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

What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

For instance, you started by trying various permutations of the inputs they've asked you to compare, such as:

. . . . .|3 - 5|

. . . . .|3| - |5|

. . . . .|5 - 3|

. . . . .|5| - |3|

. . . . .|5 - 5|

. . . . .|5| - |5|

You continued with pairs of numbers, as in the above, until you started noticing patterns, and... then what?

Please be complete. Thank you! ;-)