Inequality -DS Question

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screwdriver: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Sat Mar 05, 2011 3:35 pm

by screwdriver » Sun Jun 26, 2011 11:29 am

Hi,

Can you please help in explaining below problem?

If x is not equal to 0, is |x| less than 1?

(1)(x/|x|)< x

(2) |x| > x

Thanks

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Source: Beat The GMAT — Problem Solving | Sun Jun 26, 2011 11:29 am

akshatgupta87: Senior | Next Rank: 100 Posts; Posts: 97; Joined: Wed Sep 01, 2010 11:11 amFollowed by: 2 members; Thanked: 1 times

by akshatgupta87 » Sun Jun 26, 2011 1:25 pm

screwdriver wrote:Hi,

Can you please help in explaining below problem?

If x is not equal to 0, is |x| less than 1?

(1)(x/|x|)< x

(2) |x| > x

Thanks

Let us start with statement 2)
Plug in values for x.
say x= -1
then |-1|>-1
-> 1 > -1 ...which means |x| i.e. 1 < 1 .. No
say x=-0.8
then |-0.8|>-0.8
-> 0.8> -0.8 ...which means |x| i.e. 0.8 < 1 .. Yes
So, statement 2 is insufficient

Now, lets consider Statement 1)
(x/|x|) < x
-> x < x|x|
-> |x| > 1
therefore answer is NO.
Hence statement 1 is sufficient.
Correct Answer is A)

Thanks,
~Akshat

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Gissinggy: Newbie | Next Rank: 10 Posts; Posts: 7; Joined: Mon Jun 13, 2011 1:04 pm; Thanked: 3 times

by Gissinggy » Sun Jun 26, 2011 3:07 pm

akshatgupta87 wrote:
screwdriver wrote:Hi,

Can you please help in explaining below problem?

If x is not equal to 0, is |x| less than 1?

(1)(x/|x|)< x

(2) |x| > x

Thanks

Let us start with statement 2)
Plug in values for x.
say x= -1
then |-1|>-1
-> 1 > -1 ...which means |x| i.e. 1 < 1 .. No
say x=-0.8
then |-0.8|>-0.8
-> 0.8> -0.8 ...which means |x| i.e. 0.8 < 1 .. Yes
So, statement 2 is insufficient

Now, lets consider Statement 1)
(x/|x|) < x
-> x < x|x|
-> |x| > 1
therefore answer is NO.
Hence statement 1 is sufficient.
Correct Answer is A)

Thanks,
~Akshat

-> ->
without knowing the sign of x, you can get |x| > 1 (x is positive) or |x| < 1 (x is negative) from x < x|x|
So A is insufficient.

Correct me if i am wrong.

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goalevan: Master | Next Rank: 500 Posts; Posts: 111; Joined: Tue Dec 30, 2008 1:25 pm; Location: USA; Thanked: 28 times; GMAT Score:770

by goalevan » Sun Jun 26, 2011 4:59 pm

Question stem: x not equal to 0, |x| < 1?, or -1 < x < 1?

1) x/|x| < x, since we know |x| is positive (and not equal to 0), we can multiply:
x < x|x|
x - x|x| < 0
x(1-|x|) < 0

When we have the form xy < 0, we know that either x is positive and y is negative, or x is negative and y is positive.

Case 1: x < 0 and 1-|x| > 0, or |x| < 1, or -1 < x < 1. Combined this says that -1 < x < 0.
Case 2: x > 0 and 1-|x| < 0, or |x| > 1. Combined this means x > 1.

Taking case 1 OR case 2, -1 < x < 0 OR x > 1. The question stem requires -1 < x < 1, so this statement is insufficient.

2) |x| > x, or x < 0. x could be -1,000,000 or -1/2. Insufficient.

1 & 2) Combining the second statement with the first, we can take case 1, which stated that -1 < x < 0, sufficient.

C

That was the classical solution, but I want to present an alternative that can save a LOT of time on this type of question:

Sketch the a graphs of each expression in the statements as a function.

1) y = x/|x|, this is a straight horizontal line with y = -1 for x < 0, and y = 1 for x > 0.
Draw in the graph for y = x. You can see easily that x/|x| is less than x for -1 < x < 0 and for all x > 1, so the statement is not sufficient.

2) Draw y = |x| and y = x. If you can't pull "x is less than 0" from statement 2, this will make it clear very quickly. From the drawing you can see that any negative number satisfies this inequality, so the statement is not sufficient.

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screwdriver: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Sat Mar 05, 2011 3:35 pm

by screwdriver » Mon Jun 27, 2011 7:25 pm

Hi,

Thanks for the reply.

But as per your statement "1 & 2) Combining the second statement with the first, we can take case 1, which stated that -1 < x < 0, sufficient."

it states that -1 < x < 0 but in the question we have to prove -1 < x < 1 ?

for me both above eqn looks different. Can you explain me how combining both statement we can get the desired result.

Thanks

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amit2k9: Master | Next Rank: 500 Posts; Posts: 461; Joined: Tue May 10, 2011 9:09 amFollowed by: 3 members; Location: pune; Thanked: 36 times

by amit2k9 » Tue Jun 28, 2011 3:45 am

the question is asking for -1 < x < 1.

a x= 2 and -0.5 satisfies. not sufficient.

b all negative values satisfy. not sufficient.

a+b x = -0.5 gives -1 < -0.5 for A.

thus C it is.

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goalevan: Master | Next Rank: 500 Posts; Posts: 111; Joined: Tue Dec 30, 2008 1:25 pm; Location: USA; Thanked: 28 times; GMAT Score:770

by goalevan » Tue Jun 28, 2011 4:12 pm

Hi screwdriver.

The question asks whether x is between -1 and 1. If we know that x is between -1 and 0, as the combined statements tell us, we know that x must be between -1 and 1, no matter what its value.

Think of a few possible values: -3/4, -1/2, -1/5, -2/7, -1/1000. They are all between -1 and 0, and they are also all between -1 and 1. The combined statements actually narrow down the possible values of x further than what is required in the question stem, but it is still valid because this interval is contained within the interval in the question stem.

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screwdriver: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Sat Mar 05, 2011 3:35 pm

by screwdriver » Tue Jun 28, 2011 4:34 pm

Thanks goalevan!!

I totally understood now.

thanks for your help

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Wed Jun 29, 2011 3:02 am

Whats the OA, screwdriver?

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screwdriver: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Sat Mar 05, 2011 3:35 pm

by screwdriver » Wed Jun 29, 2011 9:37 am

Hi,

answer is C : Both statements are required

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