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Abs Value

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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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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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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by GMATGuruNY » Sun Jun 14, 2015 1:37 am
akhilsuhag wrote:Is x¹²-2x¹¹ negative?

(1) x² < |x|

(2) x-1 < -1
Questions stem:
x¹² - 2x¹¹ < 0
x¹² < 2x¹¹.

When a NONZERO value is raised to an EVEN power, the result is POSITIVE.
Thus:
If x≠0, then x¹² > 0.
Since x¹² < 2x¹¹ only if x≠0, we can safely divide each side by x¹², which must be positive:

x¹²/x¹² < 2x¹¹/x¹²
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: x² < |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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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.
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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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