It is also worth noticing that if the pairs consist of approximate numbers rounded values can be used to gather pairs whose first elements are close by some designed value:
In[4]:= realPairs = RandomReal[{1, 10}, {30, 2}]
Out[4]= {{5.21667, 6.47609}, {9.31783, 1.36685}, {1.64424,
5.90676}, {1.63877, 4.69312}, {8.20514, 9.13797}, {1.43469,
7.53501}, {7.32208, 7.86721}, {9.79302, 3.65095}, {3.37067,
1.09812}, {4.44002, 8.87933}, {2.05523, 8.75433}, {1.15523,
3.88057}, {1.87054, 2.36079}, {7.08747, 7.347}, {1.53206,
2.1205}, {6.06558, 6.74602}, {9.85321, 1.59023}, {8.23373,
8.98786}, {2.92257, 5.85836}, {4.70867, 9.13353}, {9.7615,
7.22106}, {2.21636, 5.98876}, {9.25288, 9.94328}, {8.56543,
6.00081}, {9.32182, 2.52546}, {2.91193, 3.88105}, {2.05716,
9.22982}, {2.02069, 6.72177}, {5.95095, 1.592}, {9.22202, 3.78433}}
In[6]:= GatherBy[realPairs, Round[#[[1]], .5] &]
Out[6]= {{{5.21667, 6.47609}}, {{9.31783, 1.36685}, {9.25288,
9.94328}, {9.32182, 2.52546}}, {{1.64424, 5.90676}, {1.63877,
4.69312}, {1.43469, 7.53501}, {1.53206, 2.1205}}, {{8.20514,
9.13797}, {8.23373, 8.98786}}, {{7.32208, 7.86721}}, {{9.79302,
3.65095}, {9.85321, 1.59023}, {9.7615, 7.22106}}, {{3.37067,
1.09812}}, {{4.44002, 8.87933}, {4.70867, 9.13353}}, {{2.05523,
8.75433}, {1.87054, 2.36079}, {2.21636, 5.98876}, {2.05716,
9.22982}, {2.02069, 6.72177}}, {{1.15523, 3.88057}}, {{7.08747,
7.347}}, {{6.06558, 6.74602}, {5.95095, 1.592}}, {{2.92257,
5.85836}, {2.91193, 3.88105}}, {{8.56543, 6.00081}}, {{9.22202,
3.78433}}}
