question on proving a language is regular

L is a regular language.
L_x={y | xy in L} where x is some fixed string.

How do I show that L_x is a regular language?
My thoughts so far are to show by constructing a FA.
Suppose we have a FA, call it F that accepts L.
My FA, call it F' would be constructed by copying
F and running as follows...
F' would nondeterministically skip to the q_ith state
of the FA that accepts xy where i is the length of x, |x|.
From there the FA simply runs as normal and accepts y and thus
L_x is regular.
Any advice?
Thanks!
.