Start with definition of absolute value, |x|.
It says,
|x| = x, for any x >= 0 and
|x| = -x, for any x < 0
Now, instead of x, we have the expression x+120 inside the absolute value brackets:
|x+120|
Play the same game as above - what is the absolute value here?
|x+120| = x+120 for (x+120)>=0 and
|x+120| = -(x+120) for (x+120)<0
Now, recognize that the last condition (x+120)<0 is the same as writing x<-120. That is the condition in the question. Therefore the absolute value |x+120| will equal the second expression (for that condition), namely -(x+120), or -x-120.