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

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

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18 Dec 2012, 08:31
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If x is an integer, is x|x|<2^x ?

(1) x < 0
(2) x = -10
[Reveal] Spoiler: OA

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

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18 Dec 2012, 08:35
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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.

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30 Dec 2013, 12:02
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seabhi wrote:
If x is an integer, is x |x| < 2x ?

(1) x < 0
(2) x = –10

DS from OG.

OA is D for the following reasons:
When you first see a DS question, see if there is anyway to simplify the question stem
In this case, since |x| is positive, we can divide both sides by |x| giving us a new question stem --> is x < 2?

S1: x<0, therefore x must be <2 = sufficient
S2: x = -10 and -10 < 2 = sufficient

Let me know if this helps!

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30 Dec 2013, 21:36
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bparrish89 wrote:
seabhi wrote:
If x is an integer, is x |x| < 2x ?

(1) x < 0
(2) x = –10

DS from OG.

OA is D for the following reasons:
When you first see a DS question, see if there is anyway to simplify the question stem
In this case, since |x| is positive, we can divide both sides by |x| giving us a new question stem --> is x < 2?

S1: x<0, therefore x must be <2 = sufficient
S2: x = -10 and -10 < 2 = sufficient

Let me know if this helps!

The OA is wrong here because of the following reasons:

(1) if x=-10 then -100<-20, on the other hand if x= -1, x<0 then the inequality changes from < to >, namely, -1 > -2 ; This statement is absolutely insufficient!

(2) This statement is obviously sufficient!

So, the correct answer is notD, but B

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

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27 Jul 2015, 06:43
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reza52520 wrote:
If x is an integer, is x|x|<2^x ?

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

Question : Is x|x|<2^x ?

Statement 1: x < 0

For x to be Negative LHS i.e. x|x| will always be NEGATIVE
and 2^x will be positive for any value of x
i.e. x|x|<2^x will always be true
SUFFICIENT

Statement 1: x = -10
For x =-10 LHS i.e. x|x| will always be NEGATIVE (-100)
and 2^x will be positive for given x (1/2^10)
i.e. x|x|<2^x will always be true
SUFFICIENT

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

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27 Jul 2015, 06:44
reza52520 wrote:
If x is an integer, is x|x|<2^x ?

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

Hi,
we have an equation and the RHS 2^x will be positive irrespective of value of x and LHS xlxl will depend on the value of x..
1) x is -ive .. so LHS is -ive and RHS is +ive.. suff
2) x=-10... again LHS is -ive and RHS is +ive.. suff
ans D
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Re: If x is an integer, is x|x|<2^x ? [#permalink]

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13 Dec 2015, 11:47
For,
and 2^x will be positive for any value of x

2 power x, X can be negative no?
since its x, we dont know positive or negative..
What am I missing?

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

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13 Dec 2015, 12:04
paidlukkha wrote:
For,
and 2^x will be positive for any value of x

2 power x, X can be negative no?
since its x, we dont know positive or negative..
What am I missing?

You are missing a crucial thing here. Even if x is <0, $$2^x$$ with x<0 = $$1/2^x$$ , it is still >0...(1)

Thus with x<0, |x| = -x and hence x|x| = -$$x^2$$

As, x^2 is always $$\geq$$ 0 for all x, -$$x^2$$<0 ...(2)

Thus, from (1) and (2), you get a definite "yes" for the question "is $$x|x| < 2^x$$" for x<0.

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

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24 Apr 2016, 15:22
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This question can be solved as follows.

stmt1) it says that x /x/ < 2^x. and also we are told that x< 0. So if x is zero and the abs of x is always positive then we know that x/x/ will be negative. In addition to that, we know that 2^negative number will be positive because it will be in the form of 1/2^x, it will be less than 1 but it will be greater than a negative number. So stmt1 is SUFF.

stmt2) this is a repetition of stmt1 because the left side is negative and the right side is positive. SUFF.

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

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24 Nov 2016, 06:03
I would say statement 1 is insufficient... Am I doing something incorrect?

$$x |x| < 2^x$$

$$|x| < \frac{2^x}{x}$$

For x = -1

$$|-1| < \frac{1}{2}/-1$$
$$1 < - \frac{1}{2}$$

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

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24 Nov 2016, 07:17
Drblabla wrote:
I would say statement 1 is insufficient... Am I doing something incorrect?

$$x |x| < 2^x$$

$$|x| < \frac{2^x}{x}$$

For x = -1

$$|-1| < \frac{1}{2}/-1$$
$$1 < - \frac{1}{2}$$

When you divide by negative value you should flip the sign...
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Re: If x is an integer, is x|x|<2^x ? [#permalink]

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23 Jan 2017, 09:30
1, x < 0
x = -1
-1 < $$\frac{1}{2}$$
x = -2
-4 < $$\frac{1}{4}$$
2, X = -10
-100 < $$\frac{1}{2}$$^10
D
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Re: If x is an integer, is x*lxl<2^x [#permalink]

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28 Jan 2017, 07:37
praveen27sinha wrote:
1. If x is an integer, is x*lxl<2^x

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

|x| is positive so is 2^x
so if we can find whether x is positive or negative we can find our solution.

both of the option says x is negative.

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

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17 Apr 2017, 02:20
Bunuel 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.

just for the sake of time saving:

(2) x = -10 --> we could test it! done.... next question
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Re: If x is an integer, is x|x|<2^x ? [#permalink]

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23 Aug 2017, 13:52
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If x is an integer, is x|x|<2^x ?

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

Target question: Is x|x|< 2^x ?

Given: x is an integer

Statement 1: x < 0
In other words, x is NEGATIVE
So, x|x| = (NEGATIVE)(|NEGATIVE|) = (NEGATIVE)(POSITIVE) = NEGATIVE

IMPORTANT: 2^x will be POSITIVE for all values of x.

Since x|x| must be NEGATIVE, and since 2^x must be POSITIVE, we can be certain that x|x|< 2^x
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x = -10
So, x|x| = (-10)(|-10|) = (-10)(10) = -100 = a NEGATIVE
On the other hand, 2^x = 2^(-10) = 1/(2^10) = some POSITIVE number
Since x|x| is NEGATIVE, and since 2^x must be POSITIVE, we can be certain that x|x|< 2^x
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

[Reveal] Spoiler:
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Re: If x is an integer, is x|x|<2^x ? [#permalink]

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27 Aug 2017, 08:01
Bunuel mikemcgarry IanStewart shashankism Engr2012

I know this is an OG Q, but is it not strange that st 2 is an integral part of st 1.
During actual exam I do not think to need to evaluate S2 separately since x=-10
is always going to be x<0. Let me know if my understanding is correct
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Re: If x is an integer, is x|x|<2^x ? [#permalink]

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22 Oct 2017, 16:14
Hi,

For statement 1 I tested values i.e. x = -1 or x = -2.This is more so a question regarding reciprocals and inequalities. If x = -2, then -2 (|-2|) = 2^-2. Then, this is equal to -2 (2) = 1/2^2. In the second step where I converted 2^-2 to 1/2^2 -- would I have to also flip the other side to become 1 / -2 (2) or is that wrong?

Thanks,

infinitemac

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Re: If x is an integer, is x|x|<2^x ?   [#permalink] 22 Oct 2017, 16:14
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