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# For any non-zero a and b that satisfy |ab| = ab and |a| = -a

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For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink]

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05 Feb 2012, 20:26
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75% (hard)

Question Stats:

51% (02:29) correct 49% (01:36) wrong based on 475 sessions

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For any non-zero a and b that satisfy |ab| = ab and |a| = -a, |b-4| + |ab-b| =

A. ab-4
B. 2b-ab-4
C. ab+4
D. ab-2b+4
E. 4-ab
[Reveal] Spoiler: OA

Last edited by Bunuel on 04 Dec 2012, 02:36, edited 1 time in total.
Renamed the topic and edited the question.
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06 Feb 2012, 01:23
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gmatfrenzy750 wrote:
Can someone please assist me in this questions:

For any non a and b that satisfy |ab| = ab and |a| = -a

|b-4|+ |ab-b| =

a) ab-4
b) 2b-ab-4
c) ab+4
d) ab-2b+4
e) 4-ab

Can you please check the question: I guess it should read "for any non zero a and b"

$$|a|=-a$$ means that $$a<0$$ and $$|ab|=ab$$ means that $$ab>0$$, so they have the same sign and since $$a<0$$ then $$b<0$$ too.

So, we have $$a<0$$ and $$b<0$$.

Now, $$b-4=b+(-4)=negative+negative =negative$$, so $$|b-4|=-(b-4)$$;
$$ab-b=positive-negative=positive+positive=positive$$, so $$|ab-b|=+(ab-b)$$;

Hence $$|b-4|+ |ab-b| =-(b-4)+(ab-b)=ab-2b+4$$.

For more on this topic check Absolute Value chapter of Math Book: math-absolute-value-modulus-86462.html

Hope it helps.
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Re: For any non-zero a and b that satisfy |ab|=ab and |a|=-a [#permalink]

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06 Feb 2012, 01:46
Thanks Bunnel...very good explanation
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Re: For any non-zero a and b that satisfy |ab|=ab and |a|=-a [#permalink]

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06 Feb 2012, 07:08
Thanks! It is clear now
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Re: For any non-zero a and b that satisfy |ab|=ab and |a|=-a [#permalink]

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04 Dec 2012, 02:30
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Given: |ab| = ab and |a| = -a
Question: |b-4| + |ab-b| = ?

**** Looking at |ab| = ab tells us that a and b are either both positive or negative
**** Looking at |a| = -a tells us that a must be negative
**** Combine two observations: a and b are both negative values

Let a=-1 and b=-1
|b-4| + |ab-b| = |-1-4| + |1-(-1)| = 7

Test a) ab-4 = (-1)(-1)-4 = -3
Test b) 2b-ab-4 = (2)(-1) - (1) - 4 = -7
Test c) ab+4 = 1 + 4 = 5
Test d) ab-2b+4 = 1-2(-1)+4=7 BINGO!

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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink]

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04 Jul 2013, 01:44
Expert's post
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink]

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04 Jul 2013, 05:37
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gmatfrenzy750 wrote:
For any non-zero a and b that satisfy |ab| = ab and |a| = -a, |b-4| + |ab-b| =

A. ab-4
B. 2b-ab-4
C. ab+4
D. ab-2b+4
E. 4-ab

We can also plug in values in this question

Given that |a|=-a so we we know a<0 and also |ab|=ab means b<0 because if b>0 then |ab|=-ab

take any negative value of a and b and check which option gives the same value as the above Mod eqn

a=-3, b=-2

|b-4| + |ab-b|

|-6|+ |6- (-2)|------> 6+8=14

a=-1 b =-5

|-9| + |5 - (-5)|= 19

Only Option D satisfies and hence is the answer

Thanks
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink]

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04 Jul 2013, 06:11
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gmatfrenzy750 wrote:
For any non-zero a and b that satisfy |ab| = ab and |a| = -a, |b-4| + |ab-b| =

A. ab-4
B. 2b-ab-4
C. ab+4
D. ab-2b+4
E. 4-ab

1.We need to find whether ab and b are -ve or +ve
2. Since |ab|=ab, ab is +ve
3. Since |a|=-a, a is -ve.
4. From (2) and (3), b is -ve.
5.Since b is -ve b-4 is -ve and so |b-4| becomes -(b-4)
6. Since ab is +ve and b is -ve, ab-b is +ve and |ab-b| becomes ab-b
7. (5) + (6) = -b+4+ab-b= ab-2b+4
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink]

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09 Jul 2013, 16:49
For any non-zero a and b that satisfy |ab| = ab and |a| = -a, |b-4| + |ab-b| =

|ab| = ab and |a| = -a, |b-4| + |ab-b| =

From |a| = -a we get that a is negative because:
|a| = -a and -a must be positive as it is set to an absolute value so:
|a| = -(-a)
a=a

If a is negative then from |ab| = ab we get that b must be negative as well because:
|ab| = (-a)b
(-a)b must be positive as it is set to an absolute value
|ab| = (-a)(-b)
ab=ab

So, both a and b are negative.
|b-4| + |ab-b| =

b is negative and ab is positive. Also, because b is negative we know that (ab-b) = (ab-[-b]) = (ab+b)

-(b-4) + (ab-b) =
-b+4 + ab-b

ab-2b+4

(D)
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink]

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21 Jul 2014, 07:40
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink]

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02 Nov 2015, 11:34
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a   [#permalink] 02 Nov 2015, 11:34
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