Inequalities

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Inequalities

by safina » Fri Feb 14, 2014 8:20 am
Can the experts please help me with this question?

If x is not equal to 0, is |x| less than 1?
a) x/|x|<x
b) |x|>x

Thanks!!

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by dimochka » Fri Feb 14, 2014 10:30 am
Hi safina,

The question essentially asks - is -1<x<1 (excluding 0)?

Choice A: the left hand side will be 1 when x>0 and -1 when x<0. Let's look at the possibilities:
x>1 --> Statement will be TRUE because left hand side is 1 whereas x>1.
0<x<1 --> Statement will be FALSE because left hand side is 1 whereas x is a small positive number
-1<x<0 --> Statement will be TRUE because left hand side is -1 whereas x is a smaller negative number
x<-1: Statement will be FALSE because left hand side is -1 whereas x<-1

Since the statement is TRUE for x>1 and -1<x<0, it is INSUFFICIENT to determine if |x|<1.

Choice B: |x|>x only when x is negative, otherwise they'll be equal. However, this means any negative number would work. So B is INSUFFICIENT on its own.

Together: Test out both cases for choice A being true.
x>1 - makes choice A true but makes choice B false - does not work
-1<x<0 - makes choice A true and choice B true.

Therefore for A and B to be true, -1<x<0 must be true as well. The answer is C
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by Bill@VeritasPrep » Fri Feb 14, 2014 11:26 am
safina wrote:Can the experts please help me with this question?

If x is not equal to 0, is |x| less than 1?
a) x/|x|<x
b) |x|>x

Thanks!!
I started with Statement 2. If the absolute value of x is greater than the actual value of x, then x must be negative; applying absolute value will make it positive, and a positive is always greater than a negative. However, x could be -1/2 (and we'd say yes to the question) or -10 (and we'd say no), so it is insufficient to say if |x| < 1.

From statement 1:

x/|x| < x

x < x|x|

If x is positive, we have x < x^2, which means x is greater than 1.

If x is negative, it must be between -1 and 0. For instance, -2 < -2(2) does not work, but -1/2 < -1/2 (1/2) does.

The only way to satisfy both statements is for x to between 0 and -1, so its absolute value is always less than 1. Thus, C
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by GMATGuruNY » Fri Feb 14, 2014 11:38 am
I posted an alternate approach here:

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by [email protected] » Fri Feb 14, 2014 1:22 pm
Hi safina,

This DS question can be beaten by TESTing Values and tracking the results.

We're told that X is CANNOT = 0. We're asked "is |X| < 1?" This is a YES/NO question.

Fact 1: X/|X| < X

Notice the fraction X/|X|; it will only ever equal 1 or -1, depending on the value of X. This also tells us that X COULD be > 1 or it COULD be a negative fraction.

If X = 2 then the answer to the question is NO.
If X = -1/2 then the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: |X| > X

This tells us that X MUST be negative.

If X = -1/2 then the answer to the question is YES.
If X = -2 then the answer to the question is NO.
Fact 2 is INSUFFICIENT

Combined, the only possibility that fits both Facts is that X must be a NEGATIVE FRACTION.
The answer to the question will ALWAYS be YES.
Combined, SUFFICIENT.

Final Answer: C

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by Patrick_GMATFix » Fri Feb 14, 2014 3:47 pm
Very nice question:
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Hope that helps,
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by safina » Sat Feb 15, 2014 12:03 am
thanks all! I think the most important part to solve this question is the initial rephrasing..once that is done, the problem becomes easier to solve.