This works, but with your >= becoming > and < becoming <=

$start_minute = int(($start_minute + 7.5)/15)/4;
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Let's break this down:

$start_minute + 7.5 # add 7.5 mins to number
/15                 # which quarter is the minute in?
int()               # ...0,1,2,3 or 4?
/4                  # to get desired result
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cLive ;-)

Update: oops, tye beat me to it (in a slightly different, but effectively the same, way)... :)