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

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Re: If x is an integer, is x|x|<2^x ? [#permalink] New post   13 Dec 2017, 21:48
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Hi All,

We're told that X is an integer. We're asked if X|X| < 2^X. This is a YES/NO question. We can answer it with a bit of Number Property knowledge.

1) X < 0

With Fact 1, we know that X is NEGATIVE. By definition, that means...
X|X| = (Neg)|Neg| = Negative
2^(Negative) = Positive
Thus, X|X| will ALWAYS be less than 2^X and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

2) X = -10

With the value of X, we can absolutely answer the question (we would just need to plug in that value:
Is (-10)|-10| < 2^(-10)?
The answer to the question IS yes, but we don't have to actually do that work. There would be just one answer to the question, so it doesn't really matter what that one answer is.
Fact 2 is SUFFICIENT

Final Answer:
Show Spoiler
D


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Re: If x is an integer, is x|x| < 2^x? [#permalink] New post   26 Dec 2017, 10:43
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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:
Show Spoiler
D


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Re: If x is an integer, is x|x|<2^x ? [#permalink] New post   22 Feb 2020, 04:30
Bunuel wrote:
Walkabout wrote:
If x is an integer, is x|x|<2^x ?

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


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

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 \(x*|x|<0<2^x\). Sufficient.

(2) x = -10. The same here \(x*|x|=-100<0<\frac{1}{2^{10}}\). Sufficient.

Answer: D.


Hi Bunuel,

Please correct me if I am wrong

1) x|x|<2^x
x=-1
so,

-1|1|<2^-1
-1<1/2

which will always be true for any negative value of x and also with positive values it will remain true and sufficient

2) condition 2 is already giving fix value so it will be sufficient
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Re: If x is an integer, is x|x|<2^x ? [#permalink] New post   21 May 2023, 12:46
Bunuel wrote:
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.

Answer: D.


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.




Bunuel,
Could you pls explain, -- why don't you change a sign here "x*|x|<0<2^x"

I mean, we, generally considering 2 options:
if X>0 -- we have x*x<2^x
and if X<0 -- we have -- x*x>2^x (changing a sign to an opposite)
Am I missing smth?

Thanks for help!
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Re: If x is an integer, is x|x|<2^x ? [#permalink] New post   21 May 2023, 13:26
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stepanyan wrote:
Bunuel wrote:
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.

Answer: D.


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.




Bunuel,
Could you pls explain, -- why don't you change a sign here "x*|x|<0<2^x"

I mean, we, generally considering 2 options:
if X>0 -- we have x*x<2^x
and if X<0 -- we have -- x*x>2^x (changing a sign to an opposite)
Am I missing smth?

Thanks for help!


We reverse the inequality sign when multiplying the inequality by a negative number. However, in this problem, we're merely evaluating each side of x*|x| < 2^x when x is negative. When x < 0, x*|x| = negative*positive = negative, and 2^x = 2^(negative) = positive. Therefore, we end up with the relationship x*|x| < 0 < 2^x.
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Re: If x is an integer, is x|x|<2^x ? [#permalink]
21 May 2023, 13:26
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