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1) statement 1 talks nothing about Y but however it clearly brings out the fact that the distance between B&Z is greater than the distance between A&Z. Now we know that since Y<Z, Y lies closer to A than Z and farther from B than Z. hence answers our question

2) Statement 2 : it says distance between Y&A is less than the distance between Z&B. As already stated Y lies closer to A than Z and farther to B than Z, hence this answers the question as well

I drew a number line for this which helped me conceptualize the problem (4 points A, Y, Z and B). (A) Given distance between Z and B is greater than Z and A is sufficient as Y lies between A and Z. (B) Similar approach as A. Sufficient.

Since given that \(A<Y\) (\(Y-A>0\)), then \(|Y-A|=Y-A\); Since given that \(Y<B\) (\(Y-B<0\)), then \(|Y-B|=B-Y\);

So, the question becomes: is \(Y-A<B-Y\)? --> is \(2Y<A+B\)?

(1) |Z-A| < |Z-B|. The same way as above: Since given that \(A<Z\) (\(Z-A>0\)), then \(|Z-A|=Z-A\); Since given that \(Z<B\) (\(Z-B<0\)), then \(|Z-B|=B-Z\);

So, we have that: \(Z-A<B-Z\) --> \(2Z<A+B\). Now, since \(Y<Z\), then \(2Y<2Z\), so \(2Y<2Z<A+B\). Sufficient.

(2) |Y-A| < |Z-B|. The same way as above: Since given that \(A<Y\) (\(Y-A>0\)), then \(|Y-A|=Y-A\); Since given that \(Z<B\) (\(Z-B<0\)), then \(|Z-B|=B-Z\);

So, we have that: \(Y-A<B-Z\) --> \(Y+Z<A+B\). Now, since \(Y<Z\), then \(2Y<Y+Z\) (\(2Y=Y+Y<Y+Z\) ), so \(2Y<Y+Z<A+B\). Sufficient.

Use the meaning of the absolute value: distance between two points. Rephrasing the question: we are given 4 distinct points on the number line, A, Y, Z, B, from left to right (in increasing order). The question is: is the distance between A and Y smaller than the distance between Y and B?

(1) A---Y--Z------B would visualize a typical situation, distance between A and Z less than the distance between Z and B. Since Y is between A and Z, the answer to the question "is the distance between A and Y smaller than the distance between Y and B" is definitely YES. Sufficient.

(2) A---Y-Z----B this would be a typical situation, distance between A and Y, less than the distance between Z and B. Now Z is between Y and B, so again, the distance between A and Y is smaller than the distance between Y and B. Sufficient.

Answer D
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PhD in Applied Mathematics Love GMAT Quant questions and running.

I drew a number line for this which helped me conceptualize the problem (4 points A, Y, Z and B). (A) Given distance between Z and B is greater than Z and A is sufficient as Y lies between A and Z. (B) Similar approach as A. Sufficient.

Please correct this if wrong.

You are absolutely right. Sorry, I missed your post and I have just presented a similar solution.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

For statement 1, everything works until z-a<b-z = 2z<a+b. y<z so 2y<2z ok but this does not mean 2z<a+b. we know that a<y<z<b. Now if we plug in values say 3<4<5<6 then 2(5) is not < 3+6 i.e 2z<a+b.

What could I be doing wrong here Bunuel? Could you please help me with that. Thanks in advance!

For statement 1, everything works until z-a<b-z = 2z<a+b. y<z so 2y<2z ok but this does not mean 2z<a+b. we know that a<y<z<b. Now if we plug in values say 3<4<5<6 then 2(5) is not < 3+6 i.e 2z<a+b.

What could I be doing wrong here Bunuel? Could you please help me with that. Thanks in advance!

You derived yourself that \(2Z<A+B\) and later you are negating that (blue parts).

We know from (1) that \(2Z<A+B\) (i) . We also know that \(Y<Z\), so \(2Y<2Z\) (ii).

Now, combine (i) and (ii): 2Z is less than A+B and 2Y is less than 2Z, thus 2Y is less than A+B (\(2Y<2Z<A+B\)).

Also, numbers you chose does not satisfy \(2Z<A+B\).

1) statement 1 talks nothing about Y but however it clearly brings out the fact that the distance between B&Z is greater than the distance between A&Z. Now we know that since Y<Z, Y lies closer to A than Z and farther from B than Z. hence answers our question

2) Statement 2 : it says distance between Y&A is less than the distance between Z&B. As already stated Y lies closer to A than Z and farther to B than Z, hence this answers the question as well

So choice is Either statement alone is sufficient

I've struggled with this question for 10 min., then I looked at first sentence and the light bulb turned on. It becomes quite, quite a simple task the minute you draw a straight line... thank you!
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A<Y<Z<B. So, Z is greater than Y. if 2Z < B+A then 2Y (Y being less than Z) must also be less than B+A SUFFICIENT

(2) |Y-A| < |Z-B| (Y-A) < -(Z-B) Y-A < -Z + B Y + Z < B + A

A<Y<Z<B. Y + Z is less than B + A. If Y + Z is less than B + A then 2Y must be less than B + A. 2Y is Y + Y but Y + Z is Y + a number greater than Y. If Y + a Number greater than Y is less than B + A then Y + Y must be less than Y.

I noticed it is a bit different from the way others have solved this problem. Is my way correct?

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Since given that \(A<Y\) (\(Y-A>0\)), then \(|Y-A|=Y-A\); Since given that \(Y<B\) (\(Y-B<0\)), then \(|Y-B|=B-Y\);

So, the question becomes: is \(Y-A<B-Y\)? --> is \(2Y<A+B\)?

(1) |Z-A| < |Z-B|. The same way as above: Since given that \(A<Z\) (\(Z-A>0\)), then \(|Z-A|=Z-A\); Since given that \(Z<B\) (\(Z-B<0\)), then \(|Z-B|=B-Z\);

So, we have that: \(Z-A<B-Z\) --> \(2Z<A+B\). Now, since \(Y<Z\), then \(2Y<2Z\), so \(2Y<2Z<A+B\). Sufficient.

(2) |Y-A| < |Z-B|. The same way as above: Since given that \(A<Y\) (\(Y-A>0\)), then \(|Y-A|=Y-A\); Since given that \(Z<B\) (\(Z-B<0\)), then \(|Z-B|=B-Z\);

So, we have that: \(Y-A<B-Z\) --> \(Y+Z<A+B\). Now, since \(Y<Z\), then \(2Y<Y+Z\) (\(2Y=Y+Y<Y+Z\) ), so \(2Y<Y+Z<A+B\). Sufficient.

Answer: D.

Hope it's clear.

Hi Bunuel , I am learning concepts of Mod . Just need one small clarification : given that \(A<Y\) (\(Y-A>0\)), then \(|Y-A|=Y-A\)

I want to know if Y is positive and A is negative then still Is \(|Y-A|=Y-A\) ?

Since given that \(A<Y\) (\(Y-A>0\)), then \(|Y-A|=Y-A\); Since given that \(Y<B\) (\(Y-B<0\)), then \(|Y-B|=B-Y\);

So, the question becomes: is \(Y-A<B-Y\)? --> is \(2Y<A+B\)?

(1) |Z-A| < |Z-B|. The same way as above: Since given that \(A<Z\) (\(Z-A>0\)), then \(|Z-A|=Z-A\); Since given that \(Z<B\) (\(Z-B<0\)), then \(|Z-B|=B-Z\);

So, we have that: \(Z-A<B-Z\) --> \(2Z<A+B\). Now, since \(Y<Z\), then \(2Y<2Z\), so \(2Y<2Z<A+B\). Sufficient.

(2) |Y-A| < |Z-B|. The same way as above: Since given that \(A<Y\) (\(Y-A>0\)), then \(|Y-A|=Y-A\); Since given that \(Z<B\) (\(Z-B<0\)), then \(|Z-B|=B-Z\);

So, we have that: \(Y-A<B-Z\) --> \(Y+Z<A+B\). Now, since \(Y<Z\), then \(2Y<Y+Z\) (\(2Y=Y+Y<Y+Z\) ), so \(2Y<Y+Z<A+B\). Sufficient.

Answer: D.

Hope it's clear.

Bunuel, your explanations of 'absolute value questions' using number lines have taken my ability to solve such questions to amazing heights.

I could solve this question easily using number line- this leaves me wondering why you used algebraic approach instead of the number line approach.

I want to know whether there are scenarios in which one should prefer algebraic approach to number line approach.

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