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Inequalities

by bubbliiiiiiii » Tue Jun 11, 2013 5:20 am
Q. X / |X| < X. Which of the following must be true about integer X? X is not equal to 0.

A. X > 1
B. X > -1
C. |X| < 1
D. |X| = 1
E.|X|^2 > 1

OA Later.

Please post explanations.

My working:

Q. X / |X| < X

Since X is there on both sides, irrespective of X being positive or negative we can cancel out X on both sides.

Question rephrased:

1/|X| < 1 which option must be true.

=> 1 < |X|

=> 1 < X or -1 > X

IMO A
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Pranay
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by Brent@GMATPrepNow » Tue Jun 11, 2013 6:57 am
bubbliiiiiiii wrote:Q. X / |X| < X. Which of the following must be true about integer X? X is not equal to 0.

A. X > 1
B. X > -1
C. |X| < 1
D. |X| = 1
E.|X|^2 > 1

OA Later.

Please post explanations.

My working:

Q. X / |X| < X

Since X is there on both sides, irrespective of X being positive or negative we can cancel out X on both sides.

Question rephrased:

1/|X| < 1 which option must be true.

=> 1 < |X|

=> 1 < X or -1 > X

IMO A
The mistake is highlighted above.
We have X/|X| < X
Since we have X on both sides, it's tempting to divide both sides by X (to get 1/|X| < 1).
HOWEVER, if X is negative, then we must reverse the direction of the inequality sign (to get 1/|X| > 1).

So, a complete solution must consider the case where X is positive, and the case where X is negative.

Case where X is positive
Take: X/|X| < X
Divide both sides by X to get: 1/|X| < 1
Multiply both sides by |X| to get: 1 < |X|
This means that X > 1 or X < -1
IMPORTANT: We started with the premise that X is positive, so we can rule out the possibility that X < -1, which leaves us with only X > 1

Case where X is negative
Take: X/|X| < X
Divide both sides by X to get: 1/|X| > 1
Multiply both sides by |X| to get: 1 > |X|
This means that -1 < X < 1
IMPORTANT: There are no integer values of X that satisfy this inequality.

So, all we can conclude here is that X > 1
In other words, integer x can equal 2, 3, 4, 5...

A. X > 1 This works (KEEP A)
B. X > -1 If X can be 2,3,4 etc, then X is definitely greater than -1 (KEEP B)
C. |X| < 1 this suggests that X must equal 0 - not true (ELIMINATE C)
D. |X| = 1 this suggests that X equals 1 or -1. not true (ELIMINATE D)
E.|X|^2 > 1 If X can be 2,3,4 etc, then |X| is definitely greater than 1 (KEEP E)

Looks like we have more than 1 correct answer: A, B and E

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Wed Jun 12, 2013 5:31 am, edited 2 times in total.
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by bubbliiiiiiii » Tue Jun 11, 2013 7:44 am
Thanks Brent .. I was apprehensive about that step. :)
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by GMATGuruNY » Tue Jun 11, 2013 8:13 am
Check my posts below for two different approaches:

https://www.beatthegmat.com/modulus-ps-t112185.html
https://www.beatthegmat.com/let-get-abso ... 27383.html
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
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by bubbliiiiiiii » Wed Jun 12, 2013 4:24 am
Hi Brent,

As per links posted by Mitch OA is different.
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by Brent@GMATPrepNow » Wed Jun 12, 2013 5:29 am
bubbliiiiiiii wrote:Hi Brent,

As per links posted by Mitch OA is different.
The question that Mitch answered in his links is different from the question you posed.
In your original question, X is an integer. In the questions Mitch answered, X can have any value.

This one restriction changes the question significantly.

That said, I did make an error in my solution, and I have since corrected it. The new solution shows that there are 3 correct answers, so perhaps the question was entered incorrectly.

Cheers,
Brent
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