When is the absolute value of a difference of the absolute values?

The absolute value of a difference of the absolute values, (| |a| - |b| |), is equal to the absolute value of the difference of the numbers, (|a - b|), when (a) and (b) have the same sign (both positive or both negative). In other cases, it may not hold true. Thus, the equality (| |a| - |b| | = |a - b|) is generally true when (a) and (b) are either both non-negative or both non-positive.