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On the number line, the distance between x and y is greater

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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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Re: Does z lie between x and y on a number line? [#permalink]

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jj97cornell wrote:
On a 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 a number line?

(1) xyz < 0
(2) xy < 0


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 (\(z\) lies between \(x\) and \(y\)).

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

(1)+(2) Both case A (answer YES) and case C (answer NO) satisfy the statements. Not sufficient.

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

Answer: E.

Hope it's clear.
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Re: On the number line, the distance between x and y is greater [#permalink]

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1. Statement 1 tells us that \(xyz<0\)

Which basically means that either all 3 are negative or one of them is negative. We can draw two scenarios of a number line where z is between \(x\) and \(y\) and where \(z\) is on the right of \(x\) and \(y\) on the left of \(x\) and maintain the condition that all three are negative. Hence Insufficient.

2. Statement 2 tells us that \(xy<0\)

Which basically means that either \(x\) is negative or \(y\) is negative. No statement about \(z\) so obviously insufficient.

Now together we know that if either \(x\) or \(y\) are negative, and one of them is positive, \(z\) has to be positive. So again we draw two scenarios on the number line and we find that two cases are possible, one where \(z\) is in the middle and one where it is not. Hence both statements combined are insufficient as well.

Answer. E .. I am attaching a diagram to illustrate.

[img]
Attachment:
Number%20Line.jpg
[/img]
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Number Line.jpg
Number Line.jpg [ 105.17 KiB | Viewed 68190 times ]


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Last edited by omerrauf on 03 Feb 2012, 16:04, edited 1 time in total.

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

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New post 30 Oct 2013, 01:12
Bunuel, I have a question. Among the 4 cases why are C & D a "no case".

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

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

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New post 30 Oct 2013, 20:41
Bunuel, couldn't it also be possible that the variables are arranged so x is between z and y? For example z->x->y, instead of only x->y->z? It's kind of an incidental point because I got it right, but I want to make sure I am figuring it up the proper way.

Also, when the statements xyz or xy, is that to say how the numbers are located with reference to zero, or is it saying to multiply the numbers?

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

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New post 31 Oct 2013, 00:40
Stoneface wrote:
Bunuel, couldn't it also be possible that the variables are arranged so x is between z and y? For example z->x->y, instead of only x->y->z? It's kind of an incidental point because I got it right, but I want to make sure I am figuring it up the proper way.

Also, when the statements xyz or xy, is that to say how the numbers are located with reference to zero, or is it saying to multiply the numbers?


We know that the distance between x and y is greater than the distance between x and z. This can happen in 4 ways, shown in my post. You can see there that x CAN be between z and y in cases C and D.

As for xyz it means x*y*z.
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Re: On the number line, the distance between x and y is greater [#permalink]

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New post 28 Jul 2014, 01:17
Bunuel wrote:
jj97cornell wrote:
On a 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 a number line?

(1) xyz < 0
(2) xy < 0


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 (\(z\) lies between \(x\) and \(y\)).

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

(1)+(2) Both case A (answer YES) and case C (answer NO) satisfy the statements. Not sufficient.

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

Answer: E.

Hope it's clear.


Bunuel while combining both a and b , can't we say that now out of x,y and z only one can be negative and out of x and y only one can be negative. Thus z has to be positive.
Now based on this understanding and which of x or y is negative , we notice that we can still not determine the answer. Hence insufficient

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

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New post 13 Oct 2014, 19:51
Hi Bunuel,
I could not follow No case for these 2 cases
C. y--------x--z (NO case);
D. z--x--------y (NO case)'

you responded to this question already that Z lies between x and Y . But that is the question stem. so I thought to consider those cases as well. Am i missing anything?




Bunuel wrote:
jj97cornell wrote:
On a 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 a number line?

(1) xyz < 0
(2) xy < 0


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 (\(z\) lies between \(x\) and \(y\)).

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

(1)+(2) Both case A (answer YES) and case C (answer NO) satisfy the statements. Not sufficient.

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

Answer: E.

Hope it's clear.

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

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New post 15 Sep 2015, 07:02
Hi Bunuel, does the case x---z----------y also satisfy both conditions?

with zero between x and z, we have -++, which is negative, and which contains differing signs for x & y.

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

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New post 15 Sep 2015, 08:50
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
Remember equal number of variables and independent equations ensures a solution.

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

In the original condition we have 3 variables (x,y,z) and 1 equation ( the distance between x and y is greater than the distance between x and z.). We need 2 more equations to match the number of variables and equations, and since there is 1 each in 1) and 2), C has high probabilty of being the answer.

Using both 1) & 2) together we have x=-1, y=2, z=0 which gives us answer yes, while z=-2, x=-1, y=2 gives us answer no. Therefore the conditions are not sufficient and the answer is E.

Normally for cases where we need 2 more equations, such as original conditions with 2 variable, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using ) and 2) together. (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: On the number line, the distance between x and y is greater [#permalink]

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New post 20 Apr 2016, 05:18
Using (I) and (II) alone, it is easy to figure that each statement alone is not sufficient.

Combining (I) and (II), we have xyz < 0 and xy < 0, this implies that z is positive
(if z is negative then xyz is > 0 [since xy < 0 as per (II)] and xyz > 0 violates (I). Hence z cannot be negative).
When combining (I) and (II) we can take two cases
x is -ve or y is -ve [both cannot have the same sign as this violates (II)].

We will have the below 3 cases for x is -ve and y is -ve. No single solution exists and hence both together not sufficient. Answer is E.

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

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Scenario A ) where XYZ < 0.

2 positives 1 negative
3 negatives.

For three negatives on the number the order can be z, x, y, 0...or x, z, y, 0.

So A is insufficient.

Scenario B) hwere XY < 0.

One negative one positive. So the order can go. x 0 z y...or z x 0 y. So B is insufficient.

Scenario A+B.

If X and Y are both of oppositie signs (XY<0)..then Z has to be postive in order for XYZ < 0.

So One Negative and two positives.

Order can be.....x 0 z y...or y 0 x z.

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

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New post 28 Mar 2017, 08:37
If we draw a number line we see that many combinations are possible
1) no sufficient
2) not sufficient
1+2) just tell us that z is positive and one of x and Z is negatine leaving still many combinations possible

Answer is E, not Sufficient

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

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New post 31 Jul 2017, 15:43
Here is another way to look at the problem.

According to the question, the distance on x and y is greater than x and z, since we have x as the reference point for calculating the distance.

We can only have two cases.

Case 1 - When x is in the middle of y and z

Case 2 - When x is at the extreme end i.e. on one side.

Statement 1 and Statement 2 doesn't provide us the exact values of x,y,z, therefore, Answer is E
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Re: On the number line, the distance between x and y is greater   [#permalink] 31 Jul 2017, 15:43
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