Is x negative?

Question

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.

IanStewart · Answer

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.

jlgdr · Answer

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

Saabs · Answer

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.

Bunuel · Answer

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).

vad3tha · Answer

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)

KbSharma · Answer

Can somebody please explain how to solve these inequalities in detail?
I am not getting the desired answer after solving the equations.

rohit8865 · Answer

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.

santy · Answer

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"

target760gmat · Answer

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

ANSWER: C

----
Kudos if you find the post helpful

GSBae · Answer

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.

Answer: C.

EMPOWERgmatRichC · Answer

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.

Final Answer:  C

GMAT assassins aren't born, they're made, 
Rich

praveenjain · Answer

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

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Is x negative?

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