Is |x|

Question

Is |x| < 1 ?

(1) x/|x| < x

(2) x/|x| < 1

Bunuel · Accepted Answer

Is |x| < 1 ? Is |x| < 1 --> is -1 < x < 1. (1) x/|x| < x. Two cases: A. x<0 --> \frac{x}{-x} -10 --> \frac{x}{x} 1

anupamadw · Answer

Hi Bunuel
I fail to understand above solution.
We have to find whether -1< x<1
From 1 ) we know that x/|x| < x holds true for -1<x<0 and x>1 so we know that x is not in range 0<x<1
is this not sufficient to answer NO ....?
Again from 2) We know -infinity < x<0 then answer is NO
Sorry, could you please tell me what am I missing...how are you getting YES and NO for each?

ENGRTOMBA2018 · Answer

Let me try to explain. From 1, x/|x| < x ---> 2 cases: --- when x \geq 0 --> |x| = x ---> x/x < x --> x > 1 . Thus a "no" for is |x| < 1 --- when x < 0 --> |x| = -x ---> x/-x < x --> x > -1 . For x = 0.5, then a "yes" for |x|<1 but if x = 5, then a "no" for is |x|<1. Thus NOT sufficient. From 2, x/|x| < 1 , again 2 cases, --- when x \geq 0 --> |x| = x ---> x/x < 1 --> 1<1 . Thus x \geq 0 is not a possible scenario. --- when x <0 --> |x| = -x ---> x/-x < 1 --> 1>-1 . This is true for ALL x and thus x <0 satisfies this. For x = -0.5, you get a "yes" for is |x|<1 but for x = -4, you get a "no" for is |x| <1 . Thus NOT sufficient. Alternately, if you want you can also test cases to come up with ranges for x. But this might be a bit more time consuming. For your analysis of statement 1, the text in red is not correct. Refer to the explanation mentioned above. For statement 2, you are correct that x<0 but you need to check whether -1

Shreeya24 · Answer

I have tried to use the line method to solve this problem. 1. x/!x!1. Hence not suffecient. 2. x/!x!<1 . Repeating the same logic above we get x-!x!<0. This will be true only if x<0. Still not suffeceint. Combining both statements, we can eliminate the portion x>1, which we got in statement 1. Leaving us only with -1

Probus · Answer

Hi,

There is similar question out there
https://gmatclub.com/forum/if-x-is-not- ... 86140.html

Now here Bunuel has explained two approaches for solving the question
https://gmatclub.com/forum/if-x-is-not- ... 86140.html

Hope this helps

Probus

Kinshook · Answer

Is |x| < 1 ? (1) x/|x| < x \frac{x}{|x|}- x <0 \frac{x - x|x|}{|x|} < 0 Since |x| > 0 x - x|x| < 0 x (1- |x|) < 0 x (|x| -1) > 0 If x>0; |x| > 1; x>1 If x<0; |x| < 1; -11 or -10; x-x<0; If x<0; x + x < 0; x<0 |x| may or may not be <1 NOT SUFFICIENT (1) + (2) (1) x/|x| < x If x>0; |x| > 1; x>1 If x<0; |x| < 1; -1

LoneSurvivor · Answer

do not multiply or cancel anything until and unless we know the exact sign of the x (or any other unknown stated in the question )

hiranmay · Answer

Is |x| < 1 ?
(1) x/|x| < x
(2) x/|x| < 1

Solution:
(1) x/|x| < x --> insuff
=> x < x|x|
=> x|x|-x > 0
=> x(|x|-1) > 0
` ++>0 => x >0 & |x|>1 --> no
or -->0 => x<0 &|x|<1 --> yes

(2) x/|x| < 1 --> insuff
=> x<|x|
so, x is always -ve
=> x can be -1<x<0 --> yes
or x=<-1 --> no
combing (1) & (2)=>
x< 0 from (2), so |x| <1 from (1) --> suff
Answer: C

Romil042 · Answer

The statement asks whether the value of x is between -1 and 1.

A:
x can fall into four domain,
x<-1
-1<x<0
0<x<1
1<x

for first and third the statement holds true, and not for second and fourth.

B:
It simply states that x<1.

Both the statement is insufficient. But by clubbing them still, we cannot say
because x<-1 satisfies and -1<x<0 too.

A bit confused why C is correct answer!

AndrewN · Answer

Hello, Romil042 . I just happened upon this question today and scrolled down to find your query. I think I can help clarify your confusion. I will respond inline below. So far, so good. This statement at the end is inaccurate. Did you test values or make assumptions? 1) x < -1 ; test x = -2 \frac{x}{|x|} < x \frac{(-2)}{|(-2)|} < (-2) \frac{-2}{2} < -2 -1 < -2 X Since -1 is NOT less than -2, we know that x CANNOT be less than -1 . 2) -1 < x < 0 ; test x = -0.5 \frac{x}{|x|} < x \frac{(-0.5)}{|(-0.5)|} < (-0.5) \frac{-0.5}{0.5} < -0.5 -1 < -0.5 √ We have found that x CAN be between -1 and 0 . 3) 0 < x < 1 ; test x = 0.5 \frac{x}{|x|} < x \frac{(0.5)}{|(0.5)|} < (0.5) \frac{0.5}{0.5} < 0.5 1 < 0.5 X Since 1 is NOT less than 0.5, x CANNOT be between 0 and 1 . 4) 1 < x ; test x = 2 \frac{x}{|x|} < x \frac{(2)}{|(2)|} < (2) \frac{2}{2} < 2 1 < 2 √ We have found that x CAN be greater than 1 . In short, we have two valid types of inputs based on Statement (1): x is either between -1 and 0 or it is greater than 1. Of course, we CANNOT answer the original question conclusively with this information. a) If -1 < x < 0 , |x| < 1 b) If x > 1 , |x| > 1 Thus, Statement (1) on its own is NOT sufficient. Of course, I hope you can see now that by combining the two statements, we can eliminate b) above, so it must be true that |x| < 1, and the answer is (C). Good luck with your studies. - Andrew

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