absolute value

[yakiraz]

thanks in advance

[atlantic]

I got D. Is it the OA?

Post explanation later.

[yakiraz]

yep

[ildude02]

can someone please explain the reasoning, appreciate it. I choose D as well but not sure if I used the best approach since I went about by picking numbers and backsolving. I assume there is an easier to solve it ?

[cubicle_bound_misfit]

Even I did the same and this is taking hell lot of time.
Can someone please discuss a better approach ?

regards,

[Ian Stewart]

Almost every difficult absolute value question on the GMAT is testing whether you understand that absolute value measures distance:

|c| is the distance, on the number line, from c to zero;
|c - d| is the distance, on the number line, between c and d.

The question tells us a < y < z < b. We know a is furthest to the left on the number line, b furthest to the right, and y is closer to a than z is. We can draw the points on the number line. The question asks:

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

In words, using the distance interpretation above, this just asks "Is the distance between y and a less than the distance between y and b?" In other words, "Is y closer to a than it is to b?"

1) tells us that z is closer to a than to b. Well, y is even closer to a than z is, so y must be closer to a than to b. Sufficient.

2) tells us that y is closer to a than z is to b. Well, y is even further away from b than z is, so y must be closer to a than to b. Sufficient.

D.

It's a lot easier if you draw the number line and think about what the distance statements are telling you while looking at the picture.

[Vignesh.4384]

Almost every difficult absolute value question on the GMAT is testing whether you understand that absolute value measures distance:

|c| is the distance, on the number line, from c to zero;
|c - d| is the distance, on the number line, between c and d.

The question tells us a < y < z < b. We know a is furthest to the left on the number line, b furthest to the right, and y is closer to a than z is. We can draw the points on the number line. The question asks:

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

In words, using the distance interpretation above, this just asks "Is the distance between y and a less than the distance between y and b?" In other words, "Is y closer to a than it is to b?"

1) tells us that z is closer to a than to b. Well, y is even closer to a than z is, so y must be closer to a than to b. Sufficient.

2) tells us that y is closer to a than z is to b. Well, y is even further away from b than z is, so y must be closer to a than to b. Sufficient.

D.

It's a lot easier if you draw the number line and think about what the distance statements are telling you while looking at the picture.

Hi Ian,

As per your explanation .. i consider 4 point on the number line

a = -4
y = 1
z = 2
b = 3

This follows the rule a < y < z < b.

Here if u notice the distance between y and a is greater than y and b right ?

Could you please explain this ?

[Stuart@KaplanGMAT]

Hi Ian,

As per your explanation .. i consider 4 point on the number line

a = -4
y = 1
z = 2
b = 3

This follows the rule a < y < z < b.

Here if u notice the distance between y and a is greater than y and b right ?

Could you please explain this ?

The numbers you picked violate both statements, so are impermissable for this question.

1) tells us that z is closer to a than to b. In your example, z is closer to b than to a.

2) tells us that y is closer to a than z is to b. In your example, z is closer to b than y is to a.

Remember, there are 2 steps to picking numbers in data sufficiency:

1) Pick permissible numbers. The numbers you select MUST follow the rules given in the question stem and in the statement (or statements) that you're evaluating.

2) Plug the numbers you've chosen back into the original question to see what answer you get.

The numbers you've chosen don't pass the rule in step (1), so you have to discard them and pick new numbers.

[Vignesh.4384]

Thanks Stuart.
I guess i dint read the question properly

[maihuna]

Ian,
You provided fantastic soln but can you elaborate please with an algebraic solution?

[Ian Stewart]

Ian,
You provided fantastic soln but can you elaborate please with an algebraic solution?

I wouldn't normally do this kind of question algebraically, but it is certainly possible. We need to use the fact that |x| = x if x is positive, and |x| = -x if x is negative.

Since we're given that a < y < z < b, we can easily work out whether the expressions in each absolute value are positive or negative. Note also that since y < z, then 2y < 2z must be true.

-- In the question itself, we're asked if |y-a| < |y-b|. Since y - a is positive, |y - a| = y - a. Since y - b is negative, |y - b| = b - y. So the question is just asking if y - a < b - y, or if 2y < a + b.

-- Statement 1 tells us that |z - a| < |z - b|. As above, z - a is positive and z - b is negative, so this really tells us that z - a < b - z, or 2z < a + b. Well since 2y < 2z, then 2y < a + b must be true. Sufficient.

-- Statement 2 tells us that |y - a| < |z - b|. As above, y - a is positive and z - b is negative, so this really tells us that y - a < b - z, or y + z < a + b. Now, since y < z, y + y must be less than y + z, so 2y must be less than a + b. So this statement is also sufficient.

As I said above, however, I wouldn't personally consider taking this approach to this question.

[maihuna]

Thank you very much Ian, for taking time out of your busy schedule and explaining it in great detail. Wonderful, thank you again.

I am struggling with absolute value questions a lot and may trouble you on some more questions, please bear with me.

[cubicle_bound_misfit]

maihuna,
for absolute value questions, u got the right guy to ask