it follows that $$2(x + y) = 9$$ from where $$(x + y) = 4.5$$. Now we can state that $$|x - y| = 2 \lt |x + y| = 4.5$$.

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Joined: 30 Jun 2012
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07 Dec 2014, 14:37
Why would 2) alone not be sufficient to answer the question.

From part 2 we can deduce that x= 2+y and then substitute that in the equation I 2 + y - y I = I 2 I and we substitute this is the other equation to get I 2 I > I 2+2y I. Now the least value for I 2+2y I is 0 so I2I >0 Yes sufficient. Am I missing something?
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08 Dec 2014, 04:07
rsamant wrote:
Why would 2) alone not be sufficient to answer the question.

From part 2 we can deduce that x= 2+y and then substitute that in the equation I 2 + y - y I = I 2 I and we substitute this is the other equation to get I 2 I > I 2+2y I. Now the least value for I 2+2y I is 0 so I2I >0 Yes sufficient. Am I missing something?

(2) x - y = 2 --> the question becomes: is |2| > |2y + 2|? --> is |y + 1| < 1? If y = -1, the answer is YES but if y = 2 the answer is NO.
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05 Jul 2015, 23:29
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient. S1 gives us information about $$(x - y)(x + y)$$ but does not tell how $$(x - y)$$ and $$(x + y)$$ compare to each other.

Statement (2) by itself is insufficient. S2 gives no information about $$(x + y)$$.

Statements (1) and (2) combined are sufficient. From S1 and S2 it follows that $$2(x + y) = 9$$ from where $$(x + y) = 4.5$$. Now we can state that $$|x - y| = 2 \lt |x + y| = 4.5$$.

Hi bunuel,
Why can't (1) be sufficient. Here is my reasoning:

(x-y)(x+y)=9 can be rewritten as (5-4)(5+4) or even (4-5)(4+5).
Either ways, the answer to the main question will always yield a 'no'. Doesn't this mean statement 1 is sufficient?
Math Expert
Joined: 02 Sep 2009
Posts: 43828

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06 Jul 2015, 00:01
1
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samuraijack256 wrote:
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient. S1 gives us information about $$(x - y)(x + y)$$ but does not tell how $$(x - y)$$ and $$(x + y)$$ compare to each other.

Statement (2) by itself is insufficient. S2 gives no information about $$(x + y)$$.

Statements (1) and (2) combined are sufficient. From S1 and S2 it follows that $$2(x + y) = 9$$ from where $$(x + y) = 4.5$$. Now we can state that $$|x - y| = 2 \lt |x + y| = 4.5$$.

Hi bunuel,
Why can't (1) be sufficient. Here is my reasoning:

(x-y)(x+y)=9 can be rewritten as (5-4)(5+4) or even (4-5)(4+5).
Either ways, the answer to the main question will always yield a 'no'. Doesn't this mean statement 1 is sufficient?

Why do you assume that x and y are integers? x^2 - y^2 = 9 has infinitely many solutions for x and y.

Even if you consider only integers, which is not right, you'll have more solutions:
x = ±5 and y = -4;
x = ±3 and y = 0;
x = ±5 and y = 4.
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25 Jul 2016, 20:33
Bunuel wrote:
samuraijack256 wrote:
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient. S1 gives us information about $$(x - y)(x + y)$$ but does not tell how $$(x - y)$$ and $$(x + y)$$ compare to each other.

Statement (2) by itself is insufficient. S2 gives no information about $$(x + y)$$.

Statements (1) and (2) combined are sufficient. From S1 and S2 it follows that $$2(x + y) = 9$$ from where $$(x + y) = 4.5$$. Now we can state that $$|x - y| = 2 \lt |x + y| = 4.5$$.

Hi bunuel,
Why can't (1) be sufficient. Here is my reasoning:

(x-y)(x+y)=9 can be rewritten as (5-4)(5+4) or even (4-5)(4+5).
Either ways, the answer to the main question will always yield a 'no'. Doesn't this mean statement 1 is sufficient?

Why do you assume that x and y are integers? x^2 - y^2 = 9 has infinitely many solutions for x and y.

Even if you consider only integers, which is not right, you'll have more solutions:
x = ±5 and y = -4;
x = ±3 and y = 0;
x = ±5 and y = 4.

Hi Bunuel,

For this q - What if we have to simplify the question stem and open up the modulus sign?

1) Case 1: Both sides are positives - then we get
x-y> x+y... => y<0

2) Case 2: One is negative the other is positive:
X >0 in one case and X < 0 in the other

3) Case 3: Both sides are negative:
y>0

So now if we consider option c, X-y =2. X+y = 4.5... solving gives x = 3.25 and y = 1.25.

Now what case above to consider.

Is there a generic approach of solving these questions?
Senior Manager
Joined: 08 Jun 2015
Posts: 369
Location: India
GMAT 1: 640 Q48 V29

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11 Jan 2017, 06:16
The answer has to be option C.

First statement - different combinations of x-y & x+y are possible. Hence NS
Second statement - x-y = 2. NS

Combine the two , x-y = 2, x+y = 4.5 ; hence sufficient.

Hence option C is correct.
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Joined: 30 Jun 2015
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11 Jan 2017, 12:27
Can anyone open the modulus and explain the solution algebraically?

spetznaz wrote:
The answer has to be option C.

First statement - different combinations of x-y & x+y are possible. Hence NS
Second statement - x-y = 2. NS

Combine the two , x-y = 2, x+y = 4.5 ; hence sufficient.

Hence option C is correct.
Manager
Joined: 23 Jan 2016
Posts: 220
Location: India
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11 Mar 2017, 06:52
Bunuel wrote:
Is $$|x - y| \gt |x + y|$$?

(1) $$x^2 - y^2 = 9$$

(2) $$x - y = 2$$

if we square both sides of the prompt, doesent the question become whether xy<o or whether x and y both have opposite signs? would appreciate feedback.

Thank you.
Math Expert
Joined: 02 Sep 2009
Posts: 43828

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11 Mar 2017, 07:04
OreoShake wrote:
Bunuel wrote:
Is $$|x - y| \gt |x + y|$$?

(1) $$x^2 - y^2 = 9$$

(2) $$x - y = 2$$

if we square both sides of the prompt, doesent the question become whether xy<o or whether x and y both have opposite signs? would appreciate feedback.

Thank you.

______________
Yes, that's correct.
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11 Mar 2017, 13:12

OreoShake wrote:
Bunuel wrote:
Is $$|x - y| \gt |x + y|$$?

(1) $$x^2 - y^2 = 9$$

(2) $$x - y = 2$$

if we square both sides of the prompt, doesent the question become whether xy<o or whether x and y both have opposite signs? would appreciate feedback.

Thank you.
Math Expert
Joined: 02 Sep 2009
Posts: 43828

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12 Mar 2017, 02:47
cuhmoon wrote:

OreoShake wrote:
Bunuel wrote:
Is $$|x - y| \gt |x + y|$$?

(1) $$x^2 - y^2 = 9$$

(2) $$x - y = 2$$

if we square both sides of the prompt, doesent the question become whether xy<o or whether x and y both have opposite signs? would appreciate feedback.

Thank you.

$$|x - y| \gt |x + y|$$

$$(|x - y|)^2 \gt (|x + y|)^2$$

$$x^2 - 2xy +y^2 > x^2+2xy+y^2$$

$$0>4xy$$

$$0>xy$$
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Re: M15-11   [#permalink] 12 Mar 2017, 02:47
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# M15-11

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