By sketching the graph of y=3+ | sin x | for x is greater or equal to zero and smaller or equal to 2 pi , find the range of values of k such that the equation | sin x | = k-3 does not have any real root .
y = 3 +|sin x|,
the range is 3 <= y <= 4. Or, y is between 3 and 4, inclusive.
|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.