FOR WHICH VALUES OF THE REAL NUMBER X is lx+k=lxl+k, where K is a positive real number?
I think the Question is "Real Values of X such that |X+K|=|X|+K, where K is a Positive Real Number."
Initially we have that |X+K| is less than or equal to |X|+K.
1) If X=0, then |X+K|=|X|+K holds.
2) If X>0, then |X+K|=X+K and |X|+K=X+K since |X|=X where X>0.So |X+K|=|X|+K holds.
**3)**If X<0, then For Example Take K=2 and X=-6. |X+K|=|-6+2|=4 and |X|+K=6+2=8, which are never Equal.
So |X+K|=|X|+K holds good if and only if X is a Non-Negative Real Number.
Hence the Solution Set is the Set of All Non-Negative Real Numbers.