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# Is x positive? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square.

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Is x positive? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square. [#permalink]
Bunuel
jayoptimist
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 cannot 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.

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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Is x positive? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square. [#permalink]
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jgomey
Bunuel
jayoptimist
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 cannot 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.

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:
https://gmatclub.com/forum/x2-4x-94661.html#p731476
https://gmatclub.com/forum/inequalities-trick-91482.html
https://gmatclub.com/forum/everything-is ... me#p868863
https://gmatclub.com/forum/xy-plane-7149 ... ic#p841486

Hope it helps.
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Re: Is x positive? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square. [#permalink]
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? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square. [#permalink]
Hi VeritasPrepKarishma,

Can you simplify Statement - 1 by Solving inequality method (not the one which involves number plugging)

As per my understanding {x/(x+1)} >0 will give me x>0 and x not equal to -1 ...

Thank you so much
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Is x positive? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square. [#permalink]
mihir0710
Hi VeritasPrepKarishma,

Can you simplify Statement - 1 by Solving inequality method (not the one which involves number plugging)

As per my understanding {x/(x+1)} >0 will give me x>0 and x not equal to -1 ...

Thank you so much

To solve inequalities with factors, we can use the wavy method.

The logic remains the same even when factors are divided. The sign changes in the same way. Just that when you have division, denominator cannot be 0.

So when you draw this on number line, you get that either x > 0 or x < -1

Originally posted by KarishmaB on 27 Jun 2017, 21:28.
Last edited by KarishmaB on 17 Oct 2022, 03:18, edited 1 time in total.
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Is x positive? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square. [#permalink]
Bunuel
jayoptimist
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 cannot 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.

Hope it's clear.

Is 0 a perfect square? Or is it just 1^2 and above?
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Is x positive? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square. [#permalink]
Bunuel
jayoptimist
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 cannot 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.

Hope it's clear.

Is 0 a perfect square? Or is it just 1^2 and above?

0 = 0^2, so 0 is a perfect square.
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Re: Is x positive? (1) 1/(x + 1) < 1 (2) (x - 1) is a perfect square. [#permalink]
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