If \(x\neq{0}\) and \(\frac{x}{|x|}<x\), which of the following must be true?

(A) \(x>1\)

(B) \(x>-1\)

(C) \(|x|<1\)

(D) \(|x|>1\)

(E) \(-1<x<0\)


Notice that we are asked to find which of the options MUST be true, not COULD be true.

Let's see what ranges does \(\frac{x}{|x|}< x\) give for \(x\). Two cases:

If \(x<0\) then \(|x|=-x\), hence in this case we would have: \(\frac{x}{-x}<x\) --> \(-1<x\). But remember that we consider the range \(x<0\), so \(-1<x<0\);

If \(x>0\) then \(|x|=x\), hence in this case we would have: \(\frac{x}{x}<x\) --> \(1<x\).

So, \(\frac{x}{|x|}< x\) means that \(-1<x<0\) or \(x>1\).

Only option which is ALWAYS true is B. ANY \(x\) from the range \(-1<x<0\) or \(x>1\) will definitely be more the \(-1\).

Answer: B.

As for other options:

A. \(x>1\). Not necessarily true since \(x\) could be -0.5;

C. \(|x|<1\) --> \(-1<x<1\). Not necessarily true since \(x\) could be 2;

D. \(|x|>1\) --> \(x<-1\) or \(x>1\). Not necessarily true since \(x\) could be -0.5;

E. \(-1<x<0\). Not necessarily true since \(x\) could be 2.
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The question above is exactly as it appears in CAT.

Also, notice that x=0.9, does not satisfy x/|x|<x, thus x cannot take this value.
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Consider following:
If \(x=5\), then which of the following must be true about \(x\):
A. x=3
B. x^2=10
C. x<4
D. |x|=1
E. x>-10

Answer is E (x>-10), because as x=5 then it's more than -10.

Or:
If \(-1<x<10\), then which of the following must be true about \(x\):
A. x=3
B. x^2=10
C. x<4
D. |x|=1
E. x<120

Again answer is E, because ANY \(x\) from \(-1<x<10\) will be less than 120 so it's always true about the number from this range to say that it's less than 120.

The same with original question:

If \(-1<x<0\) or \(x>1\), then which of the following must be true about \(x\):
A. \(x>1\)
B. \(x>-1\)
C. \(|x|<1\)
D. \(|x|>1\)
E. \(-1<x<0\)

As \(-1<x<0\) or \(x>1\) then ANY \(x\) from these ranges would satisfy \(x>-1\). So B is always true.

\(x\) could be for example -1/2, -3/4, or 10 but no matter what \(x\) actually is, it's IN ANY CASE more than -1. So we can say about \(x\) that it's more than -1.

Hope it's clear.
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\(\frac{x}{|x|}<x\)

Test the values for x.

(1) x = 0 ==> No! Since it is given that x is not 0
(2) x = -1 ==> -1 < -1 ==> No!
(3) x = 1 ==> 1 < 1 ==> No!
(4) x = -2 ==> -1 < -2 ==> No!
(5) x = \(\frac{-1}{4}\) ==> -1 < \(\frac{-1}{4}\) ===> Yes!
(5) x = 2 ==> 1 < 2 ==> Yes!

Our range: -1 < x < 0 or x > 1

What must always be true in that range? x > -1 always

Answer: B
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Appreciate if someone could point out where I am going wrong here.


x / |x| < x

Since x is non zero, dividing by x on both sides


1 / |x| < 1

Taking reciprocal,

|x| > 1

Then I just jumped into Choice D. Didn't even look at the others.

Await your valued views.
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But your dividing my the same number x on both sides, whether its positive or negative,
implying both sides will simultaneously be negative together or positive together which doesnt change the sign.

Right or not?
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Never multiply (or reduce) an inequality by variable (or by an expression with variable) if you don't know its sign.

So you cannot divide both parts of inequality x / |x| < x by x as you don't know the sign of this unknown: if x>0 you should write 1/|x|<1 BUT if x<0 you should write 1/|x|>1.

Hope it helps.
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A 'must be true' question! They are absolutely straight forward if you get the fundamental but they can drive you crazy if you don't.

"My answer after solving inequality is -1<x<0 or x>1" Perfect. That is the range of x for which the inequality works. So tell me, what values can x take?
-1/2, -1/3, -2/3, 1.4, 2, 500, 123498 etc...
Now the question is "which of the following must be true?"

(A) \(x>1\)
Are all these values greater than 1? No.

(B) \(x>-1\)
Are all these values greater than -1? Yes. The answer. Note that you dont have to establish that all value greater than -1 should work for the inequality. You only have to establish that all values which work for the inequality must satisfy this condition.

(C) \(|x|<1\)
Not true for all values of x.

(D) \(|x|>1\)
Not true for all values of x.

(E) \(-1<x<0\)
Not true for all values of x.
x can take values 1.4, 2, 500 etc

I wrote a post on this beautiful question sometime back:
http://www.veritasprep.com/blog/2012/07 ... -and-sets/
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I think the answer is absolutely correct. Firstly the question asks for a "MUST be true" option. Both the ranges for x, as rightly calculated above are :
x>1 OR -1<x<0. However, none of the options subscribe to MUST be true type.It is so because x could be 2 OR x could be -0.5 However, the option B, where x>-1 will always be true, irrespective of the two ranges.
Any value lying in the range -1<x<0 IS always in tandem with x>-1 AND any value for x>1 WILL also have x>-1.
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The question at NO point of time has asked "to find the rage in which all values of x would satisfy the inequality x / |x| < x", It just says which of the following MUST be TRUE. You don't assume that the given options encompass all the valid ranges. You find the valid ranges, then look for a common thread which binds them together and MUST BE TRUE, irrespective of the range(s).
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