If a<y<z<b, is abs(y-a)<abs(y-b)?

[brianmcma]

Apologies if this has already been posted, but I was not able to find it. This appeared for me as question 31 in GMAT Prep 2:

If a < y < z < b, is abs(y-a) < abs(y-b)?
1. abs(z-a) < abs(z-b)
2. abs(y-a) < abs(z-b)

Answer is D. Was wondering if there is a quicker method to solve this problem, other than simply plugging in numbers? Some sort of rules-based, or common-sense approach. Thanks!

[Ben Ku]

First, let's try to clear the absolute values.

Because we know that a < y < z < b, we know
abs(y - a) = y - a
abs(y - b) = b - y (since y - b is negative)

We can rephrase the question:
Is y - a < b - y?
or
Is 2y < a + b?

Statement (1) can be rephrased: z - a < b - z, so 2z < a + b. We also know that since y < z, then 2y < 2z. So 2y < 2z < a + b. (1) is sufficient.

Statement (2) can be rephrased: y - a < b - z, so y + z < a + b. Since y < z, we can add y to both sides: 2y < y + z. So 2y < y + z < a + b, so (2) is sufficient as well.

Hope this helps.

[dheeraj787]

Thank You ben..
This is really a beautiful approach. I was not able to see the question as you did. Will remember it. "MODULUS" is a conceptual topic and is one of my weakness area. trying to overcome with practice. If any suggestions most welcome.. Thanks

[RonPurewal]

Thank You ben..
This is really a beautiful approach. I was not able to see the question as you did. Will remember it. "MODULUS" is a conceptual topic and is one of my weakness area. trying to overcome with practice. If any suggestions most welcome.. Thanks

You may want to check the date stamps on the posts. The one to which you're responding is almost five years old. (Ben hasn't posted on here for quite a while.)

[dheeraj787]

Dear Ron,

Sorry for opening the very old post.

Regards,
Dheeraj

[RonPurewal]

No worries.

I can't tell whether you have an actual question buried in there (vs. whether you just wanted to congratulate Ben on a good post). If you do have a question, then by all means fire away.

[qqixiaofan]

Hi,

This is how I have in mind. Please could you advise if my logic is correct.

abs(y-a) is the distance between y&a on the number line;
abs(y-b) is the distance between y&a on the number line;

and we know a<y<z<b. if we draw the number line with this sequence and apply (1) and (2). the answer is obvious

(1) distance between z & a is less than the distance between z & b on the number line.
(2) distance between y&a is less than the distance between z&b on the number line.


thank you,
Xiao

Apologies if this has already been posted, but I was not able to find it. This appeared for me as question 31 in GMAT Prep 2:

If a < y < z < b, is abs(y-a) < abs(y-b)?
1. abs(z-a) < abs(z-b)
2. abs(y-a) < abs(z-b)

Answer is D. Was wondering if there is a quicker method to solve this problem, other than simply plugging in numbers? Some sort of rules-based, or common-sense approach. Thanks!

[RonPurewal]

That's a workable approach.
For this problem, in fact, that kind of approach is the only one that comes to my mind right away.