I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions or show that there are no solution for such a system using Sage.

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP:

W=[ [ [ 1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7 ], [ 1/4*x3+1/4*x4+1/4*x5+1/4*x7 ] ], [ [ 1/7*x2+2/7*x3+2/7*x4+1/7*x5+1/7*x7 ], [ 1/3*x2+1/3*x3+1/3*x4 ] ],
[ [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+1/11*x6+1/11*x7 ], [ 1/8*x1+1/8*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x6+1/8*x7 ] ],
[ [ 1/13*x1+2/13*x2+3/13*x3+2/13*x4+2/13*x5+1/13*x6+2/13*x7 ], [ 1/5*x2+1/5*x3+1/5*x4+1/5*x5+1/5*x7 ] ], [ [ 1/9*x1+2/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x7 ], [ 1/4*x1+1/4*x2+1/4*x3+1/4*x4 ] ],
[ [ 1/6*x1+1/6*x2+1/3*x3+1/6*x4+1/6*x7 ], [ 1/2*x3+1/2*x7 ] ], [ [ 1/3*x2+1/3*x3+1/3*x7 ], [ x2 ] ], [ [ 1/2*x1+1/2*x2 ], [ x1 ] ], [ [ 1/4*x3+1/4*x4+1/4*x5+1/4*x6 ], [ 1/3*x3+1/3*x4+1/3*x5 ] ],
[ [ 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x5, 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ] ],
[ [ 1/13*x1+2/13*x2+3/13*x3+3/13*x4+2/13*x5+1/13*x6+1/13*x7 ], [ 1/7*x2+2/7*x3+2/7*x4+1/7*x5+1/7*x7, 1/9*x1+1/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x6+1/9*x7,
1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/6*x5+1/12*x6+1/12*x7 ] ],
[ [ 1/15*x1+2/15*x2+4/15*x3+1/5*x4+2/15*x5+1/15*x6+2/15*x7 ], [ 1/6*x2+1/3*x3+1/6*x4+1/6*x5+1/6*x7, 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+1/11*x6+1/11*x7,
1/14*x1+1/7*x2+3/14*x3+3/14*x4+1/7*x5+1/14*x6+1/7*x7 ] ],
[ [ 1/16*x1+3/16*x2+1/4*x3+3/16*x4+1/8*x5+1/16*x6+1/8*x7 ], [ 1/13*x1+2/13*x2+3/13*x3+2/13*x4+2/13*x5+1/13*x6+2/13*x7, 1/10*x1+1/5*x2+1/5*x3+1/5*x4+1/10*x5+1/10*x6+1/10*x7,
1/10*x1+1/5*x2+3/10*x3+1/5*x4+1/10*x5+1/10*x7 ] ],
[ [ 2/17*x1+3/17*x2+4/17*x3+3/17*x4+2/17*x5+1/17*x6+2/17*x7 ], [ 1/9*x1+2/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x7, 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/12*x5+1/12*x6+1/6*x7,
1/7*x1+1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x7 ] ],
[ [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+2/11*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x7, 1/6*x1+1/6*x2+1/3*x3+1/6*x4+1/6*x7, 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ] ],
[ [ 1/7*x1+2/7*x2+2/7*x3+1/7*x4+1/7*x7 ], [ 1/3*x2+1/3*x3+1/3*x7, 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7, 1/3*x1+1/3*x2+1/3*x3 ] ], [ [ 1/4*x1+1/4*x2+1/4*x3+1/4*x7 ], [ x7, 1/2*x1+1/2*x2 ] ],
[ [ 1/3*x4+1/3*x5+1/3*x6 ], [ 1/2*x4+1/2*x5 ] ], [ [ 1/3*x3+1/3*x4+1/3*x5 ], [ 1/2*x3+1/2*x4 ] ], [ [ 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ], [ 1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ] ],
[ [ 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/6*x5+1/12*x6+1/12*x7 ], [ 1/11*x1+2/11*x2+2/11*x3+2/11*x4+2/11*x5+1/11*x6+1/11*x7 ] ],
[ [ 1/14*x1+1/7*x2+3/14*x3+3/14*x4+1/7*x5+1/14*x6+1/7*x7 ], [ 1/8*x1+1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x7 ] ], [ [ 1/10*x1+1/5*x2+3/10*x3+1/5*x4+1/10*x5+1/10*x7 ], [ 1/5*x2+2/5*x3+1/5*x4+1/5*x7 ] ],
[ [ 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/12*x5+1/12*x6+1/6*x7 ], [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+2/11*x7, 1/9*x1+2/9*x2+2/9*x3+1/9*x4+1/9*x5+1/9*x6+1/9*x7 ] ],
[ [ 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ], [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7, 1/7*x1+2/7*x2+2/7*x3+1/7*x4+1/7*x7 ] ],
[ [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7 ], [ 1/4*x1+1/4*x2+1/4*x3+1/4*x7 ] ], [ [ 1/2*x5+1/2*x6 ], [ x5 ] ], [ [ 1/2*x4+1/2*x5 ], [ x4 ] ], [ [ 1/2*x3+1/2*x4 ], [ x3 ] ],
[ [ 1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ], [ 1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x6+1/6*x7 ] ],
[ [ 1/11*x1+2/11*x2+2/11*x3+2/11*x4+2/11*x5+1/11*x6+1/11*x7 ], [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x5 ] ], [ [ 1/8*x1+1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x7 ], [ 1/3*x3+1/3*x4+1/3*x7 ] ],
[ [ 1/5*x2+2/5*x3+1/5*x4+1/5*x7 ], [ 1/2*x2+1/2*x3 ] ],
[ [ 1/9*x1+2/9*x2+2/9*x3+1/9*x4+1/9*x5+1/9*x6+1/9*x7 ], [ 1/7*x1+1/7*x2+1/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7, 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ] ],
[ [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7 ], [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7 ] ], [ [ 1/7*x1+1/7*x2+1/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ], [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7 ] ] ]

This is a list of lists.
The first entry of this list is for example [ [ 1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7 ], [ 1/4*x3+1/4*x4+1/4*x5+1/4*x7 ] ]

and the tenth entry is:
[ [ 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x5, 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (the number of lists in W) and for all k (the length of the list W[i][2]).

So in the above examples the inequalities are 1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7>= 1/4*x3+1/4*x4+1/4*x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 >= 1/4*x2+1/4*x3+1/4*x4+1/4*x5 and 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 >=1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7.