When is the absolute value of a difference equal to the difference of the absolue values?

The absolute value of a difference is equal to the difference of the absolute values when both values have the same sign. This means that if ( a ) and ( b ) are two real numbers, the equation ( |a - b| = |a| - |b| ) holds true if both ( a ) and ( b ) are non-negative or both are non-positive. In other cases, this equality does not hold.