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If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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 18 Aug 2009, 18:47
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If A<Y<Z<B, is |Y-A|< |Y-B|? (1) |Z-A| < |Z-B| (2) |Y-A| < |Z-B|
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Last edited by Bunuel on 09 Aug 2012, 01:11, edited 1 time in total.
Edited the question and added the OA.
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Re: If A<Y<Z<B, is /Y-A/< /Y-B/? [#permalink]
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yezz wrote: If A<Y<Z<B, is /Y-A/< /Y-B/?
1) /Z-A/ < /Z-B/
2) /Y-A/ < /Z-B/ Consider A,Y,Z,B lying on the straight line Given that A<Y<Z<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
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Re: If A<Y<Z<B, is /Y-A/< /Y-B/? [#permalink]
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28 Oct 2009, 17:44
I think both are insufficient Take an example A = -5 , Y= 1 , Z = 2 , B = 3 |Y- A| < | Y-B| |1+5| < |1-3| i.e. 6 < 2 ... no A = 1 , Y= 2 , Z = 3 , B = 4 |2-1| < |2-4| i.e. 1 < 2 ... yes So, A not suff similarly B - not suff I think i have mistaken somewhere , can some one correct me
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Re: If A<Y<Z<B, is /Y-A/< /Y-B/? [#permalink]
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28 Oct 2009, 20:41
D
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.
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Re: If A<Y<Z<B, is /Y-A/< /Y-B/? 1) /Z-A/ < /Z-B/ [#permalink]
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08 Aug 2012, 21:14
Bunuel
can you please clarify is this makes sense
If a < y < z < b, is |y – a| < |y – b|? (1) |z – a| < |z – b| (2) |y – a| < |z – b|
a < y < z < b
a< y implies
0< y-a or y-a >0
2) y < B (y-b ) < 0
Implies
y-b --------------- 0-------------y-b
Yes proved
2) If a < y < z < b
|y – a| < |z – b|
0 < y-a y-a >0
Z<b z-b< 0
Y< z-b implies
y-b < 0
(y-a) -------------0--------------y-b
D
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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09 Aug 2012, 01:26
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If A<Y<Z<B, is |Y-A|< |Y-B|? 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.
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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09 Aug 2012, 10:54
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yezz wrote: If A<Y<Z<B, is |Y-A|< |Y-B|?
(1) |Z-A| < |Z-B|
(2) |Y-A| < |Z-B| 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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Re: If A<Y<Z<B, is /Y-A/< /Y-B/? [#permalink]
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09 Aug 2012, 11:31
gmattokyo wrote: D
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.
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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18 Oct 2012, 05:21
Bunuel, 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!
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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23 Oct 2012, 06:52
liarish wrote: Bunuel, 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 \(2Z<A+B\). Hope it's clear.
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Re: If A<Y<Z<B, is /Y-A/< /Y-B/? [#permalink]
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06 Apr 2013, 10:56
alwynjoseph wrote: yezz wrote: If A<Y<Z<B, is /Y-A/< /Y-B/?
1) /Z-A/ < /Z-B/
2) /Y-A/ < /Z-B/ Consider A,Y,Z,B lying on the straight line Given that A<Y<Z<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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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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06 Apr 2013, 20:30
By drawing a line with the given condition, it is easy to see |y-a| < |y-b|
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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If A<Y<Z<B, is |Y-A|< |Y-B|?
Given the above, |Y-A|< |Y-B|
(Y-A) < -(Y-B)
Y-A < -Y+B
is 2Y < B+A
(1) |Z-A| < |Z-B|
(Z-A) < -(Z-B)
Z-A <-Z+b
2Z < B+A
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?
Thanks!
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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06 Dec 2013, 21:52
Drawing a line and using Bunuel's method really helps with this problem.
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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14 Jun 2015, 12:07
Bunuel wrote: If A<Y<Z<B, is |Y-A|< |Y-B|? 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 \(|Y-A|=Y-A\) ? Thanks
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If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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14 Jun 2015, 12:24
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abhinav008 wrote:
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 \(|Y-A|=Y-A\) ?
Thanks
Hello abhinav008Yes if Y is positive and A is negative then \(|Y-A|=Y-A\) more general rule (work in described case and in case when Y and A are both positive): if \(Y-A > 0\) then \(|Y-A|=Y-A\)
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If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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23 Apr 2016, 00:33
Bunuel wrote: If A<Y<Z<B, is |Y-A|< |Y-B|? 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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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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21 May 2016, 11:46
yezz wrote: If A<Y<Z<B, is |Y-A|< |Y-B|?
(1) |Z-A| < |Z-B|
(2) |Y-A| < |Z-B| An algebric way is nice, but if you think about the given equations as intervals on a line, it becomes much easier to solve. notice that in my solution, since a,c,b are intervals they are all positive.
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|? [#permalink]
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21 Jul 2016, 08:06
yezz wrote: If A<Y<Z<B, is |Y-A|< |Y-B|?
(1) |Z-A| < |Z-B|
(2) |Y-A| < |Z-B| Once one can understand that only fractions will work in this sequence the solution comes pretty easily. D.
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Re: If A<Y<Z<B, is |Y-A|< |Y-B|?
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