Is x negative?

[rsarashi]

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.

OAC

[GMATGuruNY]

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.

In each of the inequalities below, test the following list of values:
-2, -1, -1/2, 0, 1/2, 1, 2.

Statement 1: x > x³ or x² > x³ (or both)
Testing the list of values above, we get:
x > x³ is satisfied by -2 and 1/2.
x² > x³ is satisfied by -2, -1, -1/2, and 1/2.
Since can be negative or positive, INSUFFICIENT.

Statement 2: x² > x or x³ > x (or both)
Testing the list of values above, we get:
x² > x is satisfied by -2, -1, -1/2, and 2.
x³ > x is satisfied by -1/2 and 2.
Since x can be negative or positive, INSUFFICIENT.

Statements combined:
The values in red satisfy both statements.
All of the values in red are negative.
Implication:
To satisfy both statements, x must be negative.
SUFFICIENT.

The correct answer is C.

[Mo2men]

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.

Statement 1: x > x³ or x² > x³ (or both)
Case 1: x > x³
Here, it's possible that x=1/2 (in which case the answer to the question stem is NO) or that x=-2 (in which case the answer to the question stem is YES).
INSUFFICIENT.

Statement 2: x² > x or x³ > x (or both)
Case 2: x³ > x
Here, it's possible that x=2 (in which case the answer to the question stem is NO) or that x=-1/2 (in which case the answer to the question stem is YES).
INSUFFICIENT.

Statement combined:
The two inequalities in red contradict each other and thus cannot both be true.
Implication:
To satisfy both statements, the two inequalities in blue must both be true:
x > x³.
x² > x.
Only values of x such that x<-1 satisfy both inequalities.
Thus, the answer to the question stem is YES.
SUFFICIENT.

The correct answer is C.

Dear Mitch,
The fully contradiction is between :
x > x³
x³ > x

However, the both inequalities that you highlighted in red do not fully contradict each other as both inequalities in range -1 < x < 0 hold true, which proves that x is negative also because AT LEAST ONE condition in statement 1 intersects with AL LEAST ONE condition in statement 2 .
Am I right in above conclusion??

[GMATGuruNY]

x is negative because AT LEAST ONE condition in statement 1 intersects with AL LEAST ONE condition in statement 2 .
Am I right in above conclusion??

Please see my revised post above.
It illustrates what you seem to be stating:
To satisfy at least one of the two conditions in each statement, x must be negative.

[Jay@ManhattanReview]

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.

OAC

Hi rsarashi,

We have to determine whether x negative.

Statement 1: At least one of x and x^2 is greater than x^3.

Case a: If x > x^3 but x^2 < x^3, Statement is valid;
Case b: If x^2 > x^3 but x < x^3, Statement is valid;
Case c: If x < x^3 and x^2 < x^3, Statement is NOT valid

So, we must not choose an example such that x < x^3 and x^2 < x^3.

The following ranges are important to study.

1. x is negative: Say x = -2, then x^3 = (-2)^3 = -8, thus x > x^3, the answer is Yes, x is negative.
2. x is a positive integer: Say x = 2, then x^3 = (2)^3 = 8, thus x < x^3; again, x^2 = (2)^2 = 4, thus x^2 < x^3; this calls for Case c. It is not a valid example.
3. x is a positive fraction: Say x = 1/2, then x^3 = (1/2)^3 = 1/8, thus x > x^3, the answer is No, x is not necessarily negative.

Insufficient!

Statement 2: At least one of x^2 and x^3 is greater than x.

Case d: If x^2 > x but x^3 < x, Statement is valid;
Case e: If x^3 > x but x^2 < x, Statement is valid;
Case f: If x^3 < x and x^2 < x, Statement is NOT valid

So, we must not choose an example such that x^3 < x and x^2 < x.

Taking the same ranges to study as discussed in Statement 1,

4. x is negative: Say x = -2, then x^2 = (-2)^2 = 4, thus x^2 > x, the answer is Yes, x is negative.
5. x is a positive integer: Say x = 2, then x^3 = (2)^3 = 8, thus x < x^3, the answer is No, x is not necessarily negative.
6. x is a positive fraction: Say x = 1/2, then x^3 = (1/2)^3 = 1/8, thus x > x^3; again x^2 = (1/2)^2 = 1/4, thus x > x^2; this calls for Case f. It is not a valid example.

Insufficient!

Statement 1 & 2:

(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.

Keeping both the statements together only examples (1) and (4) satisfy, thus x is negative, the answer is Yes. Sufficient.

The correct answer: C

Hope this helps!

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Hi rsarashi,

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

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