Data Sufficiency

simba12123 wrote:Is X negative?


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

2. (x^2)-1<0



i simplified statement 1: to 1/(x^3) <0

I simplified statement 2: x^2 <1

I am running into trouble with statements 1 and 2. I believe that I oversimplifed the issue at hand. COnsequently, I am fumbling over picked numbers to prove sufficiency. PLease help!


qa is c

simba12123 wrote:Is X negative?


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

2. (x^2)-1<0

from 1

x^3-x^5<0

x^3<x^5.......x has the same sign -ve and x is a fraction. OR X IS +VE INTIGER..insuff

from 2
x^2<1

x can be anything +ve or -ve fraction..insuff

both of x is a fraction and x^3<x^5 thus x is -ve ........suff

C

schumi_gmat wrote:From 1,

either x<0 or x^2 >1

***Schumi, why did you separate the equation in statement #1 into two statements. I don't think you can do that. Statement #1 should give you the following information:

-1<X<0
X>1

If you put the statements you listed above, it says, X=-2 would be sufficient, since it complies with both your statements. Thus, by your breakdown, #1 would actually be sufficient, rather than insufficient.

Consider this example: x^3*X^4>0

You can't just separate these two out, or you'd get

X^3>0 (definitely positive)
X^4>0 (could be + or -)

combine these statements, and it says X is either positive or negative... insufficient. In reality, X is definitely positive, because if you combine the original statements, you get: X^7>0. The only thing that will satisfy the equation is a positive number.

rohangupta83 wrote:imo C

the way I would have done

statement 1:

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

So,

Either x^3<0 or 1-x<0 or 1+x<0

or

x<0 or 1<x or x<-1

We have 3 different possible values of x so statement 1 is insufficient


*** Why are you separating the two terms in statement #1. I don't think you can do that. Consider this example:

X^3*X^4>0

You can't just separate these two out, or you'd get

X^3>0 (X is definitely positive)
X^4>0 (X could be + or -)

Using your method, you'd say these statements are INSUFFICIENT since X could be + or -. In reality, X is definitely positive, because if you combine the original statements, you get: X^7>0. The only thing that will satisfy the equation is a positive number.

Stacey Koprince wrote:I received a PM asking me to comment.

Original problem:

Is X negative?

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

x is either negative or positive:
IF x is positive, then x^3 is positive, so the other term (1-x^2) would have to be negative. If this is the case, then:
1-x^2 < 0
1 < x^2
1 < x (or x > 1)

Stacey Koprince wrote:I received a PM asking me to comment.

Original problem:

Is X negative?

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

People have tried a couple of different things above. The particular confusion I was asked about was coming up with multiple inequalities or statements from just one starting inequality. What's going on here: people are trying to understand what properties they can deduce from the starting info - so they're actually using number property theory here.

The first thing I notice is that they ask me whether x is negative. So this seems like a pos/neg question. Then statement one tells me that some stuff multiplied together is negative - yep, this is definitely a pos/neg question.

So, let's see. (Note: I'm going to go through this first as though someone doesn't notice that (1-x^2) can be further factored - but I do recommend factoring this; see below for the explanation that includes factoring this term.)

(x^3) * (1-x^2) < 0
Two things multiplied together are negative. So one must be positive and one must be negative, because neg * pos = neg. Also, x^3 does not equal zero and (1-x^2) does not equal zero (because zero times anything is zero, not negative).

x is either negative or positive:
IF x is positive, then x^3 is positive, so the other term (1-x^2) would have to be negative. If this is the case, then:
1-x^2 < 0
1 < x^2
1 < x (or x > 1)
IF x is negative, on the other hand, then x^3 is negative, so the other term (1-x^2) would have to be positive.
If this is the case, then:
(1-x^2) > 0
1 > x^2
1 > x (or x < 1)

This doesn't tell me enough. Statement 1 is not sufficient.

You can also split the (1-x^2) term as a previous poster did (as I said above, I recommend doing this whenever you see a setup that reflects one of the three common quadratics): (1-x^2) = (1-x)(1+x)

So now you'd have (x^3) * (1-x) * (1+x) < 0
Now we've got three things multiplied together = negative. So either one is negative and the other two are positive, or all three are negative.

if x is positive, then x^3 is positive. 1+x must also be positive. By default, then, 1-x must be the negative term:
1-x < 0
1 < x (or x > 1)

If x is negative, then x^3 is negative. The other two terms could then both be positive, so 1-x > 0 and 1+x > 0. (They could also both be negative - so you could test it that way, if you prefer.)
1-x > 0 --> 1 > x (or x < 1)
1+x > 0 --> x > -1
this allows both neg and pos possibilities, so insufficient.
If you test the "both negative" possibility:
1-x < 0 --> 1 < x (or x > 1)
1+x < 0 --> x < -1
again, this allows both pos and neg possibilities, so insufficient.

Stacey Koprince wrote:I received a PM asking me to comment.

Original problem:

Is X negative?

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

People have tried a couple of different things above. The particular confusion I was asked about was coming up with multiple inequalities or statements from just one starting inequality. What's going on here: people are trying to understand what properties they can deduce from the starting info - so they're actually using number property theory here.

The first thing I notice is that they ask me whether x is negative. So this seems like a pos/neg question. Then statement one tells me that some stuff multiplied together is negative - yep, this is definitely a pos/neg question.

So, let's see. (Note: I'm going to go through this first as though someone doesn't notice that (1-x^2) can be further factored - but I do recommend factoring this; see below for the explanation that includes factoring this term.)

(x^3) * (1-x^2) < 0
Two things multiplied together are negative. So one must be positive and one must be negative, because neg * pos = neg. Also, x^3 does not equal zero and (1-x^2) does not equal zero (because zero times anything is zero, not negative).

x is either negative or positive:
IF x is positive, then x^3 is positive, so the other term (1-x^2) would have to be negative. If this is the case, then:
1-x^2 < 0
1 < x^2
1 < x (or x > 1)
IF x is negative, on the other hand, then x^3 is negative, so the other term (1-x^2) would have to be positive.
If this is the case, then:
(1-x^2) > 0
1 > x^2
1 > x (or x < 1)

This doesn't tell me enough. Statement 1 is not sufficient.

You can also split the (1-x^2) term as a previous poster did (as I said above, I recommend doing this whenever you see a setup that reflects one of the three common quadratics): (1-x^2) = (1-x)(1+x)

So now you'd have (x^3) * (1-x) * (1+x) < 0
Now we've got three things multiplied together = negative. So either one is negative and the other two are positive, or all three are negative.

if x is positive, then x^3 is positive. 1+x must also be positive. By default, then, 1-x must be the negative term:
1-x < 0
1 < x (or x > 1)

If x is negative, then x^3 is negative. The other two terms could then both be positive, so 1-x > 0 and 1+x > 0. (They could also both be negative - so you could test it that way, if you prefer.)
1-x > 0 --> 1 > x (or x < 1)
1+x > 0 --> x > -1
this allows both neg and pos possibilities, so insufficient.
If you test the "both negative" possibility:
1-x < 0 --> 1 < x (or x > 1)
1+x < 0 --> x < -1
again, this allows both pos and neg possibilities, so insufficient.

Stacey Koprince wrote:I received a PM asking me to comment.

Original problem:

Is X negative?

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

People have tried a couple of different things above. The particular confusion I was asked about was coming up with multiple inequalities or statements from just one starting inequality. What's going on here: people are trying to understand what properties they can deduce from the starting info - so they're actually using number property theory here.

The first thing I notice is that they ask me whether x is negative. So this seems like a pos/neg question. Then statement one tells me that some stuff multiplied together is negative - yep, this is definitely a pos/neg question.

So, let's see. (Note: I'm going to go through this first as though someone doesn't notice that (1-x^2) can be further factored - but I do recommend factoring this; see below for the explanation that includes factoring this term.)

(x^3) * (1-x^2) < 0
Two things multiplied together are negative. So one must be positive and one must be negative, because neg * pos = neg. Also, x^3 does not equal zero and (1-x^2) does not equal zero (because zero times anything is zero, not negative).

x is either negative or positive:
IF x is positive, then x^3 is positive, so the other term (1-x^2) would have to be negative. If this is the case, then:
1-x^2 < 0
1 < x^2
1 < x (or x > 1)
IF x is negative, on the other hand, then x^3 is negative, so the other term (1-x^2) would have to be positive.
If this is the case, then:
(1-x^2) > 0
1 > x^2
1 > x (or x < 1)

This doesn't tell me enough. Statement 1 is not sufficient.