I'd probably approach it like this...
data = {0., 0.533, 0.617, 0.819, 1.131, 1.455, 1.511, 1.7};
x = .56;
BlockMap[Between[x, #] &, data, 2, 1]
(* {False, True, False, False, False, False, False} *)
To get the position of True:
FirstPosition[BlockMap[Between[x, #] &, data, 2, 1], True]
(* {2} *)
Take the first element or apply Sequence, if you want a plain integer.
Here's another alternative:
SplitBy[data, LessThan[x]]
(* {{0., 0.533}, {0.617, 0.819, 1.131, 1.455, 1.511, 1.7}} *)
The length of the first list is what you're after:
Length@First@SplitBy[data, LessThan[x]]
(* 2 *)
Probably should consider how to handle edge cases where x is not within any interval or where the data isn't sorted or etc.