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Sub 505 Level|   Absolute Values|   Algebra|   Inequalities|                        
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If x is an integer, is x|x|<2^x ? 13 Sep 2019, 23:14
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If x is an integer, is x|x| < \(2^x\) ?

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

Edit: Formating
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Given: x is an integer

Asked: Is x|x| < \(2^x\) ?

(1) x < 0
|x| = -x
-x^2 < 2^x
-x^2 < 0
2^x >0
-x^2 < 2^x
\(x|x| < 2^x\)
SUFFICIENT

(2) x = -10
|x| = 10
x|x| = -10 * 10 = -100
\(2^x = 2^{-10} = \frac{1}{1024}\)
\(-100 < \frac{1}{1024}\)
\(x|x| < 2^x\)
SUFFICIENT

IMO D
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If x is an integer, is x|x|<2^x ? 22 Feb 2020, 04:30
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Bunuel
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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.
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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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If x is an integer, is x|x|<2^x ? 22 Feb 2020, 14:25
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Farina
Bunuel
Walkabout
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
Show more

Hi Farina,

Most of what you wrote is correct. This DS question is based on a couple of Number Properties:

when X is NEGATIVE....
X|X| will ALWAYS be NEGATIVE
2^(X) will ALWAYS be POSTIIVE

This means that the answer to the question is ALWAYS YES and Fact 1 is SUFFICIENT.

However, what you noted about POSITIVE values is not true. For example....

when X = 1...
1|1| = 1
2^1 = 2
so the answer to the question "is X|X| < 2^X?".... is YES.

when X = 2...
2|2| = 4
2^2 = 4
so the answer to the question "is X|X| < 2^X?".... is NO.

This means that when X is POSITIVE, the answer to the question changes and this situation would be INSUFFICIENT.

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If x is an integer, is x|x|<2^x ? 21 May 2023, 12:46
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Bunuel
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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
-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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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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If x is an integer, is x|x|<2^x ? 21 May 2023, 13:26
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Bunuel
cmugeria
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
-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!
Show more

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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If x is an integer, is x|x|<2^x ? 23 Jul 2024, 16:59
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