Absolute value sum

[abhirup1711]

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]

|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]

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

From definition,

• If (a - b) > 0, |a - b| = (a - b)
  If (a - b) < 0, |a - b| = -(a - b) = (b - a)

As |a - b| = b - a, (a - b) must be less than zero.
Hence, a < b

So, III must be true.

Option C and E are our possible correct answer.
From given information we cannot algebraically conclude that II must be true.
Hence, only III must be true.

If you are not confident, pick some numbers and try to prove the same.
Consider the following example,

• a = 0 and b = 1 ---> |a - b| = |0 - 1| = 1 = (b - 1) but (a + b) = 1 > 0
  Hence, II must not be true.

The correct answer is C.

[GMATGuruNY]

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.

[Brent@GMATPrepNow]

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