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

Question

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

Bunuel · Accepted Answer

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 implies that a ≤ 0. Since a is non-zero, this gives a < 0.
|ab| = ab implies that ab ≥ 0. Since both a and b are non-zero, this gives ab > 0.
Given a < 0 and ab > 0, we can further infer that b < 0.

Now, let's evaluate |b - 4| and |ab - b|:

Since b is negative, b - 4 = negative - 4 = negative, so |b - 4| = -(b - 4) = 4 - b.
Since ab is positive and b is negative, ab - b = positive - negative = positive + positive = positive. Therefore, |ab - b| = ab - b.

Thus, |b - 4| + |ab - b| = (4 - b) + (ab - b) = ab - 2b + 4.

Answer: D.

Hope it helps.

subhashghosh · Answer

|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

gamezfreak · Answer

Thanks Bunnel...very good explanation

MSoS · Answer

Thanks! It is clear now

mbaiseasy · Answer

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!

Answer: D

SVaidyaraman · Answer

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

GMATinsight · Answer

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

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

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