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# If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9

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Manager
Joined: 19 Feb 2019
Posts: 115
Location: India
Concentration: Marketing, Statistics
GMAT 1: 650 Q46 V34
GPA: 3
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

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08 Dec 2019, 19:13
Quote:
x is positive

(1) |x-1| >2
x>3. Sufficient

(2) |x-2|>3
x>5 Sufficient

D is correct.

Hi
when you say |x-1|>2 why is mod sign not applicable for 2??
Senior Manager
Joined: 27 Feb 2014
Posts: 345
Location: India
GMAT 1: 570 Q49 V20
GPA: 3.97
WE: Engineering (Education)
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

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08 Dec 2019, 20:40
devavrat wrote:
Quote:
x is positive

(1) |x-1| >2
x>3. Sufficient

(2) |x-2|>3
x>5 Sufficient

D is correct.

Hi
when you say |x-1|>2 why is mod sign not applicable for 2??

|2| is same as 2 only.
Intern
Joined: 10 Dec 2019
Posts: 43
If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

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16 Feb 2020, 03:21
1
Bunuel wrote:
rohitgoel15 wrote:
If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4
(2) (x - 2)^2 > 9

Can someone point a mistake in my method?
(1)
Taking one of the equations:
(x - 1)^2 > 4
x^2 + 1 - 2x > 4
x^2 + 1 - 2x - 4 > 0
x^2 - 3x + 1x - 3 > 0
(x-3) (x+1) > 0
x > 3 and x > -1

while in the explanation given the ans is coming out to be x > 3 and x < -1

If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4 --> $$(x+1)(x-3)>0$$ --> roots are -1 and 3. Now, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: $$x<-1$$ or $$x>3$$. Since given that $$x$$ is positive then only one range is valid: $$x>3$$. Sufficient.

(2) (x - 2)^2 > 9 --> $$(x+1)(x-5)>0$$ --> roots are -1 and 5. Again, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: $$x<-1$$ or $$x>5$$. Since given that $$x$$ is positive then only one range is valid: $$x>5$$. Sufficient.

Solving inequalities:
http://gmatclub.com/forum/x2-4x-94661.html#p731476 (check this one first)
http://gmatclub.com/forum/inequalities-trick-91482.html
http://gmatclub.com/forum/data-suff-ine ... 09078.html
http://gmatclub.com/forum/range-for-var ... me#p873535
http://gmatclub.com/forum/everything-is ... me#p868863

Another approach:

If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4 --> since both sides of the inequality are non-negative then we can take square root from both parts: $$|x-1|>2$$. $$|x-1|$$ is just the distance between 1 and $$x$$ on the number line. We are told that this distance is more than 2: --(-1)----1----3-- so, $$x<-1$$ or $$x>3$$. Since given that $$x$$ is positive then only one range is valid: $$x>3$$. Sufficient.

(2) (x - 2)^2 > 9 --> $$|x-2|>3$$. The same here: $$|x-2|$$ is just the distance between 2 and $$x$$ on the number line. We are told that this distance is more than 3: --(-1)----2----5-- so, $$x<-1$$ or $$x>5$$. Since given that $$x$$ is positive then only one range is valid: $$x>5$$. Sufficient.

Hope it helps.

Hi Bunuel

Considering (1) (x - 1)^2 > 4 --> $$(x+1)(x-3)>0$$ --> roots are -1 and 3. Now, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: $$x<-1$$ or $$x>3$$.

Well I get confused still to understand why can't we write x>-1 and x>3 and why are we changing polarity in case of negative value is to determine the range or what exactly?
Math Expert
Joined: 02 Sep 2009
Posts: 61484
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9  [#permalink]

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16 Feb 2020, 03:27
Anurag06 wrote:
Bunuel wrote:
rohitgoel15 wrote:
If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4
(2) (x - 2)^2 > 9

Can someone point a mistake in my method?
(1)
Taking one of the equations:
(x - 1)^2 > 4
x^2 + 1 - 2x > 4
x^2 + 1 - 2x - 4 > 0
x^2 - 3x + 1x - 3 > 0
(x-3) (x+1) > 0
x > 3 and x > -1

while in the explanation given the ans is coming out to be x > 3 and x < -1

If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4 --> $$(x+1)(x-3)>0$$ --> roots are -1 and 3. Now, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: $$x<-1$$ or $$x>3$$. Since given that $$x$$ is positive then only one range is valid: $$x>3$$. Sufficient.

(2) (x - 2)^2 > 9 --> $$(x+1)(x-5)>0$$ --> roots are -1 and 5. Again, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: $$x<-1$$ or $$x>5$$. Since given that $$x$$ is positive then only one range is valid: $$x>5$$. Sufficient.

Solving inequalities:
http://gmatclub.com/forum/x2-4x-94661.html#p731476 (check this one first)
http://gmatclub.com/forum/inequalities-trick-91482.html
http://gmatclub.com/forum/data-suff-ine ... 09078.html
http://gmatclub.com/forum/range-for-var ... me#p873535
http://gmatclub.com/forum/everything-is ... me#p868863

Another approach:

If x is positive, is x > 3 ?

(1) (x - 1)^2 > 4 --> since both sides of the inequality are non-negative then we can take square root from both parts: $$|x-1|>2$$. $$|x-1|$$ is just the distance between 1 and $$x$$ on the number line. We are told that this distance is more than 2: --(-1)----1----3-- so, $$x<-1$$ or $$x>3$$. Since given that $$x$$ is positive then only one range is valid: $$x>3$$. Sufficient.

(2) (x - 2)^2 > 9 --> $$|x-2|>3$$. The same here: $$|x-2|$$ is just the distance between 2 and $$x$$ on the number line. We are told that this distance is more than 3: --(-1)----2----5-- so, $$x<-1$$ or $$x>5$$. Since given that $$x$$ is positive then only one range is valid: $$x>5$$. Sufficient.

Hope it helps.

Hi Bunuel

Considering (1) (x - 1)^2 > 4 --> $$(x+1)(x-3)>0$$ --> roots are -1 and 3. Now, ">" sign indicates that the solution lies to the left of a smaller root and to the right of the larger root: $$x<-1$$ or $$x>3$$.

Well I get confused still to understand why can't we write x>-1 and x>3 and why are we changing polarity in case of negative value is to determine the range or what exactly?

x > -1 or x > 3 does not make sense. Which is it? Is it x > -1? Could x be, say 1, or because of x > 3 it cannot be.

The method used in the solution is explained in details in the links given there. Please go through them. Hope it helps.
_________________
Re: If x is positive, is x > 3 ? (1) (x - 1)^2 > 4 (2) (x - 2)^2 > 9   [#permalink] 16 Feb 2020, 03:27

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