In this paper we present relatively simple proofs of the following new upper bounds: c^N, where c < 2 is a constant and N is the number of variables, for MAX-SAT for formulas with constant clause density; 2 ^K/6, where K is the number of clauses, for MAX-2-SAT; 2 ^N/6.7 for (n, 3)-MAX-2-SAT. All bounds are proved by the splitting method. The main two techniques are combined complexity measures and clause learning.