Probably you thought only x>2 meets the given condition that x > x / lxl. However x>-1 also meets that requirement.

To cover all the possibility x has to be >-1.
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If x is a -ve fraction, x/lxl < x is also true.

Suppose x = -0.5:
x/lxl < x
-0.5/l-0.5l < -0.5.
-1 < -0.5

However x = 0 and x = any +ve fraction to 1 doesont fit to the equation but to cover all the possibilities, x must be >-1.

A is not correct because it is leaving the possibilities that x could be greater than 1 and equal to 2.
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Agreed....
I did not note that x is an integer..
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Thanks Guys.

The answer was B. I got A.

May be the club answer is wrong.

Vote for A too. Very frequent inequality with absolute value question in DS.
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I also go with ans A since for integer x to satisfy the equation,
x > 1.

If x/|x|<x, which of the following must be true about integer x? (x is not equal to 0)

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

First let's solve inequality:

\frac{x}{|x|}< x multiply both sides of inequality by |x| (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too) --> x<x|x| --> x(|x|-1)>0:

Either x>0 and |x|-1>0, so x>1 or x<-1 --> x>1;
Or x<0 and |x|-1<0, so -1<x<1 --> -1<x<0 --> but as we are told that x is an integer then this range is out as there is no integer in this range.

So, finally we have that given inequality \frac{x}{|x|}< x with a restriction that x is an integer is true for: x>1 --> which means that x could be: 2, 3, 4, 5, 6, ...

Now, we are asked to find out which of the following must be true about x, (note the given options are not supposed to give solutions for inequality we have).

A. x > 2 --> not always true as x could be 2;
B. x > -1 --> always true: we have that x>2 (x is more than 2), so it must also be more than -1;
C. |x| < 1 --> not true;
D. |x| = 1 --> not true;
E. |x|^2 > 1 --> always true: as x>2 then any x from this range when squared will be more than 1.

So options B and E are always true. Question needs revision.

There is similar question at: ps-inequality-13943-40.html which does not say that x is an integer and for it the answer is B (there is my solution on page 3).

To elaborate more. Question uses the same logic as in the examples below:

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.

Or:
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. |x|^2>1

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.

On the other hand for example A is not always true as it says that x>1, which is not always true as x could be -1/2 and -1/2 is not more than 1.

Hope it's clear.
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Not so, B is always true. Again, \frac{x}{|x|}< x with a restriction that x is an integer means that x>1 --> x could be: 2, 3, 4, 5, 6, ...

Now, I ask you is x>-1? YES, as ANY x from the range x>1 is definitely more than -1.

Is x>-1,000,000? YES. Is x>0.5? YES. Is x>1? YES. Is x>1.9? YES. All these statement are always true.

It seems that you are confused with the logic of the question, please see the examples in the end of my previous post for examples.

Hope it's clear.
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Why x>2 doesn't satisfy \frac{x}{|x|}< x? If x=3 then \frac{3}{|3|}=1<3. Also I don't get the "subset" part at all.

Anyway:
Forget about \frac{x}{|x|}< x and x=integer.



We have x=integer and x>1 (as above means exactly that).

x could be: 2, 3, 4, 5, 6, ... It's given to be true.

Now, option B says x>-1. Is it true? Is 2 more than -1? Is 3 more than -1? ... All possible x-es are more than -1. So B is always true.
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x>-1 and x not equal to 0 and 1 - This is given. => x could be 2 , 3, 4,5,6,7----- so on
Every value of x should satisfy the answer.

Q1. Which of the following is always true.

1. X>2 -> is this always true? -> This is always true expect for x = 2.

If I take x = 2 using the inequality x/|x| < x , x =2 does not satisfy x>2.

I hope it makes sense.

Bunnel my point is every value that satisfy x/|x| < x should satisfy the answer choice.

Since 2 satisfies x/|x| < x but no x>2, x>2 can not be the answer...

Subset reason : All the values of x that satisfy x/|x| < x should be a subset of x>2
=> if x/|x| < x satisfy n values, then all those n values should be satisfied by x>2.
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