Absolute value sum

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abhirup1711: Senior | Next Rank: 100 Posts; Posts: 66; Joined: Wed Jul 13, 2011 2:27 pmFollowed by: 2 members

by abhirup1711 » Tue Apr 16, 2013 2:06 am

If a â‰ 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?

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Source: Beat The GMAT — Problem Solving | Tue Apr 16, 2013 2:06 am

Zarrolou: Junior | Next Rank: 30 Posts; Posts: 26; Joined: Sat Feb 09, 2013 1:30 pm; Thanked: 4 times

by Zarrolou » Tue Apr 16, 2013 2:10 am

|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

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Anju@Gurome: GMAT Instructor; Posts: 511; Joined: Wed Aug 11, 2010 9:47 amFollowed by: 86 members; Location: Delhi, India; Thanked: 344 times

by Anju@Gurome » Tue Apr 16, 2013 2:18 am

abhirup1711 wrote:If a â‰ 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.

Anju Agarwal
Quant Expert, Gurome

Backup Methods : General guide on plugging, estimation etc.
Wavy Curve Method : Solving complex inequalities in a matter of seconds.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Apr 16, 2013 5:36 am

abhirup1711 wrote:If a â‰ 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 â‰¥ 0
a â‰¤ b.
Here, it is given that aâ‰ 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: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Apr 16, 2013 5:37 am

abhirup1711 wrote:If a â‰ 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 â‰ 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

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