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Is x positive?

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Is x positive? [#permalink] New post 28 Jul 2012, 18:39
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Is x positive?

(1) 1/(x+1) < 1
(2) (x-1) is a perfect square.

What is wrong with the following line of thought for condition 1

1/(x+1) < 1

1 < (x+1)

1 -1 < x+1-1

0 < x Hence x>0.
[Reveal] Spoiler: OA
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Re: Is X positive? [#permalink] New post 28 Jul 2012, 19:46
jayoptimist wrote:
Is X positive?

1) 1/(x+1) < 1
2) (x-1) is a perfect square.

What is wrong with the following line of thought for condition 1

1/(x+1) < 1

1 < (x+1)

1 -1 < x+1-1

0 < x Hence x>0.

Hi

For statement 1:
x can be +ve or -ve for positive ( 1,2 ) canbe true for negative values (-3) satisfies the condition, hence 1 is insufficient

for statement 2:
(x-1) is a perfect square hence positive
=> x-1>0 or x>1, hence positive

Thus correct answer is B
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Re: Is x positive? [#permalink] New post 29 Jul 2012, 00:02
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jayoptimist wrote:
Is x positive?

(1) 1/(x+1) < 1
(2) (x-1) is a perfect square.

What is wrong with the following line of thought for condition 1

1/(x+1) < 1

1 < (x+1)

1 -1 < x+1-1

0 < x Hence x>0.


We can not multiple \frac{1}{x+1}< 1 by x+1 since we don't know whether this expression (x+1) is positive or negative, for the first case we should keep the same sign but for the second case we should flip the sign (when multiplying by negative number we should flip the sign of the inequity).

Is x positive?

(1) 1/(x+1) < 1. If x=10 then the answer is YES, but if x=-10 then the answer is NO. Not sufficient.

Or if you want to solve this inequality then: \frac{1}{x+1}< 1 --> 1-\frac{1}{x+1}>0 --> \frac{x}{x+1}>0 --> x>0 or x<-1, hence x could be positive as well as negative. Not sufficient.

(2) (x-1) is a perfect square. Given: x-1=\{perfect \ square\} --> x=\{perfect \ square\}+1. Now, since perfect square is more than or equal to zero, then x=\{perfect \ square\}+1=non-negative+1=positive. Sufficient.

Answer: B.

Hope it's clear.
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Re: Is x positive? [#permalink] New post 10 Jan 2013, 01:34
Bunuel wrote:
jayoptimist wrote:
Is x positive?

(1) 1/(x+1) < 1
(2) (x-1) is a perfect square.

What is wrong with the following line of thought for condition 1

1/(x+1) < 1

1 < (x+1)

1 -1 < x+1-1

0 < x Hence x>0.


We can not multiple \frac{1}{x+1}< 1 by x+1 since we don't know whether this expression (x+1) is positive or negative, for the first case we should keep the same sign but for the second case we should flip the sign (when multiplying by negative number we should flip the sign of the inequity).

Is x positive?

(1) 1/(x+1) < 1. If x=10 then the answer is YES, but if x=-10 then the answer is NO. Not sufficient.

Or if you want to solve this inequality then: \frac{1}{x+1}< 1 --> 1-\frac{1}{x+1}>0 --> \frac{x}{x+1}>0 --> x>0 or x<-1, hence x could be positive as well as negative. Not sufficient.

(2) (x-1) is a perfect square. Given: x-1=\{perfect \ square\} --> x=\{perfect \ square\}+1. Now, since perfect square is more than or equal to zero, then x=\{perfect \ square\}+1=non-negative+1=positive. Sufficient.

Answer: B.

Hope it's clear.


Bunuel,

How did you arrive at x<-1 in the explanation below?

"\frac{x}{x+1}>0 --> x>0 or x<-1, hence x could be positive as well as negative. Not sufficient."
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Re: Is x positive? [#permalink] New post 10 Jan 2013, 03:16
Expert's post
jgomey wrote:
Bunuel wrote:
jayoptimist wrote:
Is x positive?

(1) 1/(x+1) < 1
(2) (x-1) is a perfect square.

What is wrong with the following line of thought for condition 1

1/(x+1) < 1

1 < (x+1)

1 -1 < x+1-1

0 < x Hence x>0.


We can not multiple \frac{1}{x+1}< 1 by x+1 since we don't know whether this expression (x+1) is positive or negative, for the first case we should keep the same sign but for the second case we should flip the sign (when multiplying by negative number we should flip the sign of the inequity).

Is x positive?

(1) 1/(x+1) < 1. If x=10 then the answer is YES, but if x=-10 then the answer is NO. Not sufficient.

Or if you want to solve this inequality then: \frac{1}{x+1}< 1 --> 1-\frac{1}{x+1}>0 --> \frac{x}{x+1}>0 --> x>0 or x<-1, hence x could be positive as well as negative. Not sufficient.

(2) (x-1) is a perfect square. Given: x-1=\{perfect \ square\} --> x=\{perfect \ square\}+1. Now, since perfect square is more than or equal to zero, then x=\{perfect \ square\}+1=non-negative+1=positive. Sufficient.

Answer: B.

Hope it's clear.


Bunuel,

How did you arrive at x<-1 in the explanation below?

"\frac{x}{x+1}>0 --> x>0 or x<-1, hence x could be positive as well as negative. Not sufficient."


We have \frac{x}{x+1}>0.

The roots are -1, and 0 (equate the expressions to zero to get the roots and list them in ascending order), this gives us 3 ranges: x<-1, -1<x<0, and x>0.

Now, test some extreme value: for example if x is very large number then all two terms (x and x+1) will be positive which gives the positive result for the whole expression, so when x>0 the expression is positive. Now the trick: as in the 3rd range expression is positive then in 2rd it'll be negative and finally in 1st it'll be positive again: + - +. So, the ranges when the expression is positive are: x<-1 (1st range) and x>0 (3rd range).

For more check:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863
xy-plane-71492.html?hilit=solving%20quadratic#p841486

Hope it helps.
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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.


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Re: Is x positive? [#permalink] New post 11 Jan 2013, 17:33
The light bulb clicked!!!!

Got it so the condition is met for all numbers less than -1 or greater than 0, but any number between 0 and -1 (Fraction) will result in a negative number.

But even without thinking about the possibility of a fraction, Statement 1 is insufficient because x>0 or x<-1 right?

I know this is a silly, but I didn't realize I could find the roots by doing this:

x/x+1>0

so

x=0

and

x+1=0...x=-1


Is my thinking correct?
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Re: Is x positive? [#permalink] New post 07 Jun 2014, 23:04
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Re: Is x positive?   [#permalink] 07 Jun 2014, 23:04
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