Is x negative?

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

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).

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)

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.

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"

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

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Kudos if you find the post helpful

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:

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

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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