Quote:
How would the equation be exactly transformed given that either a or b or both are negative?
I'd like to stress that this is a
hypothetical situation, because in the question we show that
b >
a > 0.
So let's try the proposed conditions.
a + 3 + b + b – a + ab = ?1)
b < 0
In this case we can simplify 
b = 
b, because 
b > 0. But we can NOT simplify any other absolute value, because we know nothing about
a.
2)
a < 0
In this case we can NOT simplify any absolute value, because

a + 3 can be
a + 3 (if
a is from 3 to 0)
OR

a + 3 can be 
a – 3 (if
a is less than 3)
Any other absolute value contains
b and we know nothing about it.
3)
a < 0 and
b < 0
In this case we can simplify:

b = 
b, because 
b > 0;

ab =
ab, because
ab > 0.
We can NOT simplify:

a + 3 for the same reasons as in 2)

b –
a can be
b –
a (if
b >
a)

b –
a can be 
b +
a (if
b <
a).
You may try to plug some numbers in the absolute values we could NOT simplify:
3)
a < 0 and
b < 0
a = 1 < 0 ,
b = 4 < 0

a + 3 = 1 + 3 = 1 + 3 = 2
in this case we used 
a + 3 =
a + 3, because (
a = 1 is between 3 and 0)

b –
a = 4 – (1) = (4) + (1) = 3
in this case we used 
b –
a = 
b +
a, because (
b = 4 < 1 =
a)
a = 4 < 0 ,
b = 1 < 0

a + 3 = 4 + 3 = (4) – 3 = 1
in this case we used 
a + 3 = 
a – 3, because (
a = 1 is less than 3)

b –
a = 1 – (4) = 1 – (4) = 3
in this case we used 
b –
a =
b –
a, because (
b = 1 > 4 =
a)