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# If x is an integer, is x|x| < 2^x?

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Intern
Joined: 08 Jan 2007
Posts: 42
If x is an integer, is x|x| < 2^x?  [#permalink]

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17 Jan 2007, 15:27
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If x is an integer, is x|x| < 2^x?

(1) x < 0
(2) x = -10
Manager
Joined: 04 Nov 2006
Posts: 158

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17 Jan 2007, 19:03
1
1) If x<0, then x*|x| is always negative. 2^x is positive. So, this statement is sufficient.

2) Just plug it in. Sufficient.

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Joined: 28 Feb 2006
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17 Jan 2007, 19:04
D for me

I couldn't figure out using statement 1 so I ignored it to start with. Using statement 2, x|x| will always be < 2^x. So the answer is either B or D.

Going back to statement 1, I plugged in negative values such as -1, -2 and also fractions and realised that if x is not an integer, the result need not always be true but if x is an integer, x|x| will always be < 2^x. And the question stem does say that x is an integer.

Hence D for me.
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19 Jan 2007, 01:08
If you determine that A satisfies the question then B would have to also.
Intern
Joined: 22 Dec 2009
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DS Questions - absolute integers - Help understand logic  [#permalink]

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13 Sep 2010, 15:41
1
If X is an integer is X |x| < 2^X

1. X<0
2. X=-10

I solved it - using two options
-10 |10| < 1/2^10 AND -10 -|10| < 1/2^10. This method gives two solutions and therefore not sufficient. However my logic is wrong. Please explain why there are not two options. I have come across questions where one is required to use the two options. why not in this case? thanks
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Joined: 02 Sep 2009
Posts: 52344
Re: DS Questions - absolute integers - Help understand logic  [#permalink]

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13 Sep 2010, 15:56
cmugeria wrote:
If X is an integer is X |x| < 2^X

1. X<0
2. X=-10

I solved it - using two options
-10 |10| < 1/2^10 AND -10 -|10| < 1/2^10. This method gives two solutions and therefore not sufficient. However my logic is wrong. Please explain why there are not two options. I have come across questions where one is required to use the two options. why not in this case? thanks

If x is an integer, is x*|x|<2^x

This is YES/NO data sufficiency question: In a Yes/No Data Sufficiency question, each statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no".

Now, you should notice that the RHS (right hand side) of the expression is always positive ($$2^x>0$$), but the LHS is positive when $$x>0$$ ($$x>0$$ --> $$x*|x|=x^2$$), negative when $$x<0$$ ($$x<0$$ --> $$x*|x|=-x^2$$) and equals to zero when $$x={0}$$.

(1) x<0 --> according to the above $$x*|x|<0<2^x$$, so the answer to the question "is x*|x|<2^x" is YES. Sufficient.

(2) x=-10, the same thing here $$x*|x|=-100<0<\frac{1}{2^{10}}$$, so the answer to the question "is x*|x|<2^x" is YES. Sufficient.

cmugeria wrote:
-10 |10| < 1/2^10 AND -10 -|10| < 1/2^10.

When $$x=-10$$ then $$|x|=|-10|=10$$ and $$x*|x|=-10*10=-100$$.

Hope it's clear.
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Re: DS Questions - absolute integers - Help understand logic  [#permalink]

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13 Sep 2010, 17:20
To give you another way of looking at it...

In an inequality you can't move variables from side to side using multiplication or division unless you know their sign since the inequality expression will change direction if you multiply or divide both sides by a negative number.

Both statements tell you that X is negative, so you know that you can now divide both sides by X, giving you a definite equation that |x| > (2^x)/X
Intern
Joined: 22 Dec 2009
Posts: 20
Re: DS Questions - absolute integers - Help understand logic  [#permalink]

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13 Sep 2010, 18:08
Thank you for the explanation.

Please help me understand what the difference (in regards to having two solutions in terms of absolute and non absolute values) is between the two questions
is x*|x|<2^x and the question
|x+1|= x*|3x-2|what are the possible values for x from advanced equations of MGMAT Equations and inequalities - the answer is 1/4 and 3/2

Maybe i am overanalyzing the questions
Math Expert
Joined: 02 Sep 2009
Posts: 52344
Re: DS Questions - absolute integers - Help understand logic  [#permalink]

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13 Sep 2010, 18:45
cmugeria wrote:
Thank you for the explanation.

Please help me understand what the difference (in regards to having two solutions in terms of absolute and non absolute values) is between the two questions
is x*|x|<2^x and the question
|x+1|= x*|3x-2|what are the possible values for x from advanced equations of MGMAT Equations and inequalities - the answer is 1/4 and 3/2

Maybe i am overanalyzing the questions

I don't quite understand your question.

Original question asks whether $$x*|x|<2^x$$ is true, it's YES/NO DS question, it doesn't ask for specific value of $$x$$. AGAIN: In a Yes/No Data Sufficiency question, each statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no". As EACH statement ALONE gives the definite answer YES x*|x| is less than 2^x then EACH statement ALONE is sufficient to answer the question which means than answer is D .

Another one $$|x+1|=|3x-2|$$ (I believe it's $$|x+1|=|3x-2|$$ and not |x+1|= x*|3x-2| as you wrote, as solutions you provided 1/4 and 3/2 satisfy the first equation and not the second one), seems to be another type of DS question, the one which asks for a certain value of an unknown. For this type of questions statement is sufficient if it gives single numerical value of this unknown. So as $$|x+1|=|3x-2|$$ has two solutions $$x=\frac{1}{4}$$ and $$x=\frac{3}{2}$$ then this statement (if this is the only thing we know for certain statement) is not sufficient, as it does not give single numerical value of $$x$$.

For more absolute values and inequalities:

Check Walker's topic on ABSOLUTE VALUE: math-absolute-value-modulus-86462.html

For practice check collection of 13 tough inequalities and absolute values questions with detailed solutions at: inequality-and-absolute-value-questions-from-my-collection-86939.html

700+ PS and DS questions (also have some inequalities and absolute values questions with detailed solutions):
tough-problem-solving-questions-with-solutions-100858.html

700-gmat-data-sufficiency-questions-with-explanations-100617.html

Hope it helps.
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Re: If x is an integer, is x|x| < 2^x?  [#permalink]

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10 Jul 2017, 10:11
cucrose wrote:
If x is an integer, is x|x| < 2^x?

(1) x < 0
(2) x = -10

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

There is 1 variable. Thus D is the answer most likely.

Condition 1) $$x < 0$$
Since $$|x| = -x$$ if $$x < 0$$, the question $$x|x| < 2^x$$ is equivalent to $$-x^2 < 2^x$$.
We have the left hand side $$-x^2 < 0$$ and the right hand side $$2^x > 0$$ all times.
Thus $$-x^2 < 0 < 2^x$$.
This is sufficient.

Condition 2) $$x = -10$$
Since $$x = -10$$ is negative, by the same logic of the condition 1), this condition is also sufficient.

Therefore, the answer is D as expected.

-> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 13366 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If x is an integer, is x|x| < 2^x? [#permalink] ### Show Tags 26 Dec 2017, 09:43 Hi All, We're told that X is an INTEGER. We're asked if X|X| is less than 2^X. This is a YES/NO question. This DS question is built around a couple of Number Properties, so you can actually solve it without doing much math. 1) X < 0 This Fact tells us that X is NEGATIVE, which means.... X|X| = (negative)|negative| = (-)(+) = ALWAYS negative 2^X = 2^(negative) = 1/[2^(positive)] = ALWAYS positive Thus, the answer to the question is ALWAYS YES. Fact 1 is SUFFICIENT 2) X = -10 Since this tells us the only value of X, there will be only one answer to the given question (the answer happens to be YES). Fact 2 is SUFFICIENT Final Answer: GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: If x is an integer, is x|x| < 2^x? &nbs [#permalink] 26 Dec 2017, 09:43
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