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xyz < 0. All this tells us that either one or three of the numbers is negative and none of them are zero.

So, you can have x = 1, y = 8, z = -3, where the distance between XY is greater than the distance between XZ. Here, z does not lie between x and y.

But you can also have x = -1, y = -8, z = -3, where the distance between XY is greater than the distance between XZ, but where z lies between the two on the number line. Insufficient.

Statement 2:

xy < 0

All this tells us is that either x or y is negative and neither is zero. Taking x = -1, y = 8, z = -3, where the distance between XY is greater than the distance between XZ. Here, z does not lie between them on the number line.

But, taking x = 1, y = -8, z = -3, you fulfill the distance requirement and z falls between x and y on the number line. Insufficient.

Both Statements:

Taking both statements together, we learn that either x or y is negative and everything else is positive. Taking x = -1, y = 8, z = 2, we find that z lies between the points on the number line and fulfills the distance requirement. However, taking x = 8, y = -1, z = 10, z no longer lies between the two points but XY is still greater than XZ. Still insufficient.

xyz < 0. All this tells us that either one or three of the numbers is negative and none of them are zero.

So, you can have x = 1, y = 8, z = -3, where the distance between XY is greater than the distance between XZ. Here, z does not lie between x and y.

But you can also have x = -1, y = -8, z = -3, where the distance between XY is greater than the distance between XZ, but where z lies between the two on the number line. Insufficient.

Statement 2:

xy < 0

All this tells us is that either x or y is negative and neither is zero. Taking x = -1, y = 8, z = -3, where the distance between XY is greater than the distance between XZ. Here, z does not lie between them on the number line.

But, taking x = 1, y = -8, z = -3, you fulfill the distance requirement and z falls between x and y on the number line. Insufficient.

Both Statements:

Taking both statements together, we learn that either x or y is negative and everything else is positive. Taking x = -1, y = 8, z = 2, we find that z lies between the points on the number line and fulfills the distance requirement. However, taking x = 8, y = -1, z = 10, z no longer lies between the two points but XY is still greater than XZ. Still insufficient.

So, answer E.

Thanks for your detailed explanations, very helpful and yes the OE is E

Bunuel, the expert, is there a better way to solve this problem. I just took the Prep test and took me a long time to test each number. Is theer a quicker way to do this?

Bunuel, the expert, is there a better way to solve this problem. I just took the Prep test and took me a long time to test each number. Is theer a quicker way to do this?

This is a hard problem. Below is another way of solving it:

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?

The distance between x and y is greater than the distance between x and z, means that we can have one of the following four scenarios: A. y--------z--x (YES case) B. x--z--------y (YES case) C. y--------x--z (NO case) D. z--x--------y (NO case)

The question asks whether we have scenarios A or B (z lie between x and y ).

(1) xyz <0 --> either all three are negative or any two are positive and the third one is negative. We can place zero between y and z in case A (making y negative and x, z positive), then the answer would be YES or we can place zero between y and x in case C, then the answer would be NO. Not sufficient.

(2) xy<0 --> x and y have opposite signs. The same here: We can place zero between y and x in case A, then the answer would be YES or we can place zero between y and x in case C, then the answer would be NO. Not sufficient.

(1)+(2) Cases A (answer YES) and case C (answer NO) both work even if we take both statement together, so insufficient.

A. y----0----z--x (YES case) --> xyz<0 and xy<0; C. y----0----x--z (NO case) --> xyz<0 and xy<0

Re: On the number line, the distance between x and y is greater [#permalink]

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Best technique is to draw and visualize their positions in the numberline...

1. xyz < 0 Try x and y and z as all negative... <---(z)---x--(z)---y-----0------------>

Just this scenario is already giving us two possibilities... INSUFFICIENT

2. xy < 0 This means x and y has opposite signs...

<----(z)---x--(z)--0-------y------>

Just this scenario is already giving us two possibilities... INSUFFICIENT

Now, let us combine... If x and y has opposite signs then for xyz to be negative z must be positive...

scenario 1: <---------x-----0-(z)-------y--------------> YES z is in between scenario 2: <---------y-----0-----(z)-------x------(z)-------> NO z is not in between

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

We are given that on the number line, the distance between x and y is greater than the distance between x and z. We need to determine whether z lies between x and y on the number line.

Statement One Alone:

xyz < 0

Using the information in statement one, we have two possible cases:

Case 1: Exactly one variable (either x, y, or z) is negative

Case 2: All three variables are negative.

Even with this information, we cannot determine whether z lies between x and y.

For example, for Case 1, if x = -1, y = 2, and z = 1, then z falls between x and y. However, for Case 2, if x = 1, y = 4, and z = -1, then z does not fall between x and y. Statement one alone is not sufficient. We can eliminate answer choices A and D.

Statement Two Alone:

xy < 0

Using the information from statement two, we know that exactly one of the values x or y is negative and the other is positive. However, without knowing anything about z, we cannot determine whether z falls between x and y. Statement two alone is not sufficient. We can eliminate answer choice B.

Statements One and Two Together:

From the information in statements one and two, we know that z must be positive and exactly one of the values x or y is negative. However, we still we cannot determine whether z falls between x and y or outside x and y.

For example, if x = -1, y = 2, and z = 1, then z falls between x and y. However, if x = 1, y = -2, and z = 2, then z does not fall between x and y. The two statements together are still not sufficient.

Answer: E
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