Absolute value

Expert replies

Uri: Master | Next Rank: 500 Posts; Posts: 229; Joined: Tue Jan 13, 2009 6:56 am; Thanked: 8 times; GMAT Score:700

by Uri » Mon Apr 20, 2009 9:11 am

If x≠0, is |x| <1?
(1) x^2 < 1
(2) |x| < (1/x)

Please explain your logic.

OA: [spoiler](D)[/spoiler]
Source: GMAT Plus Book
I don't understand how [spoiler](2)[/spoiler] can be sufficient. Please help me.

Like this post

Source: Beat The GMAT — Data Sufficiency | Mon Apr 20, 2009 9:11 am

iamcste: Legendary Member; Posts: 940; Joined: Tue Aug 26, 2008 3:22 amFollowed by: 1 member; Thanked: 55 times

Re: Absolute value

by iamcste » Mon Apr 20, 2009 9:49 am

Uri wrote:If x≠0, is |x| <1?
(1) x^2 < 1
(2) |x| < (1/x)

Please explain your logic.

OA: [spoiler](D)[/spoiler]
Source: GMAT Plus Book
I don't understand how [spoiler](2)[/spoiler] can be sufficient. Please help me.

Second statment means X has to be between 0 and 1

Qtn Is X between -1 and 1?

Yes, any value of X that suffices qtn 2 is between that range and no other value outside that range suffices qtn 2

In short as I dont have any value of x outside range -1 to 1 that sufficies qtn 2, Statement is sufficient

Did you think that x is not between 0 and -1 hence its not sufficient?

Stmt 1 is sufficient

[spoiler]
Choose D[/spoiler]

Like this post

Uri: Master | Next Rank: 500 Posts; Posts: 229; Joined: Tue Jan 13, 2009 6:56 am; Thanked: 8 times; GMAT Score:700

by Uri » Tue Apr 21, 2009 1:21 am

i made a very silly mistake....and i am very much prone to making this mistake.
for (2), i simplified as
x|x|<1
but this will be true only when x is positive

you have done it in back-solving method. although this makes sense, i would like to know whether there is any other way to solve this problem

Like this post

iamcste: Legendary Member; Posts: 940; Joined: Tue Aug 26, 2008 3:22 amFollowed by: 1 member; Thanked: 55 times

by iamcste » Tue Apr 21, 2009 2:12 am

Uri wrote:i made a very silly mistake....and i am very much prone to making this mistake.
for (2), i simplified as
x|x|<1
but this will be true only when x is positive

you have done it in back-solving method. although this makes sense, i would like to know whether there is any other way to solve this problem

try to use test data, -2,-1,-1/2,0,1/2. 1 and 2

see the relationship only for for 1/2, you will see that relationship is true...

Like this post

austin: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Oct 23, 2008 9:12 amFollowed by: 1 member; Location: India; Thanked: 9 times

by austin » Tue Apr 21, 2009 3:50 am

If x≠0, is |x| <1?
(1) x^2 < 1
(2) |x| < (1/x)

statement 1: x^2 < 1=> -1<x<1 => |x| < 1 sufficient
statement 2: |x| < (1/x).. for x > 1 and x <-1, this does NOT hold true.. it holds true only when -1<x<1 => |x|<1... sufficient

D

NOTE: DO NOT CROSS-MULTIPLY VARIABLES ...we can cross multiply only when we know the sign of that quantity!!!!

Like this post

iamcste: Legendary Member; Posts: 940; Joined: Tue Aug 26, 2008 3:22 amFollowed by: 1 member; Thanked: 55 times

by iamcste » Tue Apr 21, 2009 4:17 am

austin wrote: statement 2: |x| < (1/x).. for x > 1 and x <-1, this does NOT hold true.. it holds true only when -1<x<1 => |x|<1... sufficient

D

are you sure, that this holds true for value of x between 0 and -1?

I dont think so.

Like this post

austin: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Oct 23, 2008 9:12 amFollowed by: 1 member; Location: India; Thanked: 9 times

by austin » Tue Apr 21, 2009 4:29 am

x = 1/2

|x| = 1/2; 1/x = 1/(1/2) = 2

=> |x| < (1/x); satisfies the given condition

Like this post

iamcste: Legendary Member; Posts: 940; Joined: Tue Aug 26, 2008 3:22 amFollowed by: 1 member; Thanked: 55 times

by iamcste » Tue Apr 21, 2009 4:48 am

austin wrote:x = 1/2

|x| = 1/2; 1/x = 1/(1/2) = 2

=> |x| < (1/x); satisfies the given condition

Dude, I am talking about a value between 0 and -1

1/2 is not between 0 and -1

Okie, what I ideally meant is IMO stmt 2 means that x has to be between 0 and 1 and not -1 to 1 as you mentioned

so pls recheck this once

Like this post

austin: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Oct 23, 2008 9:12 amFollowed by: 1 member; Location: India; Thanked: 9 times

by austin » Tue Apr 21, 2009 7:21 am

iamcste,

You are right...

Like this post

cubicle_bound_misfit: Master | Next Rank: 500 Posts; Posts: 246; Joined: Mon May 19, 2008 7:34 am; Location: Texaco Gas Station; Thanked: 7 times

by cubicle_bound_misfit » Tue Apr 21, 2009 7:31 am

FOR STMT 2.

TAKE
X = -1/2

THEN |X| > 1/X

ALSO FOR x> 1 |x| > 1/x

relation is only true

for 0<x<1 for all the values in this range |x| <1 hence Suff

Cubicle Bound Misfit

Like this post

maihuna: Legendary Member; Posts: 1578; Joined: Sun Dec 28, 2008 1:49 amFollowed by: 9 members; Thanked: 82 times; GMAT Score:720

by maihuna » Tue Apr 21, 2009 10:03 am

1. x^2 < 1 for -1<x<1 => |x| <1 Suff

2. |x| < 1/x
trick here is we dont know sign of x but we do know the sign of |x|

so 1/|x| > x
or 1 > x|x|
so 1 > -x^2 or x^2> -1 or x^2+1>0 true as x^2 will be always positive
and 1>x^2 or x^2<1 that means |x| <1
so sufficient

ans: D

Like this post

schumi_gmat: Master | Next Rank: 500 Posts; Posts: 377; Joined: Wed Sep 03, 2008 9:30 amFollowed by: 2 members; Thanked: 15 times

by schumi_gmat » Tue Apr 21, 2009 10:38 am

@ Cubicle

for 0<x<1 for all the values in this range |x| <1 hence Suff

We have to prove lxl < 1 i.e -1<x<1

I agree that the equation is true for 0<x<1.
But how is it possible that it is sufficient to prove -1<x<1. Isnt it partially true?

Can you explain what I am doing wrong here?

Like this post

ketkoag: Legendary Member; Posts: 876; Joined: Thu Apr 10, 2008 8:14 am; Thanked: 13 times

by ketkoag » Tue Apr 21, 2009 11:42 am

statement 2 doesn't hold true when x is between -1 and 0.
IMO : D

Like this post

iamcste: Legendary Member; Posts: 940; Joined: Tue Aug 26, 2008 3:22 amFollowed by: 1 member; Thanked: 55 times

by iamcste » Thu Apr 23, 2009 10:59 am

maihuna wrote:1. x^2 < 1 for -1<x<1 => |x| <1 Suff

2. |x| < 1/x
trick here is we dont know sign of x but we do know the sign of |x|

so 1/|x| > x
or 1 > x|x|
so 1 > -x^2 or x^2> -1 or x^2+1>0 true as x^2 will be always positive
and 1>x^2 or x^2<1 that means |x| <1
so sufficient

ans: D

Maihuna, I feel this is incorrect. you also move x while moving mod x

Also, Stmt 2 is satisied only between 0 and 1 and NOT between -1 and 1, as you conclude.

Pls let me know if I get you incorrect.

Like this post

Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Thu Apr 23, 2009 2:41 pm

schumi_gmat wrote:@ Cubicle
for 0<x<1 for all the values in this range |x| <1 hence Suff
We have to prove lxl < 1 i.e -1<x<1

I agree that the equation is true for 0<x<1.
But how is it possible that it is sufficient to prove -1<x<1. Isnt it partially true?

Can you explain what I am doing wrong here?

You've misinterpreted the question.

You've read the question as "could x be every number that satisfies |x| < 1?"

However, the question is actually "is |x| < 1?"

In order for a statement to be sufficient, it has to give us a definite "yes" or definite "no" answer to the question.

From (2) we can derive 0<x<1. In other words, x must be a positive fraction.

Is the absolute value of every positive fraction less than 1? Definitely YES! Therefore, (2) is sufficient.

Let's look at a much simpler question to illustrate the conceptual error that you made:

Q: Is x greater than 0?

(1) x = 5

For this question, you probably wouldn't even think twice. If x=5, then it's definitely greater than 0... sufficient.

However, if we follow the line of thinking you employed for the original question, we'd say:

"5 doesn't satisfy every possible value that's greater than 0, therefore (1) isn't sufficient."

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

Kaplan Exclusive: The Official Test Day Experience | Ready to Take a Free Practice Test? | Kaplan/Beat the GMAT Member Discount
BTG100 for $100 off a full course

Like this post

Post new topic Post Reply

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review