Alternatively one can do it like:
DeleteCases[{{-2, 3}, {0, 3}, {2, 3}, {2, -1}, {-1,1.2}}, {_?(LessThan[1]), _}]
Select[{{-2, 3}, {0, 3}, {2, 3}, {2, -1}, {-1, 1.2}}, !LessThan[1][First[#]] &]
I always tend to favour Select when a function has to be done, and Cases/DeleteCases when it is a 'pure simple pattern' (without Condition or PatternTest).