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# Is x negative?

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17 Feb 2013, 12:16
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Is x negative?

(1) At least one of x and x^2 is greater than x^3.
(2) At least one of x^2 and x^3 is greater than x.
[Reveal] Spoiler: OA

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Re: Is x negative? At least one of x and x^2 is greater x^3 [#permalink]

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17 Feb 2013, 14:50
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If x > 1, then it is always true that x < x^2 < x^3.

If 0 < x < 1, then it is always true that x^3 < x^2 < x.

From the above, you can see that neither statement is sufficient alone, since in each case, x can be positive. Notice from the above that if x is positive, x^2 is never the largest of the three expressions x, x^2 and x^3. Since Statement 1 guarantees that x^3 is not the largest of the three expressions, and Statement 2 guarantees that x is not the largest of the three expressions, then using both statements, the only possibility is that x^2 is the largest of the three expressions. Since that can't happen when x is positive, x must be negative, and the answer is C.
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04 Feb 2014, 07:28
PraPon wrote:
Is x negative?

(1) At least one of x and x^2 is greater than x^3.
(2) At least one of x^2 and x^3 is greater than x.

I second the approach by IanStewart although I followed a slightly different approach

Statement 1

X could be either a fraction or a negative number

Statement 2

X could be either a positive or a negative number

Statement 1 and 2 together

X has to be a negative number

C

Just my 2c

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J
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13 Jul 2014, 02:30
Can someone please explain me what "at least one of" means in the statements? I really have no idea how to convert this sentence into an equation.
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13 Jul 2014, 05:32
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Saabs wrote:
Can someone please explain me what "at least one of" means in the statements? I really have no idea how to convert this sentence into an equation.

At least one of x and x^2 is greater than x^3 means that x>x^3 OR x^2>x^3 OR x>x^3 and x^2>x^3 (so, x is greater than x^3 or x^2 is greater than x^3 or both are greater than x^3).
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05 Aug 2014, 23:43
PrashantPonde wrote:
Is x negative?

(1) At least one of x and x^2 is greater than x^3.
(2) At least one of x^2 and x^3 is greater than x.

Thanks for the question. C is the answer.

(1) x <1 for (1) to happen
(2) x<0 or x>1 for (2) to happen

Combine (1) and (2), x<0 , so C is the answer.

(You can solve it by doing a little bit algebra x^2> x^3 -> x^2 - x^3>0 -> x^2 (1-x)> 0, so x<1)
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09 Feb 2016, 12:27
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09 Feb 2016, 16:53
Can somebody please explain how to solve these inequalities in detail?
I am not getting the desired answer after solving the equations.
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13 Mar 2016, 06:50
KbSharma wrote:
Can somebody please explain how to solve these inequalities in detail?
I am not getting the desired answer after solving the equations.

for statement(1) consider 2 cases
x > x^2 > x^3
1/2 > 1/4 > 1/8 &
-2 > -4 > -8.

for statement(2) consider 2 cases
x < x^2 < x^3
-1/2 < -1/4 < -1/8 &
2 < 4 < 8.

combining both statements we see -ve values from all 4 cases satisfying that x is -ve.

Last edited by rohit8865 on 09 Jul 2016, 00:44, edited 2 times in total.
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07 May 2016, 13:36
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I documented the behavior of a variable in different regions. Uploading its image as I think it would be helpful.

Although, I believe, memorizing how every power of 'x' behaves in each region would be pointless, noticing the patterns such as the ones mentioned below would be useful.
-- The behavior of odd powers of 'x' in the region " -1 < x < 0" is exactly same as that of even powers in the region "x < -1"
-- The behavior of even powers of 'x' in the region " -1 < x < 0" is exactly same as that of odd powers in the region "x < -1"
Attachments

Behavior of 'X' in different regions.png [ 31.35 KiB | Viewed 740 times ]

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09 Jul 2016, 00:23
You can see how x, x^2 and x^3 behaves from the graph attached.. We can then answer the qn accordingly

Is x negative?

(1) At least one of x and x^2 is greater than x^3.
(2) At least one of x^2 and x^3 is greater than x.

Lets define the regions : A <-1 , -1< B <0 , 0<C<1 , 1<D;

Blue line - x ,Red line - x^2 , Green line - x^3

(1) So the region can be either of A , B or C.. It can be either positive or negative
(2) So the region can be either of A , B or D .. It can be either positive or negative

Each is insufficient. Now combine both (1) and (2)...

We get the regions A and B ... which are negative

----
Kudos if you find the post helpful
Attachments

plots.jpg [ 70.3 KiB | Viewed 318 times ]

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12 Jul 2016, 08:25
Just draw out the number line and write out the order of the three functions in each region:

$$x^3<x<x^2$$     $$x<x^3<x^2$$    $$x^3<x^2<x$$    $$x<x^2<x^3$$
<------------------|------------------|------------------|------------------>
-1                       0                      1

(1) At least one of x and x^2 is greater than x^3.

This can be true in regions 1,2, and 3, x can be positive or negative. INSUFFICIENT

(2) At least one of x^2 and x^3 is greater than x.

This can be true in regions 1,2, and 4, x can be positive or negative. INSUFFICIENT

Taking the two together, this is true in regions 1 and 2, which means x is negative. SUFFICIENT.

Is x negative?   [#permalink] 12 Jul 2016, 08:25
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