inequalities

[beater]

Is x negative?

1) (x^3) (1-(x^2)) < 0
2) (x^2) -1 < 0

[Gmatss]

Is x negative?

1) (x^3) (1-(x^2)) < 0
2) (x^2) -1 < 0

1.) To satisfy this condition,

either x^3 is neg or (1-(x^2) is negative.
so, -1<x<0 or x>1 it can be positive or negative. so hence insuff

2.) To satisfy this condition

-1<x<1 so can be either positive or negative, insuff

Taking 1 and 2, we know x has to be less than 1 and only negative number satisfy both conditions. therefore C is the answer.

[lunarpower]

worth commenting a little more on this one:

1.) To satisfy this condition,

either x^3 is neg or (1-(x^2) is negative.
so, -1<x<0 or x>1 it can be positive or negative. so hence insuff

this is true inasmuch as "or" is used here in the sense of exclusive "or". in other words, exactly one of those two things must be true to satisfy the statement, but not both.

a clearer way to express the result is: x^3 and (1 - x^2) have opposite signs.
once you have the fact that x^3 is negative for x < 0 and positive for x > 0, as well as the fact that (1 - x^2) is negative for x < -1, x > 1 and positive for -1 < x < 1, the result follows.

[Stockmoose16]

worth commenting a little more on this one:

1.) To satisfy this condition,

either x^3 is neg or (1-(x^2) is negative.
so, -1<x<0 or x>1 it can be positive or negative. so hence insuff

this is true inasmuch as "or" is used here in the sense of exclusive "or". in other words, exactly one of those two things must be true to satisfy the statement, but not both.

a clearer way to express the result is: x^3 and (1 - x^2) have opposite signs.
once you have the fact that x^3 is negative for x < 0 and positive for x > 0, as well as the fact that (1 - x^2) is negative for x < -1, x > 1 and positive for -1 < x < 1, the result follows.

Ron,

Can you please explain why you're separating out the terms in the first statement to determine sufficiency? I didn't think that was allowed.

Consider this example:

Is X positive?

(1) X^3*X^4>0

If you separate out the statements, you'd get:

X^3>0 (which means x is definitely +)
X^4>0 (which means x is - or +)

... this statement would be insufficient, since the combined statements would show X could be either positive or negative.

But here's the problem, X is CLEARLY positive. If you combine the original statement, you get X^7>0. Therefore, X can only be positive to satisfy the statement.

Can you please explain why you were allowed to separate the terms in your original example, but you wouldn't be able to do so in my example?

[logitech]

worth commenting a little more on this one:

1.) To satisfy this condition,

either x^3 is neg or (1-(x^2) is negative.
so, -1<x<0 or x>1 it can be positive or negative. so hence insuff

this is true inasmuch as "or" is used here in the sense of exclusive "or". in other words, exactly one of those two things must be true to satisfy the statement, but not both.

a clearer way to express the result is: x^3 and (1 - x^2) have opposite signs.
once you have the fact that x^3 is negative for x < 0 and positive for x > 0, as well as the fact that (1 - x^2) is negative for x < -1, x > 1 and positive for -1 < x < 1, the result follows.

Ron,

Can you please explain why you're separating out the terms in the first statement to determine sufficiency? I didn't think that was allowed.

Consider this example:

Is X positive?

(1) X^3*X^4>0

If you separate out the statements, you'd get:

X^3>0 (which means x is definitely +)
X^4>0 (which means x is - or +)

... this statement would be insufficient, since the combined statements would show X could be either positive or negative.

But here's the problem, X is CLEARLY positive. If you combine the original statement, you get X^7>0. Therefore, X can only be positive to satisfy the statement.

Can you please explain why you were allowed to separate the terms in your original example, but you wouldn't be able to do so in my example?

in your analysis, there is something you are missing.

X^3>0 (which means x is definitely +) This is correct but since you have to use the SAME + x for the X^4 ..it can only be positive too.

Please look at the attached diagram