# Solving linear inequality systems using Sage

Given a matrix A with rational entries, for example the 36x7 matrix:

```
A=[ [ 1/10, 1/10, -1/20, -1/20, -1/20, 1/10, -3/20 ], [ 0, -4/21, -1/21, -1/21, 1/7, 0, 1/7 ], [ -3/88, 5/88, 1/44, 5/88, -3/88, -3/88, -3/88 ], [ 1/13, -3/65, 2/65, -3/65, -3/65, 1/13, -3/65 ],
[ -5/36, -1/36, -1/36, -1/36, 1/9, 0, 1/9 ], [ 1/6, 1/6, -1/6, 1/6, 0, 0, -1/3 ], [ 0, -2/3, 1/3, 0, 0, 0, 1/3 ], [ -1/2, 1/2, 0, 0, 0, 0, 0 ],
[ 0, 0, -1/12, -1/12, -1/12, 1/4, 0 ], [ 0, -5/36, -1/36, -1/36, -1/36, 1/9, 1/9 ], [ 1/13, 1/91, -5/91, -5/91, 1/91, 1/13, -6/91 ], [ 1/15, -1/30, -1/15, 1/30, -1/30, 1/15, -1/30 ],
[ -3/208, 7/208, 1/52, 7/208, -3/104, -3/208, -3/104 ], [ 1/153, -7/153, 2/153, -7/153, 1/153, 1/17, 1/153 ], [ 1/11, -3/44, 1/44, -3/44, 1/11, 0, -3/44 ], [ 1/7, -1/21, -1/21, 1/7, 0, 0, -4/21 ],
[ 1/4, 1/4, 1/4, 0, 0, 0, -3/4 ], [ 0, 0, 0, -1/6, -1/6, 1/3, 0 ], [ 0, 0, -1/6, -1/6, 1/3, 0, 0 ], [ 0, -1/56, -1/28, 3/28, -1/56, -1/56, -1/56 ],
[ -1/132, -1/66, 3/44, -1/66, -1/66, -1/132, -1/132 ], [ -3/56, 1/56, -1/28, -1/28, 1/56, 1/14, 1/56 ], [ 1/10, 0, -1/10, 0, 1/10, 0, -1/10 ], [ -1/132, -1/66, -1/44, -1/66, -1/132, 1/12, -1/66 ],
[ -1/24, 1/12, 1/12, -1/24, -1/24, 0, -1/24 ], [ -1/20, -1/20, -1/20, 1/5, 0, 0, -1/20 ], [ 0, 0, 0, 0, -1/2, 1/2, 0 ], [ 0, 0, 0, -1/2, 1/2, 0, 0 ],
[ 0, 0, -1/2, 1/2, 0, 0, 0 ], [ 0, -1/42, 5/42, -1/42, -1/42, -1/42, -1/42 ], [ -6/55, -1/55, -1/55, -1/55, -1/55, 1/11, 1/11 ], [ 1/8, 1/8, -1/12, -1/12, 1/8, 0, -5/24 ],
[ 0, -3/10, -1/10, 1/5, 0, 0, 1/5 ], [ -2/63, 5/63, 5/63, -2/63, -2/63, -2/63, -2/63 ], [ -1/30, -1/30, -1/30, -1/30, 1/6, 0, -1/30 ], [ -1/42, -1/42, -1/42, -1/42, -1/42, 1/7, -1/42 ] ]
```

Is there an easy way using Sage (or another computer algebra system) to check whether there is a solution to to the inequality systems Ax>0 (meaning every entry of the vector of Ax is strictly positive) in rational (or even real) numbers?

So the input should be the matrix A and the output should be whether there is a solution of Ax>0 or not in the rationals numbers (and if possible give a solution if there is one).

sage: Polyhedron?

Thank you for that comment. I am not sure what you mean. Can you give an example for the much smaller list W=[ [ [ 1/2

x1+1/2x2 ], [ x2 ] ], [ [ 1/2x2+1/2x3 ], [ x3 ] ], [ [ 1/3x1+1/3x2+1/3x3 ], [ 1/2x2+1/2*x3 ] ] ] using Sage?