Absolute value sum

Question

If a ≠ b and |a-b| = b-a, which of the following statements must be true ?

I. a < 0
II. a + b < 0
III. a < b

A) None
(B) I only
(C) III only
(D) I and II
(E) II and III

Can an expert please provide a simple explanation?

Zarrolou · Answer

|a-b| can have 2 values:
1)a>b a-b>0
a-b=b-a

2)a<b
-a+b=b-a

As you can see in case 2 the equality is true. So if we know that
|a-b|=b-a
we must be in the second case.

keep in mind that a must be different from b
Otherwise both could be 0 and the right answer would be different (None in this case)

C

Anju@Gurome · Answer

abhirup1711 wrote:If a ≠ b and |a-b| = b-a, which of the following statements must be true ?

I. a < 0
II. a + b < 0
III. a < b

A) None
(B) I only
(C) III only
(D) I and II
(E) II and III

We know that |something| > 0
So, if |a - b| = b - a it must be true that b - a > 0
Since we're told that a ≠ b, we can conclude that b - a > 0
Now take this inequality and add a to both sides to get b > a
This is all that we can conclude about a and b.

Now check the answer choices.

Our conclusion matches III, so the correct answer is C (III only) or E (II and III).

IMPORTANT: At this point, we need only check II (a + b < 0 )

If all we know is that b > a, we cannot conclude that a + b is greater than 0. So, we know that II need not be true.

For further evidence, consider these 2 possible cases.
Case a: a = -2 and b = 1, in which case b + a < 0
Case b: a = 1 and b = 2, in which case b + a > 0

Since only III must be true, the correct answer is C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

GMATGuruNY · Answer

abhirup1711 wrote:If a ≠ b and |a-b| = b-a, which of the following statements must be true?

I. a < 0
II. a + b < 0
III. a < b

A) None
(B) I only
(C) III only
(D) I and II
(E) II and III

Can an expert please provide a simple explanation?

Be definition:
|a-b| is the DISTANCE between a and b.

|a-b| = b-a implies the following:
The DISTANCE between a and b is equal to the DIFFERENCE between b and a.

A DISTANCE must be greater than or equal to 0.
A DIFFERENCE can be negative, 0, or positive.
For the DISTANCE between two values to be equal to the DIFFERENCE between the two values, the DIFFERENCE -- like the DISTANCE -- must be greater than or equal to 0:
b-a ≥ 0
a ≤ b.
Here, it is given that a≠b, so a<b.

Thus, statement III must be true.
Eliminate any answer choice that does not include statement III (A, B and D).

The only difference between C and E is that the latter answer choice includes statement II.
Try to show that statement II does NOT have to be true.
If we plug a=2 and b=3 into |a-b| = b-a, we get:
|2-3| = 3-2
1 = 1.
Since it does not have to be true that a+b < 0, eliminate E.

The correct answer is C.

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Brent@GMATPrepNow · Answer

abhirup1711 wrote:If a ≠ b and |a-b| = b-a, which of the following statements must be true?

I. a < 0
II. a + b < 0
III. a < b

A) None
(B) I only
(C) III only
(D) I and II
(E) II and III

Can an expert please provide a simple explanation?

Be definition:
|a-b| is the DISTANCE between a and b.

|a-b| = b-a implies the following:
The DISTANCE between a and b is equal to the DIFFERENCE between b and a.

A DISTANCE must be greater than or equal to 0.
A DIFFERENCE can be negative, 0, or positive.
For the DISTANCE between two values to be equal to the DIFFERENCE between the two values, the DIFFERENCE -- like the DISTANCE -- must be greater than or equal to 0:
b-a ≥ 0
a ≤ b.
Here, it is given that a≠b, so a<b.

Thus, statement III must be true.
Eliminate any answer choice that does not include statement III (A, B and D).

The only difference between C and E is that the latter answer choice includes statement II.
Try to show that statement II does NOT have to be true.
If we plug a=2 and b=3 into |a-b| = b-a, we get:
|2-3| = 3-2
1 = 1.
Since it does not have to be true that a+b < 0, eliminate E.

The correct answer is C.

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