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akhilsuhag: Master | Next Rank: 500 Posts; Posts: 351; Joined: Mon Jul 04, 2011 10:25 pmFollowed by: 4 members; Thanked: 57 times

by akhilsuhag » Sat Jun 13, 2015 11:47 am

Is x^12-2x^11 negative?

(1) x^2 < |x|

(2) x-1 < -1

OA [spoiler](B)[/spoiler]

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Source: Beat The GMAT — Data Sufficiency | Sat Jun 13, 2015 11:47 am

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Sat Jun 13, 2015 12:50 pm

We can first work out when the answer to the question will be 'yes'. If x^12 - 2x^11 is negative, then

x^12 - 2x^11 < 0

We can safely divide by x^10 on both sides without worrying about whether to reverse the inequality, since x^10 cannot be negative (it is an even power of x), so we have

x^2 - 2x < 0
x(x-2) < 0

If this product is negative, then one factor is positive, and the other factor is negative. Since x-2 is clearly smaller than x, then x-2 is the negative factor, and x is the positive factor. So the answer to the question is 'yes' only when 0 < x < 2.

Since S2 tells us x < 0 (adding 1 to both sides), S2 is sufficient, since it guarantees the answer to the question is 'no'. Statement 1 is not sufficient, since x might be 1/2 or -1/2.

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theCEO: Master | Next Rank: 500 Posts; Posts: 363; Joined: Sun Oct 17, 2010 3:24 pmFollowed by: 3 members; Thanked: 115 times

by theCEO » Sat Jun 13, 2015 1:16 pm

akhilsuhag wrote:Is x^12-2x^11 negative?

(1) x^2 < |x|

(2) x-1 < -1

OA [spoiler](B)[/spoiler]

x^12 - 2x^11
x^11(x - 2)
is x^11(x - 2) negative? Yes or no answer

statement 1
x^2 < |x|
therefore x is a negative or positive fraction eg. x = -1/4 or 1/4

if x = negative fraction :
x^11 = negative : negative^odd power = negative fraction
(x - 2) = negative fraction - positive number = negative mixed number
x^11( x - 2) = negative x negative = positive

if x = positive fraction
x^11 = positive : positive ^odd power = positive fraction
(x - 2) = negative: positive fraction - positive number = negative mixed number
x^11( x - 2) = positive x negative = negative

Statement isn't satisfactory since we can have a positive or negative outcome

statement 2
x - 1 < -1
x < 0
therefore x is negative for eg. X = -1/4 or -2
x^11: negative^odd power = negative for eg. -2^3 = -8
(x - 2): negative - positive = negative

x^11( x - 2) = negative x negative = positive
therefore x^12 - 2x^11 = positive number
statement is sufficient

answer = b

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Jun 14, 2015 1:37 am

akhilsuhag wrote:Is xÂ¹Â²-2xÂ¹Â¹ negative?

(1) xÂ² < |x|

(2) x-1 < -1

Questions stem:
xÂ¹Â² - 2xÂ¹Â¹ < 0
xÂ¹Â² < 2xÂ¹Â¹.

When a NONZERO value is raised to an EVEN power, the result is POSITIVE.
Thus:
If xâ‰ 0, then xÂ¹Â² > 0.
Since xÂ¹Â² < 2xÂ¹Â¹ only if xâ‰ 0, we can safely divide each side by xÂ¹Â², which must be positive:

xÂ¹Â²/xÂ¹Â² < 2xÂ¹Â¹/xÂ¹Â²
1 < 2/x
2/x > 1.

2/x > 1 only if x is POSITIVE but LESS THAN 2.
Questions stem, rephrased:
Is 0 < x < 2?

Statement 1: xÂ² < |x|
If x = 1/2, then 0 < x < 2.
If x = -1/2, then x < 0.
INSUFFICIENT.

Statement 2: x-1 < -1
Thus:
x < 0.
Here, x < 0, so the answer to the rephrased question stem is NO.
SUFFICIENT.

The correct answer is B.

Last edited by GMATGuruNY on Sun Jun 14, 2015 2:19 am, edited 1 time in total.

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Sun Jun 14, 2015 2:08 am

GMATGuruNY wrote:
Statement 2: x-1 < -1
Thus:
x < -2

Mitch - I think you mean "x < 0" at the end there, and not "x < -2", so you might want to edit your post.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Jun 14, 2015 2:22 am

Ian Stewart wrote: Mitch - I think you mean "x < 0" at the end there, and not "x < -2", so you might want to edit your post.

Thanks, Ian.
The post has been edited.

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