Is Z > 0

(1) (Z+1)(Z)(Z-1) < 0
There are 3 transition points -1, 0 and 1. The expression will be negative if Z < -1 or 0 < Z < 1. If you are not sure how we got this, check: http://www.veritasprep.com/blog/2012/06 ... e-factors/

So Z may be positive or negative in this range. Not sufficient.

(2) |Z|< 1
-1 < Z < 1
Z may be positive or negative in this range. Not sufficient.

Which range of Z satisfies both inequalities? 0 < Z < 1
In this range, Z is always positive. Sufficient.

Answer (C)
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Could anyone share their workings for this one? I started with (2) because it'd easier to rule out. For (1) I started by expanding the formula out then picked smart numbers. The OA, however, did it by just picking smart numbers without any prework.

I understand how to pick the right smart numbers BUT what I can't work out is what should I look for to determine whether I need to expand or leave the equation as is before I plug in smart numbers?

Hi ColdSushi,

This DS question is loaded with Number Property rules; you'll actually see Number Properties on many DS questions, so it's a good area to spend some extra time on.

We're asked if X > 0. This is a YES/NO question.

Fact 1: (Z+1)(Z)(Z-1) < 0

This looks like a great opportunity to TEST VALUES. Before randomly TESTing a value though, I want to note the Number Property patterns in this question. Since the product is LESS than 0, we cannot have a product that equals 0.

Looking at the three parentheses, we can see that the product will equal 0 if Z is -1, 0 or 1. This means that we CANNOT TEST any of those values.

The first TEST should be something fairly obvious. We need a negative product, so let's choose a negative value.

IF...
Z = -2
(-1)(-2)(-3) = -6, which is < 0
The answer to the question is NO.

Notice how in the first Test we had the product of 3 negative numbers. We can also get a negative product with just 1 negative number, so we should look for that option next.

IF....
Z = 1/2
(3/2)(1/2)(-1/2) = -3/8, which is < 0
The answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: |Z| < 1

This means that -1 < Z < 1

IF...
Z = 1/2
The answer to the question is YES

IF...
Z = -1/2
The answer to the question is NO
Fact 2 is INSUFFICIENT

Combined, we know....
(Z+1)(Z)(Z-1) < 0
-1 < Z < 1

Since Z is greater than -1, there's no way for (Z+1) to be negative. By extension, we can't end up with the product of 3 negative numbers, so the only way to satisfy the first Fact is if we have JUST 1 negative number (which will be the (Z-1) piece of the product. Z itself must be positive....which will give us...

(Z+1)(Z)(Z-1)
(+)(+)(-)
The answer to the question is ALWAYS YES
Combined, SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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By the way, establishing a statement using "smart numbers" isn't the best idea.
It is easy to negate a statement using numbers but establishing isn't always advisable.

Say, I am given that |Z|< 1. Is Z always positive?
I can negate it by plugging in a single value Z = -1/2. Here Z is negative but 1/2 < 1 holds. So I know that Z is not always positive in this case. Great.

But if I want to establish using both statements that Z must always be positive, I may miss out on testing some important numbers since it must hold for all values of Z. So be careful when you use "smart numbers" in DS questions.
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Option A: (z + 1)(z)(z - 1) < 0
implies 0<Z<1 OR Z<-1 NOT SUFFICIENT.

Option B: |z| < 1 implies -1<Z<1 NOT SUFFICIENT.

Combining Together

0<Z<1 SUFFICIENT.

ANSWER C.

Thanks very much guys - really appreciate your explanations.

Is z > 0?

(1) (z + 1)(z)(z - 1) < 0
(2) |z| < 1

answer is C

1 not suff z=0.5 and z =-2
2 not suff -1<z<1

combining we get
sufficient in the range of 0<z<1

Statement 1:

-1 0 1
From 1 we can deduce
Z<-1
-1<z<0
0<z<1
z>1
Many cases So we cannot say whether Z>0.Not Sufficient
Statement 2:
From 2 we can get:
-1<z<1
positive or negative so not sufficient

1+2
here still 2 cases can be possible right?
Because we have

-1<z<0
0<z<1
in statement 1 .
These 2 cases are satisfied in both statements and two possible solutions.
Please help me where I am going wrong
Bunuel

Hi SonGoku,

You made a minor mistake when dealing with Fact 1. There are a couple of different 'ranges' of values that will fit...

(Z + 1)(Z)(Z - 1) < 0

Z < -1 and 0 < Z < 1 are both possibilities.

However, -1 < Z < 0 is NOT an option. With a bit more work, you'll find that the ONLY possibilities that will fit BOTH Facts are when Z is a positive fraction.

GMAT assassins aren't born, they're made,
Rich
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Thanks for the reply
But why can’t we deduce-1<z<0 from statement 1 ?

Hi SonGoku,

Consider what would happen to each of the 3 parentheses if -1 < Z < 0 (meaning that Z is a negative fraction)....

(Z + 1)(Z)(Z - 1) < 0

With any negative fraction in place for Z, we would end up with a product that is...

( + )( - )( - ) = POSITIVE

For example, if Z = -0.5.....
(-0.5 + 1)(-0.5)(-0.5 - 1)
(+0.5)(-0.5)(-1.5) = +.375

This does NOT fit what we were told; the product is supposed to be NEGATIVE. Thus, Z cannot be in the range -1 < Z < 0.

GMAT assassins aren't born, they're made,
Rich
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Question: Is z>0?
(1) (z+1)(z)(z-1)<0
(2) |z| <1

Both the statements are insufficient independently however they are sufficient together. (C)

But in explanation of trying to prove (2) insufficient to strike option B and D, they have tested three cases using -2, ½, -½.
On testing each of the three cases they have considered the following:
a) -2: 2<1 Not considered
b) ½: ½<1 so z>0 Yes
c) -½: -½<1 so z>0 No

But my doubt is that in the third case shouldn't the answer be yes as the modulus changes the sign of any value inside it to positive? Wont |-½| = ½?
But in the answer they have considered |-½|= -½

Please clarify. Thanks in advance.

For statement one

We have z (z-1) (z +1) < 0

3 different values multipled it means either all are negative or only one is negative. This way we cannot tell whether z is negative or z - 1 etc..

For statement two

z < 1
- z < 1 this then becomes z > -1

-1 < z < 1

Pick z = 1/2 or - 1/2

|1/2| = 1/2 < 1 (satisfies the statement) but is z > 0? Yes.

| -1/2 | = 1/2 < 1(satisfies the statement)

But is z > 0? No.

I think something went off on the explanation.

Now combined we know z is between 1 and -1

So for the first statement to be negative z has to be a positive number less than 1

The reason is

1/2 * (1/2 - 1) * (1/2 + 1)

1/2 * (- 1/2) * (3/2) < 0

If z was negative then it would be -1/2

This way it becomes -1/2 * (-3/2) * 1/2 which invalidates the statement.

Since z has to be a positive number less than 1.

We have an answer.

Hope this helps!

Posted from my mobile device

Thank you so much. Your explanation makes a lot of sense. There is a typo in the book. Thanks a lot!

target is z>0
#1
(z + 1)(z)(z - 1) < 0
possible when z=1/2 , -2,-3
insufficient
#2
|z| < 1
range of values of z
-1<z<1
insufficient clearly
from 1 &2
z = 1/2 only possible
hence sufficient option c

Hi VeritasKarishma
Thank you for the explanation.
I have a small doubt, though.
Taking two statement together, I chose option E. (Please don't mind the grammatical error in my sentence.)
I could not internalize this part: (Which range of Z satisfies both inequalities? 0 < Z < 1). Why did we not took other unsatisfied range into consideration and thus chose option E. Please shed some light. Thank you.

This has to do with understanding the way DS questions are formulated.
You are asked: Is Z > 0?

Hint 1 gives to you is: (Z+1)(Z)(Z-1) < 0
You can rewrite this as Z < -1 or 0 < Z < 1.
So you know now that either Z is less than -1 or between 0 and 1. Can you say whether Z is positive? No. It may or may not be.

Hint 2 given to you is: |Z|< 1
You can rewrite this as -1 < Z < 1.
So you know now that Z lies between -1 and 1. Can you say whether Z is positive? No. It may or may not be.

Using both the hints (statements), you know that "either Z is less than -1 or between 0 and 1" and also that "Z lies between -1 and 1". Both statements need to be satisfied.
"Z must be either less than -1 or between 0 and 1" AND "Z must lie between -1 and 1".
Both can hold only if Z lies between 0 and 1 (e.g. say if Z = 0.5, it works. If Z = -0.5 or -2 etc, it doesn't work. One or the other condition is not met.)

So Z must lie between 0 and 1 only.
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