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Re: range of root - GMAT Club test - M24 [#permalink]
15 Jan 2011, 13:47

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Expert's post

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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 \(x>1\).

Re: If x/|x|, which of the following must be true for all [#permalink]
03 Nov 2014, 10:55

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If x/|x|, which of the following must be true for all [#permalink]
23 Apr 2015, 00:44

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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 takeawayfrom 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.

Re: If x/|x|, which of the following must be true for all [#permalink]
23 Apr 2015, 03:36

Expert's post

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.

Re: If x/|x|, which of the following must be true for all [#permalink]
24 Apr 2015, 10:21

Expert's post

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.

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