\(b<0\), let's assume b=-k where k>0

\(15<|b-11|<16\)
putting b=-k we get
\(15<|-k-11|<16\) = \(15<|-(k+11)|<16\)
And we know that |-x| = |x|
=> \(15<|k+11|<16\)
=> \(15<k+11<16\) [ As k+11>0 and we know that |x| = x if x>0]
Subtracting 11 from all sides we get
=> \(15-11 < k+11-11 <16-11\)
=> \(4 < k < 5\)
Putting k =-b
=> \(4 < -b < 5\)
Multiplying with -1 on all sides and reversing the inequalities we get
=> \(-4 > b > -5\)
=> \(-5 < b < -4\)
So, b can be any number between -5 and -4

Quantity A = |b|
Since b lies between -5 and -4 => |b| will lie between |-5| and |-4| = 5 and 4

Quantity B = 4.5

Definitely, |b| can be > , = or < 4.5
=> we need to know the exact value of b to answer the problem

So, answer will be D
Hope it helps!

To learn about Absolute value and Inequalities watch the following videos