The preconditioned Chebyshev iteration is examined in which at each step the linear system involving the preconditioner is solved inexactly by an inner iteration. The tolerance used in the inner iteration is allowed to decrease from one outer iteration to the next. When the tolerance converges to zero, the asymptotic convergence rate is the same as for the exact method. The sequence of tolerance values that yields the lowest cost to achieve a specific accuracy is then determined. Numerical examples illustrate the methodology and verify the results of the paper.