One way to show that checking the feasibility of a linear system of inequalities is as hard as linear programming is via the reduction given by the ellipsoid method. An even easier way is to guess the optimal solution and introduce it as a constraint via binary search. Both of these reductions are polynomial, but not strongly polynomial (i.e they depend on the number of bits in the coefficients of the inequalities). Is there a strongly polynomial reduction from LP optimization to LP  feasibility ?
Equivalence of feasibility checking and optimization for linear systems
ds.algorithms;linear programming