We present a mean-field solution of the dynamics of a Greenberg-Hastings neural network with both excitatory and inhibitory units. We analyze the dynamical phase transitions that appear in the stationary state as the model parameters are varied. Analytical solutions are compared with numerical simulations of the microscopic model defined on a fully connected network. We found that the stationary state of this system exhibits a first-order dynamical phase transition (with the associated hysteresis) when the fraction of inhibitory units f is smaller than some critical value f_{t}≲1/2, even for a finite system. Moreover, any solution for f<1/2 can be mapped to a solution for purely excitatory systems (f=0). In finite systems, when the system is dominated by inhibition (f>f_{t}), the first-order transition is replaced by a pseudocritical one, namely a continuous crossover between regions of low and high activity that resembles the finite size behavior of a continuous phase transition order parameter. However, in the thermodynamic limit (i.e., infinite-system-size limit), we found that f_{t}→1/2 and the activity for the inhibition dominated case (f≥f_{t}) becomes negligible for any value of the parameters, while the first-order transition between low- and high-activity phases for f<f_{t} remains.