We study the moving bright solitons in the weak attractive Bose-Einstein condensate with a spin-orbit interaction.By solving the coupled nonlinear Schr(o)dinger equation with the variational method and the imaginary time evolution method,two kinds of solitons (plane wave soliton and stripe solitons) are found in different parameter regions.It is shown that the soliton speed dominates its structure.The detuning between the Raman beam and energy states of the atoms decides the spin polarization strength of the system.The soliton dynamics is also studied for various moving speed and we find that the shape of individual components can be kept when the speed of soliton is low.