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by rommysingh » Fri Aug 07, 2015 5:54 pm
On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line?

(1) xyz<0
(2) xy<0
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by Ian Stewart » Fri Aug 07, 2015 8:04 pm
rommysingh wrote:On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line?

(1) xyz<0
(2) xy<0
Neither statement is sufficient alone, as we'll see when we combine them. Using both, we know xy < 0, and (xy)(z) < 0, so z must be positive. We also know that x is closer to z than it is to y, and that x and y have opposite signs (because xy < 0). But that still leaves us with possible number lines like this one:

-----y-----0-------z--x-------

where the answer to the question is 'yes', and this one:

-----y------0--------x---z-----

where the answer to the question is 'no', so the two statements together are not sufficient and the answer is E.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by [email protected] » Sat Aug 08, 2015 11:01 am
Hi rommysingh,

This question can be solved in a number of different ways. You can use a combination of TESTing VALUES and Number Properties to get to the solution:

We're told that, on a number line, the distance between X and Y is GREATER than the distance between X and Z. We're asked if Z is BETWEEN X and Y on the number line. This is a YES/NO question.

Fact 1: (X)(Y)(Z) < 0

This tells us that the three variables are either all negative OR 1 negative and 2 positives.

IF...
X = 1
Y = -2
Z = 2
Then the answer to the question is NO.

IF...
X = 2
Y = -2
Z = 1
Then the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: (X)(Y) < 0

This tells us that one variable is positive and the other is negative. Unfortunately it tells us NOTHING about Z. As it stands though, the TESTs that I did in Fact 1 fit Fact 2 as well....

IF...
X = 1
Y = -2
Z = 2
Then the answer to the question is NO.

IF...
X = 2
Y = -2
Z = 1
Then the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, we already have two TESTs that fit BOTH Facts and produce different answers (a NO and a YES), so there's no more work needed.
Combined, INSUFFICIENT

Final Answer: E

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Rich
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