This paper investigates the achievable rate region of the multiple-input single-output (MISO) interference channel aided by intelligent reflecting surfaces (IRSs). We exploit the the additional design degree of freedom provided by the coordinated IRSs to enhance the desired signal and suppress interference so as to enlarge the achievable rate region of the interference channel. To this end, we jointly optimize the active transmit beamforming at the transmitters and passive reflective beamforming at the IRSs, subject to the constant modulus constraints of reflective beamforming vectors. To address the non-convex optimization problem, we propose an iterative algorithm to optimize the transmit beamforming via second-order cone program (SOCP) and the reflective beamforming via the semi-definite relaxation (SDR). Numerical results demonstrate that the performance of the IRS-aided interference channel with the proposed algorithm can significantly outperform the conventional interference channel without IRS.