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# For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +

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Math Expert
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For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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22 Jul 2015, 01:10
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Question Stats:

66% (02:13) correct 34% (02:09) wrong based on 929 sessions

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For any a and b that satisfy $$|a – b| = b – a$$ and $$a > 0$$, then $$|a + 3| + |-b| + |b – a| + |ab| =$$

A. $$-ab + 3$$

B. $$ab + 3$$

C. $$-ab + 2b + 3$$

D. $$ab + 2b – 2a – 3$$

E. $$ab + 2b + 3$$

Kudos for a correct solution.

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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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22 Jul 2015, 03:12
6
9
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

Given |a-b| = b-a ---> This can only happen when a-b<0 ---> b>a

Also given a>0 , thus b>a>0

Now per question, |a + 3| = a+3 as a >0 and thus a+3 >0
|-b| = b as b>0
|b – a| = b-a as b>a
|ab| = ab as both a,b >0

Thus |a + 3| + |-b| + |b – a| + |ab| = a+3+b+b-a+ab = 3+2b+ab. E is the correct answer.
##### General Discussion
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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22 Jul 2015, 04:05
6
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

Solution -
Given that |a – b| = b – a, so b – a > 0, or b > a. Also given that a > 0, so b > 0.

|a + 3| + |-b| + |b – a| + |ab| =
a + 3 + b + b – a + ab =ab + 2b + 3
ANS E
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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Updated on: 22 Jul 2015, 04:51
1
1
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

Such Question require an intense observation of the given Information step-by-step

Observation-1: |a – b| = b – a

which is possible only when signs of a and b are Same
Since Given a > 0
so we figure out that a and b are both positive

Observation-2: |a – b| must be Non-Negative and so should be the value of b-a which is possible only when absolute value of b is greater than or equal to absolute value of a

Now you may choose the values of a and b based on above observations

e.g. b = 3 and a=1 and check the value of given functions and options

|a + 3| + |-b| + |b – a| + |ab| = |1 + 3| + |-3| + |3 – 1| + |1*3| = 12

A. -ab + 3 = -1*3 + 3 = 0 not equal to 12 hence, INCORRECT
B. ab + 3 = 1*3 + 3 = 6 not equal to 12 hence, INCORRECT
C. -ab + 2b + 3 = -1*3+2*3+3 = 6 not equal to 12 hence, INCORRECT
D. ab + 2b – 2a – 3 = 1*3 + 2*3 - 2*1 - 3 = 4 not equal to 12 hence, INCORRECT
E. ab + 2b + 3 = 1*3 + 2*3 + 3 = 12 CORRECT
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Originally posted by GMATinsight on 22 Jul 2015, 04:33.
Last edited by GMATinsight on 22 Jul 2015, 04:51, edited 1 time in total.
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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22 Jul 2015, 04:39
1
GMATinsight wrote:
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

Such Question require an intense observation of the given Information step-by-step

Observation-1: |a – b| = b – a

which is possible only when signs of a and b are Same
Also, Since Given a > 0 so we figure out that a and b are both positive

Observation-2: |a – b| must be Non-Negative and so should be the value of b-a which is possible only when absolute value of b is greater than or equal to absolute value of a

Now you may choose the values of a and b based on above observations

e.g. b = 3 and a=1 and check the value of given functions and options

|a + 3| + |-b| + |b – a| + |ab| = |1 + 3| + |-3| + |3 – 1| + |1*3| = 12

GMATinsight, the text in red above is not necessarily true.

|a-b| = b-a ONLY when a-b <0 or b>a . now b > a by being (3,-1) (opposite signs) or by (-4,-2) or by (3,1) (in these 2 cases, same signs)
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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22 Jul 2015, 04:50
Engr2012 wrote:
GMATinsight wrote:
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

Such Question require an intense observation of the given Information step-by-step

Observation-1: |a – b| = b – a

which is possible only when signs of a and b are Same
Also, Since Given a > 0 so we figure out that a and b are both positive

Observation-2: |a – b| must be Non-Negative and so should be the value of b-a which is possible only when absolute value of b is greater than or equal to absolute value of a

Now you may choose the values of a and b based on above observations

e.g. b = 3 and a=1 and check the value of given functions and options

|a + 3| + |-b| + |b – a| + |ab| = |1 + 3| + |-3| + |3 – 1| + |1*3| = 12

GMATinsight, the text in red above is not necessarily true.

|a-b| = b-a ONLY when a-b <0 or b>a . now b > a by being (3,-1) (opposite signs) or by (-2,-4) or by (3,1) (in these 2 cases, same signs)

a and b will always have same sign for |a – b| = b – a to be true for given a >0

The word "also" was disturbing the meaning probably, so removed it.

and b may have opposite sign only when a is negative which was out of question due to given information a>0
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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22 Jul 2015, 05:31
GMATinsight wrote:

GMATinsight, the text in red above is not necessarily true.

|a-b| = b-a ONLY when a-b <0 or b>a . now b > a by being (3,-1) (opposite signs) or by (-2,-4) or by (3,1) (in these 2 cases, same signs)

a and b will always have same sign for |a – b| = b – a to be true for given a >0

The word "also" was disturbing the meaning probably, so removed it.

and b may have opposite sign only when a is negative which was out of question due to given information a>0

Yes, exactly. To me, you were mentioning about a>0 as an additional information after the statement. This seemed incorrect. Thanks for the clarification.
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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22 Jul 2015, 11:52
1
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

|a-b| = -(a-b), which means that a-b is <=0.
Since a>0, then b>a>0.

let a = 1 and b = 2

|a + 3| + |-b| + |b – a| + |ab| = 4+2+1+2 = 9

Sub the value of a and b in answer options

E. 9

Hence Option E
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Posts: 2067
Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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23 Jul 2015, 06:22
Engr2012 wrote:
GMATinsight wrote:
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

Such Question require an intense observation of the given Information step-by-step

Observation-1: |a – b| = b – a

which is possible only when signs of a and b are Same
Also, Since Given a > 0 so we figure out that a and b are both positive

Observation-2: |a – b| must be Non-Negative and so should be the value of b-a which is possible only when absolute value of b is greater than or equal to absolute value of a

Now you may choose the values of a and b based on above observations

e.g. b = 3 and a=1 and check the value of given functions and options

|a + 3| + |-b| + |b – a| + |ab| = |1 + 3| + |-3| + |3 – 1| + |1*3| = 12

GMATinsight, the text in red above is not necessarily true.

|a-b| = b-a ONLY when a-b <0 or b>a . now b > a by being (3,-1) (opposite signs) or by (-2,-4) or by (3,1) (in these 2 cases, same signs)

Hi Engr2012,
I can understand that both should have the same signs. But I think the example of negative signs should be revered. i.e. (-2,-4) to (-4,-2) to satisfy that b>a as -2>-4. I think there is a typo.
Math Expert
Joined: 02 Sep 2009
Posts: 53020
Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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26 Jul 2015, 11:12
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

800score Official Solution:

We know that |a – b| = b – a, so b – a > 0, or b > a. We also know that a > 0, so b > 0.

Knowing that both a and b are positive we can easily simplify:
|a + 3| + |-b| + |b – a| + |ab| =
a + 3 + b + b – a + ab =
ab + 2b + 3

The right answer is choice (E).
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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27 Jul 2015, 09:43
Mo2men wrote:

Hi Engr2012,
I can understand that both should have the same signs. But I think the example of negative signs should be revered. i.e. (-2,-4) to (-4,-2) to satisfy that b>a as -2>-4. I think there is a typo.

Yes, thanks. It was a typo.
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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27 Jul 2015, 16:37
1
Hi All,

While this prompt looks complex, it can be solved rather easily by TESTing VALUES.

We're told that |A-B| = B-A. While there are LOTS of values that will fit this equation, the easiest 'option' is to make A=B. We're also told that A > 0.

IF...
A=1
B=1

We're asked for the value of |A+3| + |-B| + |B-A| + |AB|...

|1+3| +|-1| + |0| + |1| = 6

Answer A: -AB + 3 = 2 NOT a match
Answer B: AB + 3 = 4 NOT a match
Answer C: -AB + 2B + 3 = 4 NOT a match
Answer D: AB + 2B – 2A – 3 = -2 NOT a match
Answer E: AB + 2B + 3 = 6 This IS a MATCH

GMAT assassins aren't born, they're made,
Rich
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Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 13562 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + [#permalink] ### Show Tags 27 Jul 2015, 16:43 1 Bunuel wrote: Bunuel wrote: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| = A. -ab + 3 B. ab + 3 C. -ab + 2b + 3 D. ab + 2b – 2a – 3 E. ab + 2b + 3 Kudos for a correct solution. 800score Official Solution: We know that |a – b| = b – a, so b – a > 0, or b > a. We also know that a > 0, so b > 0. Knowing that both a and b are positive we can easily simplify: |a + 3| + |-b| + |b – a| + |ab| = a + 3 + b + b – a + ab = ab + 2b + 3 The right answer is choice (E). Hi All, This explanation is not completely correct. A and B can be the SAME value. Be mindful of these types of Number Property details that you might miss - while it would not impact your solution to this particular question, it could cost you points in other areas (especially in DS). GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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27 Jul 2015, 16:49
EMPOWERgmatRichC wrote:
Bunuel wrote:
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

800score Official Solution:

We know that |a – b| = b – a, so b – a > 0, or b > a. We also know that a > 0, so b > 0.

Knowing that both a and b are positive we can easily simplify:
|a + 3| + |-b| + |b – a| + |ab| =
a + 3 + b + b – a + ab =
ab + 2b + 3

The right answer is choice (E).

Hi All,

This explanation is not completely correct. A and B can be the SAME value. Be mindful of these types of Number Property details that you might miss - while it would not impact your solution to this particular question, it could cost you points in other areas (especially in DS).

GMAT assassins aren't born, they're made,
Rich

Hi EMPOWERgmatRichC

But I have a question about your statement "A and B can be the SAME value".

IMHO, this is not true. We have been given that |a – b| = -(a-b) . This is similar to |x| = -x for all x<0. We do not include equality with the negative value. Please clarify.
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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27 Jul 2015, 17:02
Hi Engr2012,

The easiest way to prove the point is to TEST values for A and B that are the SAME.... You can either use physical values or substitute in a new variable....

IF...
A=B=2

|A-B| = B-A
|2-2| = 0
2-2 = 0
0 = 0

Thus, we have proof that A CAN = B

Or, using a variable....
A=B=X

|X-X| = X-X
|X-X| = 0
X-X = 0
0 = 0

Again, proof that A CAN = B

GMAT assassins aren't born, they're made,
Rich
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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07 Jul 2017, 05:07
Hi balamoon, I have a ques on the step below:
Solution -
Given that |a – b| = b – a, so b – a > 0, or b > a. Also given that a > 0, so b > 0.

|a + 3| + |-b| + |b – a| + |ab| =
a + 3 + b + b – a + ab =ab + 2b + 3

here |-b| becomes just b. How did that happen? The rest of the modulus openings retained thier signs but here it changed. Can anyone explain how?

Thank you !

ANS E
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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08 Aug 2017, 06:58
balamoon wrote:
Bunuel wrote:
For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| + |-b| + |b – a| + |ab| =

A. -ab + 3
B. ab + 3
C. -ab + 2b + 3
D. ab + 2b – 2a – 3
E. ab + 2b + 3

Kudos for a correct solution.

Solution -
Given that |a – b| = b – a, so b – a > 0, or b > a. Also given that a > 0, so b > 0.

|a + 3| + |-b| + |b – a| + |ab| =
a + 3 + b + b – a + ab =ab + 2b + 3
ANS E

why b-a>0
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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08 Aug 2017, 07:49
Hi balamoon, I have a ques on the step below:
Solution -
Given that |a – b| = b – a, so b – a > 0, or b > a. Also given that a > 0, so b > 0.

|a + 3| + |-b| + |b – a| + |ab| =
a + 3 + b + b – a + ab =ab + 2b + 3

here |-b| becomes just b. How did that happen? The rest of the modulus openings retained thier signs but here it changed. Can anyone explain how?

Thank you !

ANS E

This is because b>a>0; If b = 3, -b = -3 & |-3|= 3
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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13 Sep 2017, 09:41
I interpreted it like this:

|a-b|= b-a

This can also be written as:
|b-a|= b-a

That means b-a is positive. (a>0)
b-a>0
b>0

Rest is same.
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Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +  [#permalink]

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13 Oct 2017, 08:41
Bunuel wrote:
For any a and b that satisfy $$|a – b| = b – a$$ and $$a > 0$$, then $$|a + 3| + |-b| + |b – a| + |ab| =$$

A. $$-ab + 3$$

B. $$ab + 3$$

C. $$-ab + 2b + 3$$

D. $$ab + 2b – 2a – 3$$

E. $$ab + 2b + 3$$

Kudos for a correct solution.

Given |a-b| = b - a
b>a, for |a-b| = positive
a>0
therefore, a+3>0

Let a = 1, b=2
4+ 2 + 1 + 2 = 9
ab + 2b + 3 = 2+4+3 = 9

Therefore, (E) ab + 2b + 3
Re: For any a and b that satisfy |a – b| = b – a and a > 0, then |a + 3| +   [#permalink] 13 Oct 2017, 08:41

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