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Let $aøplusb=max(a,b)$ and $aøtimesb=a+b$ for $a,b\in ℝ$. Max-algebra is an analogue of linear algebra developed on the pair of operations $(øplus,øtimes)$ extended to matrices and vectors. The system of equations $Aøtimesx=b$ and inequalities $Cøtimesxłeqd$ have each been studied in the literature. We consider a problem consisting of these two systems and present necessary and sufficient conditions for its solvability. We also develop a polynomial algorithm for solving max-linear program whose constraints are max-linear equations and inequalities.