Inequality

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tapanmittal: Senior | Next Rank: 100 Posts; Posts: 31; Joined: Fri Apr 10, 2015 4:04 am

by tapanmittal » Sun Aug 23, 2015 3:33 am

If X/|X| < X Which of the following must be true about X ?

a) X>1

b) X>-1

c) |X| < 1

d) |X| = 1

e) |X|^2 > 1

OA after discussion

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Source: Beat The GMAT — Problem Solving | Sun Aug 23, 2015 3:33 am

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 Aug 23, 2015 3:36 am

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by [email protected] » Sun Aug 23, 2015 9:55 am

Hi tapanmittal,

This question can be solved by TESTing VALUES.

We're told that X/|X| < X. We're asked which of the following MUST be true?

Let's see what happens when we TEST different values for X....

IF...X = 1
1/|1| < 1 which is NOT correct. Thus, X CANNOT = 1.

IF...X = 2
2/|2| < 2 which IS correct. Thus, X CAN = 2.

With these first two TESTs, we can eliminate Answers C and D.

Given the absolute value sign, we should also consider NEGATIVE possibilities...

IF...X = -2
-2/|-2| < -2 which is NOT correct. Thus, X CANNOT = -2.

IF...X = -1/2
(-1/2)/|-1/2| < -1/2 which IS correct. Thus, X CAN = -1/2.

With these two TESTs, we can eliminate Answers A and E.

Final Answer: B

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Sun Aug 23, 2015 11:08 am

Since |x| can never be negative, it's safe to multiply both sides of the inequality by |x|.

This gives us

x < x * |x|

Now we have two cases.

Case I:: x > 0

In this case, we divide both sides by x, and we get

1 < |x|

Case II:: x < 0

In this case, we divide both sides by x again, but this time the sign FLIPS, since we're dividing by a negative.

1 > |x|

Remember that x is a negative number. If x is negative, |x| = -x. So we REALLY have

1 > -x

or

-1 < x

Since x is negative, we can write this as -1 < x < 0, and we're done!

So our solution sets are

x > 1
and
0 > x > -1

The only answer that describes both sets is B. (Note that we can't have 1 â‰¥Â x â‰¥Â 0, but B doesn't say that we do; it only says that x, whatever it is, must be greater than -1 - it doesn't say that x can take any value greater than -1.)

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