In this article, we investigate the solution of a new class of the absolute value equation (NCAVE) A1x − |B1x − c| = d. Based on spectral radius condition, singular value condition and row and column W-property, some necessary and sufficient conditions for unique solvability for NCAVE are gained. Some new results for the unique solvability of the new generalized absolute value equation (NGAVE) A1x − |B1x| = d are also obtained.