The dynamics of a spin-1 Ising system containing biquadratic interactions near equilibrium states is formulated by the method of thermodynamics of irreversible processes. From the expression for the entropy production, generalized forces and fluxes are determined. The kinetic equations are obtained by introducing kinetic coefficients that satisfy the Onsager relation. By solving these equations a set of relaxation times is calculated and examined for temperatures near the phase transition temperatures, with the result that one of the relaxation times approaches infinity near the second-order phase transition temperature on either side, whereas it is sharply cusped at the first-order phase transition temperature. On the other hand, the other relaxation time has a cusp at the second-order phase transition temperature but displays a different behavior at the first-order phase transition, just a jump discontinuity. The behavior of both relaxation times is also investigated at the tricritical point. Moreover, the phase transition behaviors of the relaxation times are also obtained analytically via the critical exponents. Results are compared with conventional kinetic theory in the random-phase or generalized molecular-field approximation and a very good overall agreement is found.