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

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kannn
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For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   08 May 2011, 05:59
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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
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D
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beyondgmatscore
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   08 May 2011, 07:08
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kannn 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


Since \(|a| = -a\), a is negative. Similarly, since \(|ab| = ab\), a and b are the same sign so b is also negative.

Lets see \(|b - 4|\), since b and -4 are both negative \(|b - 4|\) will actually equal \(|b| + 4\)

Similarly, \(|ab-b|\) would equal \(ab + |b|\), so expression becomes \(ab + 2|b| + 4\)

Since \(|b| = -b\), the expression reduces to \(ab -2b + 4\)

So, D is the answer
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   21 Jul 2015, 10:04
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kannn 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


|a| = -a Suggests that 'a' is Negative

|ab| = ab Suggests that 'ab' is Positive

'ab' is Positive Suggests that 'a' and 'b' have same Sign i.e. either both positive or both negative

But since 'a' is Negative therefore 'b' is Negative too.

Since b is negative so |b - 4| = -b+4

Since ab is Positive and b is Negative so |ab - b| = ab - b

i.e. |b - 4| + |ab - b| = -b+4 + ab - b = ab - 2b + 4

Answer: Option D
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   Updated on: 15 May 2011, 21:28
I tried two values

a is negative bcos of the constraint |a| = -a. Since a is negative b is negative bcos of absolute value |ab| = ab constraint. Absolute value is non negative.

Now put a = -1 and b = -2. Only D is left.

Hence the answer is D. I will like to see some algebraic explanations though.

kannn 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

Originally posted by gmat1220 on 08 May 2011, 06:25.
Last edited by gmat1220 on 15 May 2011, 21:28, edited 1 time in total.
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   08 May 2011, 07:30
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|ab| = ab and |a| = -a

=> b is also negative.

So

|b-4| + |ab -b|

= -(b-4) (as b-4 is negative, so expression is totally negative) + ab - b ( as ab is positive and -b is positive, so expression is totally positive)


= 4 - b + ab - b

= ab - 2b + 4

Answer - D
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   08 May 2011, 09:18
using values as both a and b are negative.

a = -1 and b = -2 also a = -2 and b = -3.

gives d.
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   08 May 2011, 13:01
Given that |a|=-a which means a is -ve
Also |ab|=ab which means both a,b are either both -ve or both +ve
since a is negative, b is also negative.

Considering this
|b-4| will always be a -ve expression. So if we remove mod signs, it becomes 4-b
|ab-b| will always be a +ve expression since ab is +ve and b is -ve. To remove mod signs it will be +ve expression. ab-b

So |b-4| + |ab-b| = 4-b+ab-b
= 4-2b+ ab
= D
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   08 May 2011, 14:11
slight variation of @beyondgmatscore approach

|a| = -a => a<0

ab>0 => a<0, b<0

now we have |b-4| + |ab-b|

as b<0, we know |b|=-b =>b= -|b|

given expression = |-|b|-4| +|ab +|b||

=|b|+4+ab+|b|= ab-2b+4 ( as |b|= -b)

Answer is D.
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   04 Aug 2015, 11:30
hey guys, I think I am missing something here...

howcome |a| = -a? I thought that an absolute value can never be negative?
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   04 Aug 2015, 11:45
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RussianDude wrote:
hey guys, I think I am missing something here...

howcome |a| = -a? I thought that an absolute value can never be negative?


There are 3 things when we talk about "absolute values"

1. They can not be negative. Example, |-6| = 6 and |7| = 7, |0| =0 etc.
2. |a| = -a ONLY IF a <0 . So lets say, a =-6, then |-6| = 6 (a POSITIVE) value.
3. |a| = a for a \(\geq\)0

What you are saying is correct that |a| can never be NEGATIVE but as per #2 above, |a| = -a when a < 0 . Thus when a <0 then -a will be >0 (positive).

Hope this helps.
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   04 Aug 2015, 11:49
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RussianDude wrote:
hey guys, I think I am missing something here...

howcome |a| = -a? I thought that an absolute value can never be negative?


-a will be positive for negative values of a

E.g. if a=-2, then |a| = 2= -(-2) = -a

I.e. if|a| = -a then a must be negative
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   04 Aug 2015, 12:08
thanks guys, it is clear now. Kudos for you :-)
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   25 Nov 2015, 19:18
solved it as well through assigning variables
if |a| = -a that means that a is negative and the only case when |ab|=ab is when b is negative as well
suppose b=-3 and a=-2

by plugging in numbers, only D works
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   21 Jun 2019, 00:03
|ab| = ab means that the product of a and b must be positive which is possible only when both are of the same signs.
It is also given that |a|= -a which implies that a is negative, and so b must also be negative.
Now |b - 4| will be negative since b < 0 and |ab - b| will be positive since ab is +ve and b is -ve so |+ve - -ve| = +ve.
So we get -b+4+ab-b = ab-2b+4 => Option D is the right choice.
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   21 Jun 2019, 00:45
If you read the question carefully
Quote:
For any non-zero a and b that satisfy

The underlined part of the above statement tells us that plugging in values for a and b, which satisfy the below conditions, into the five options will reveal the correct answer.
Quote:
|ab| = ab and |a| = -a

Since |a| = -a, a is a negative quantity. Since |ab| = ab, and a is a negative quantity, b must be a negative quantity as well.
The question setter will definitely anticipate a lot of students using this approach, and using -1 and -1. Luckily plugging in [-1,-1] does reveal that option (D) is correct. I'd go (and went) with [-2,-2]
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Re: For any non-zero a and b that satisfy |ab| = ab and |a| = -a [#permalink] New post   06 Jun 2023, 00:08
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