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Senior Manager  Joined: 27 Jun 2012
Posts: 350
Concentration: Strategy, Finance
Schools: Haas EWMBA '17
Is x negative?  [#permalink]

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14 00:00

Difficulty:   85% (hard)

Question Stats: 52% (02:15) correct 48% (01:56) wrong based on 544 sessions

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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.

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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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8
6
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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SVP  Joined: 06 Sep 2013
Posts: 1573
Concentration: Finance
Re: Is x negative?  [#permalink]

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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

Cheers
J
Intern  Joined: 13 May 2013
Posts: 24
Is x negative?  [#permalink]

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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.
Math Expert V
Joined: 02 Sep 2009
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Re: Is x negative?  [#permalink]

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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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Re: Is x negative?  [#permalink]

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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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Intern  B
Joined: 17 Dec 2015
Posts: 41
Re: Is x negative?  [#permalink]

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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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Joined: 05 Mar 2015
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Is x negative?  [#permalink]

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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.

Originally posted by rohit8865 on 13 Mar 2016, 06:50.
Last edited by rohit8865 on 09 Jul 2016, 00:44, edited 2 times in total.
Intern  Joined: 22 May 2010
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Re: Is x negative?  [#permalink]

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1
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 6308 times ]

Intern  Joined: 15 Jun 2016
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Location: India
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GMAT 1: 730 Q50 V39 Is x negative?  [#permalink]

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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 5878 times ]

Current Student B
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GMAT 1: 760 Q49 V45 GPA: 3.5
Is x negative?  [#permalink]

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4
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.

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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Is x negative?  [#permalink]

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Hi All,

We're asked if X is NEGATIVE. This is a YES/NO question. We can answer it by TESTing VALUES or using Number Properties.

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

With this Fact, we know that X can either be....
-ANY negative value (since X^2 would be positive and X^3 would be negative) and the answer to the question would be YES.
-ANY positive fraction (re: 0 < X < 1) - since squaring/cubing a positive fraction makes it smaller - and the answer to the question would be NO.
Fact 1 is INSUFFICIENT

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

With this Fact, we know that X can either be....
-ANY negative value (since X^2 would be positive and X would be negative) and the answer to the question would be YES.
-ANY value greater than 1 (since squaring/cubing those values makes the result BIGGER than X) and the answer to the question would be NO.
Fact 2 is INSUFFICIENT

Combined, there's only one group of values that 'fit' both Facts: NEGATIVES. Thus, the answer to the question is ALWAYS YES.

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Re: Is x negative?  [#permalink]

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to satisfy the statement (1) , X has to be <1 , i.e. X<1
to satisfy the statement (2) , X has to either >1 or <0 i.e. X>1 or X<0
But X<0 only condition which satisfy both the statement hence X always be negative Re: Is x negative?   [#permalink] 27 Dec 2018, 05:33
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