GMAT Prep: Absolute values DS

Question

Explanation please

hobbit · Answer

answer is d

there is, of course, an algebraic solution to this... but i think it is better in this case to visualize, using the interpretation of |a-b| as the distance between a and b.

so you have 4 points on a line in that order: a,y,z,b

you need to find whether the distance between a and y is less that y and b.

st1) says that the distance between z and a is less than distance between z and b. seems to be sufficient... the distance between a and y is definitely less than the one between a and z. and the distance from y and b is definitely larger than the one from b to z... (just because of their ordering on the line....) so just by visualizing it - it is enough.

st2) here is even clearer.... the distance between y and b is certainly larger than the distance between z and b. plug that fact in st2.... and you get the answer to stem


hence D is the naswer...

mbagal1 · Answer

I get B. I found it easy to look at a,b, y,z almost like they are points on a number line. easier for me to visualize that way.

a<y<z<b

|y-a| < |y-b|?

1) |z-a| < |z-b|

we know from question that z-a > z-y (or y-a >0) and b-y > b-z, but that really does not tell us much about y-b.

2)|y-a|< |z-b|

we know from question that b-z > b-y or can be written as: |z-b| < |y-b|.

Since it is give that |y-a| < |z-b|, we know that |y-a| < |y-b|

mbagal1 · Answer

hmm, I haven't been able to visualize why statement 1 is sufficient...yet.

can you explain mathematically why this has to be true?

Thanks.

hobbit · Answer

here is st1 proven mathematically:

from a < y < z we get easily |z-a| > |y-a|
from y < z < b we get easily |y-b| = |b-y| > |b-z| = |z-b|
st1 gives us |z-a| < |z-b|

and now we create a chain:
|y-a|<|z-a|<|z-b|<|y-b|
hence |y-a|<|y-b| as required....

hence st1 sufficient.

Andr359 · Answer

ItÂ´s D.

The easiest way to solve this sort of problems is using a line and plotting a, y, z and b and remembering that |-| is the distance between two of them.

subhen · Answer

Hobbit,

Excellent explanation. Btw I have seen your other posts and they are very useful to understand absolute values. Thanks a ton.

I think the problem becomes easier if we start viewing absolute values on the number line and break it up into cases. Like Hobbit has already mentioned
Case I: 
a<y<z implies 
|y - a| < |z - a| i.e. dist of y from a is less than dist of z from a.

Case II:
y<z<b implies
|z - b| < |y - b| i.e. dist of z from b is less than dist of y from b.

St 1: |y - a| < |z - b|
Hence from Case II, |y - a| < |y - b|

St2: |z - a| < |z - b|
From Case I and Case II, |y - a| < |y - b|

Hence D is the answer.

GMAT Prep: Absolute values DS

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