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Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x

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Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x [#permalink]

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New post 05 Sep 2017, 01:30
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Re: Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x [#permalink]

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New post 05 Sep 2017, 01:45
x a positive number?

(1) (x – 2)^2 > 2
Assume x=1 (1-2)^2 > 2 No...
but x=0.1 positive (0.1-2)^2 = (1.9)^2 > 2... True.....
Not sufficient

(2) 2^x > 3^x
Assume x=1 => 2>3.. False
x=0.1 => 2^0.1 > 3^0.1 => False.. All positive powers will give answer "False"
so for 2^x > 3^x ... x is not positive
Sufficient

Answer: B

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Re: Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x [#permalink]

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New post 05 Sep 2017, 02:16
Bunuel wrote:
Is x a positive number?

(1) (x – 2)^2 > 2
(2) 2^x > 3^x



Bunuel,

I have a doubt here.

For statement 1,
I took ((x – 2)^2- (\sqrt{2}) ^2> 0

Now, can we not take (a-b)(a+b)>0 seperately to find out the solution?

Thanks

Kudos [?]: 53 [0], given: 504

Expert Post
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Joined: 02 Sep 2009
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Re: Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x [#permalink]

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New post 05 Sep 2017, 02:28
KS15 wrote:
Bunuel wrote:
Is x a positive number?

(1) (x – 2)^2 > 2
(2) 2^x > 3^x



Bunuel,

I have a doubt here.

For statement 1,
I took ((x – 2)^2- (\sqrt{2}) ^2> 0

Now, can we not take (a-b)(a+b)>0 seperately to find out the solution?

Thanks


Well yes but it will give the same answer.

Probably the easiest way to deal with the first statement would be to tests numbers or to take the square root from both sides:

\(|x-2|>\sqrt{2}\);

\(x-2>\sqrt{2}\) or \(-(x-2)>\sqrt{2}\)

\(x>2+\sqrt{2}\) or \(x<2-\sqrt{2}\)
_________________

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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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Kudos [?]: 135268 [0], given: 12679

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Re: Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x [#permalink]

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New post 05 Sep 2017, 02:34
Bunuel wrote:
KS15 wrote:
Bunuel wrote:
Is x a positive number?

(1) (x – 2)^2 > 2
(2) 2^x > 3^x



Bunuel,

I have a doubt here.

For statement 1,
I took ((x – 2)^2- (\sqrt{2}) ^2> 0

Now, can we not take (a-b)(a+b)>0 seperately to find out the solution?

Thanks


Well yes but it will give the same answer.

Probably the easiest way to deal with the first statement would be to tests numbers or to take the square root from both sides:

\(|x-2|>\sqrt{2}\);

\(x-2>\sqrt{2}\) or \(-(x-2)>\sqrt{2}\)

\(x>2+\sqrt{2}\) or \(x<2-\sqrt{2}\)


Can you show this using the formula a^2-b^2 i.e (a-b)(a+b) and then solving? I know it is easy to solve using numbers but not able to get using this formula

Thanks

Kudos [?]: 53 [0], given: 504

Senior Manager
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Joined: 02 Jul 2017
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Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x [#permalink]

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New post 05 Sep 2017, 02:42
KS15 wrote:
Can you show this using the formula a^2-b^2 i.e (a-b)(a+b) and then solving? I know it is easy to solve using numbers but not able to get using this formula

Thanks


KS15

(x – 2)^2 > 2
=> (x – 2)^2 - 2 >0
=>\((x – 2)^2 - \sqrt{2} >0\)
=> \((x-2+\sqrt{2}) (x-2-\sqrt{2}) >0\)

This will term either both term are positive or both terms are negative
both positive :
\(x-2+\sqrt{2} > 0\) and \(x-2-\sqrt{2} > 0\)
\(x>2-\sqrt{2}\) and \(x >2+\sqrt{2}\)

OR

both negative :
\(x-2+\sqrt{2} < 0\) and \(x-2-\sqrt{2} < 0\)
\(x<2-\sqrt{2}\)and \(x <2+\sqrt{2}\)

Combining all 4 equations we can write :

\(x >2+\sqrt{2}\) OR \(x<2-\sqrt{2}\)

Last edited by Nikkb on 05 Sep 2017, 02:52, edited 3 times in total.

Kudos [?]: 106 [0], given: 66

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42544

Kudos [?]: 135268 [0], given: 12679

Re: Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x [#permalink]

Show Tags

New post 05 Sep 2017, 02:45
KS15 wrote:
Bunuel wrote:
KS15 wrote:

Bunuel,

I have a doubt here.

For statement 1,
I took ((x – 2)^2- (\sqrt{2}) ^2> 0

Now, can we not take (a-b)(a+b)>0 seperately to find out the solution?

Thanks


Well yes but it will give the same answer.

Probably the easiest way to deal with the first statement would be to tests numbers or to take the square root from both sides:

\(|x-2|>\sqrt{2}\);

\(x-2>\sqrt{2}\) or \(-(x-2)>\sqrt{2}\)

\(x>2+\sqrt{2}\) or \(x<2-\sqrt{2}\)


Can you show this using the formula a^2-b^2 i.e (a-b)(a+b) and then solving? I know it is easy to solve using numbers but not able to get using this formula

Thanks


\((x – 2)^2 > 2\)

\((x – 2)^2 -2> 0\)

\((x – 2-\sqrt{2})(x – 2+\sqrt{2})> 0\).

For the product to be positive both multiples must have the same sign.

When both are positive:
\(x – 2-\sqrt{2}> 0\) and \(x – 2+\sqrt{2}> 0\).
\(x > 2+\sqrt{2}\) and \(x > 2-\sqrt{2}\).

For both above to be true simultaneously, \(x > 2+\sqrt{2}\) should hold (so x must be more than the larger number).

When both are negative:
\(x – 2-\sqrt{2}< 0\) and \(x – 2+\sqrt{2}< 0\).
\(x < 2+\sqrt{2}\) and \(x < 2-\sqrt{2}\).

For both above to be true simultaneously, \(x < 2-\sqrt{2}\) should hold (so x must be less than the smaller number).

So, \(x>2+\sqrt{2}\) or \(x<2-\sqrt{2}\).

Hope it's clear.
_________________

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?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 135268 [0], given: 12679

Re: Is x a positive number? (1) (x – 2)^2 > 2 (2) 2^x > 3^x   [#permalink] 05 Sep 2017, 02:45
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