In my sketch on paper of the graph of

y = 3 +|sin x|,

the range is 3 <= y <= 4. Or, y is between 3 and 4, inclusive.

The

|sin x| = k -3

can be re-written as

k = 3 +|sin x|

For that to not have real roots, the curve must not touch the x-axis.

So k must not be anywhere from y=3 to y=4, inclusive.

Therefore, (k < 3) U (k > 4) ----------answer.