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If x≠0, is |x|<1? (1) x^2/|x| > x

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If x≠0, is |x|<1? (1) x^2/|x| > x [#permalink]

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New post 18 Feb 2017, 05:24
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If \(x ≠ 0\), is \(|x|<1\)?

(1) \(\frac{x^2}{{|x|}} > x\)

(2) \(\frac{x}{{|x|}} < x\)

I have a hard time to answer this question. Could someone please help to explain?

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Re: If x≠0, is |x|<1? (1) x^2/|x| > x [#permalink]

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New post 18 Feb 2017, 07:50
1
ziyuenlau wrote:
If \(x ≠ 0\), is \(|x|<1\)?

(1) \(\frac{x^2}{{|x|}} > x\)

(2) \(\frac{x}{{|x|}} < x\)

I have a hard time to answer this question. Could someone please help to explain?


Hi,

is \(|x|<1\)? => is \(-1 < x < 1\)?

St 1:
Case1: x>0 => |x| = x.
\(\frac{x^2}{{|x|}} = \frac{x^{2}}{x} = x > x\) => No solution.

Case2: x<0 => |x| = -x
\(\frac{x^2}{{|x|}} = \frac{x^{2}}{-x} = -x > x\) => All negative values will satisfy this equation.

Hence, not sufficient.

St 2:
Case 1: x>0 => |x| = x

\(\frac{x}{{|x|}} = \frac{x}{x} = 1 < x\) => solution x>1.

case 2: x<0 => |x| = -x

\(\frac{x}{{|x|}} = \frac{x}{-x} = -1 < x\) => solution -1 < x < 0.

From this statement, we have two sets of solutions. Hence, not sufficient.

By St 1 and st 2, we have
-1< x < 0. => Unique answer. Answer C.

Hope this helps.
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Re: If x≠0, is |x|<1? (1) x^2/|x| > x [#permalink]

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New post 18 Feb 2017, 08:06
If \(x ≠ 0\), is \(|x|<1\)?

The question asks whether -1 < x < 1 (x ≠ 0).

(1) \(\frac{x^2}{{|x|}} > x\) --> reduce the LHS by |x|: |x| > x. This implies that x is negative. Not sufficient.

(2) \(\frac{x}{{|x|}} < x\):

If x < 0, we'll have x/(-x) < x --> -1 < x. Since we consider x < 0, then we'll have -1 < x < 0.
If x > 0, we'll have x/x < x --> 1 < x.

So, we have that \(\frac{x}{{|x|}} < x\) is true for -1 < x < 0 and x > 1. Not sufficient.

(1)+(2) Intersection of the ranges from (1) and (2) is -1 < x < 0. Thus the answer to the question is YES. Sufficient.

Answer: C.
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Re: If x≠0, is |x|<1? (1) x^2/|x| > x [#permalink]

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New post 28 Feb 2017, 01:41
Bunuel wrote:
If \(x ≠ 0\), is \(|x|<1\)?

The question asks whether -1 < x < 1 (x ≠ 0).

(1) \(\frac{x^2}{{|x|}} > x\) --> reduce the LHS by |x|: |x| > x. This implies that x is negative. Not sufficient.

(2) \(\frac{x}{{|x|}} < x\):

If x < 0, we'll have x/(-x) < x --> -1 < x. Since we consider x < 0, then we'll have -1 < x < 0.
If x > 0, we'll have x/x < x --> 1 < x.

So, we have that \(\frac{x}{{|x|}} < x\) is true for -1 < x < 0 and x > 1. Not sufficient.

(1)+(2) Intersection of the ranges from (1) and (2) is -1 < x < 0. Thus the answer to the question is YES. Sufficient.

Answer: C.


Hello Bunuel, could you please elaborate more on the (1) condition, I can't grasp how did you get that x is -ve.
Thx
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Re: If x≠0, is |x|<1? (1) x^2/|x| > x [#permalink]

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New post 28 Feb 2017, 05:52
Alexey1989x wrote:
Bunuel wrote:
If \(x ≠ 0\), is \(|x|<1\)?

The question asks whether -1 < x < 1 (x ≠ 0).

(1) \(\frac{x^2}{{|x|}} > x\) --> reduce the LHS by |x|: |x| > x. This implies that x is negative. Not sufficient.

(2) \(\frac{x}{{|x|}} < x\):

If x < 0, we'll have x/(-x) < x --> -1 < x. Since we consider x < 0, then we'll have -1 < x < 0.
If x > 0, we'll have x/x < x --> 1 < x.

So, we have that \(\frac{x}{{|x|}} < x\) is true for -1 < x < 0 and x > 1. Not sufficient.

(1)+(2) Intersection of the ranges from (1) and (2) is -1 < x < 0. Thus the answer to the question is YES. Sufficient.

Answer: C.


Hello Bunuel, could you please elaborate more on the (1) condition, I can't grasp how did you get that x is -ve.
Thx


We have |x| > x.

Now, if x were non-negative, then |x| would be equal to x, for example, if x=2, then |x| = |2| = 2 = x. Only for negative x we can have |x| > x. For example, consider x = -2: |x| = |-2| = 2 > x = 2.
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New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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Re: If x≠0, is |x|<1? (1) x^2/|x| > x   [#permalink] 28 Feb 2017, 05:52
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