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Intern  Joined: 06 Nov 2010
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If x/|x|, which of the following must be true for all  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 59% (01:38) correct 41% (01:55) wrong based on 630 sessions

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If $$\frac{x}{|x|} \lt x$$, which of the following must be true about $$x$$? ($$x \ne 0$$)

A. $$x\gt 2$$
B. $$x \in (-1,0) \cup (1,\infty)$$
C. $$|x| \lt 1$$
D. $$|x| = 1$$
E. $$|x|^2 \gt 1$$

M24

Originally posted by praveenvino on 15 Jan 2011, 11:44.
Last edited by Bunuel on 09 Jul 2013, 08:56, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 65765
Re: range of root - GMAT Club test - M24  [#permalink]

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praveenvino wrote:
X/|X| < X . Which of the following must be true for all ?

a. X > 1
b. X is an element in (-1,0) U (1,inf)
c. |X| < 1
d. |X| = 1
e. |X|^2 > 1

Can some one explain how X can be zero for the above condition?

x is in the denominator so it can not equal to zero as division be zero is undefined.

Correct form of this question is below (m09 q22, discussed here: m09-q22-69937.html):

If $$\frac{x}{|x|} \lt x$$, which of the following must be true about $$x$$? ($$x \ne 0$$)
A. $$x\gt 2$$
B. $$x \in (-1,0) \cup (1,\infty)$$
C. $$|x| \lt 1$$
D. $$|x| = 1$$
E. $$|x|^2 \gt 1$$

$$\frac{x}{|x|}< x$$
Two cases:
A. $$x<0$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$. But as we consider the range $$x<0$$ then $$-1<x<0$$

B. $$x>0$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$.

So the given inequality holds true in two ranges $$-1<x<0$$ and $$x>1$$.

For more check: math-absolute-value-modulus-86462.html

Hope it helps.
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Senior Manager  Joined: 07 Aug 2011
Posts: 488
GMAT 1: 630 Q49 V27
Re: If x/|x|, which of the following must be true for all  [#permalink]

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praveenvino wrote:
If $$\frac{x}{|x|} \lt x$$, which of the following must be true about $$x$$? ($$x \ne 0$$)

A. $$x\gt 2$$
B. $$x \in (-1,0) \cup (1,\infty)$$
C. $$|x| \lt 1$$
D. $$|x| = 1$$
E. $$|x|^2 \gt 1$$

M24

x < x*|x|
x-x*|x|< 0
roots of this equation are : -1,0,1
rest is explained in the attached image ...
Attachments gmatclub.jpg [ 16.88 KiB | Viewed 7029 times ]

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If x/|x|, which of the following must be true for all  [#permalink]

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AverageGuy123 wrote:
Brunel, Can you please explain why option E is not feasible?

Dear AverageuGuy123

As Bunuel explained above,

Either -1 < x < 0
Or x > 1

Now, |x| as you know, represents the magnitude of x. Option E says that |x|^2 must be greater than 1.

Let's first consider the case when -1 < x < 0

A possible value of x in this case is -0.5
So, what is the value of |x|^2? It is equal to 0.25

Is it greater than 1? NO

Let's now consider the case when x > 1

A possible value of x in this case is 2.
So, what is the value of |x|^2? It's 4.

Is it greater than 1? YES

So, as we see, that |x|^2 CAN BE greater than 1. But can we say that |x|^2 MUST BE greater than 1? NO, because |x|^2 is not greater than 1 for all possible values of x.

So, the key takeaway from this discussion is that:

we need to be careful whether the question is asking about MUST BE TRUE statements or about CAN BE TRUE statements.

Hope this helped! - Japinder
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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Given: x/|x| < x
since |x| >= 0 always
multiply both LHS and RHS by |x|
x < x|x|
=> x - x|x| < 0
=> x(1 - |x|) < 0
if x > 0, then 1 - |x| < 0 to hold the above inequality => |x| > 1 => x(since x is positive in this case) > 1
if x < 0, then 1 - |x| > 0 => |x| < 1 => -1 < x < 0 (to hold the above inequality)

Option B captures the above range perfectly
Intern  Joined: 06 Nov 2010
Posts: 18
Re: range of root - GMAT Club test - M24  [#permalink]

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Thanks Bunuel. X not equals zero condition was actually missing in the question in m24. Thanks for your help.
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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A must be true too.
If x>1 satisfy x/|x|<x
then x>2 will do too.
can anyone explain choice A? thanks!
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Posts: 65765
Re: If x/|x|, which of the following must be true for all  [#permalink]

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pinguuu wrote:
A must be true too.
If x>1 satisfy x/|x|<x
then x>2 will do too.
can anyone explain choice A? thanks!

x > 2 is NOT necessarily true. Consider x = -1/2.
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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Brunel, Can you please explain why option E is not feasible?
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Joined: 16 Oct 2010
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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AverageGuy123 wrote:
Brunel, Can you please explain why option E is not feasible?

You can plug in numbers to eliminate options.

"which of the following must be true about x" means that every acceptable value of x must lie in the range given in the correct option. The acceptable values of x are the values for which x/|x| < x.

A. x>2
Must x be greater than 2?

This should make you check for 2.
2/|2| < 2
1 < 2 (True)
So 2 is an acceptable value of x. But 2 is not greater than 2.
So this option is not correct. This also makes you eliminate options (C) and (D).

E. |x|^2>1
Must x be greater than 1 or less than -1?

Check for 1/2
(1/2)/|1/2| < 1/2
1 < 1/2 (False)

Check for -1/2
(-1/2)/|-1/2| < -1/2
-1 < -1/2 (True)

So x = -1/2 is an acceptable value but it does not lie in this range. Hence option (E) is also incorrect.

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Re: If x/|x|, which of the following must be true for all  [#permalink]

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Hi All,

This question can be dealt with in a variety of ways. It's actually really susceptible to TESTing VALUES, which we can use to determine possibilities and eliminate answers.

We're told that X/|X| < X. The question asks what must be TRUE about X.

While this inequality looks complicated, you can quickly prove some things about X....

IF....
X = 1
1/|1| is NOT < 1
So X CANNOT be 1
Eliminate D.

IF.....
X = 2
2/|2| IS < 2
So X CAN be 2
Eliminate A and C.

IF....
X = -2
-2/|-2| is NOT < -2
So X CANNOT be -2
Eliminate E.

GMAT assassins aren't born, they're made,
Rich
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Re: If x/|x|, which of the following must be true for all  [#permalink]

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_________________ Re: If x/|x|, which of the following must be true for all   [#permalink] 28 May 2020, 11:47

# If x/|x|, which of the following must be true for all   