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Since everything is linear your solution set is a polyhedron. For simplicity, lets take it closed (use <= instead of <). Then:

sage: P = Polyhedron(eqns=[(-10,1,1,1)], ieqs=[(0,-1,1,0), (0,0,-1,1)])
sage: P.Hrepresentation()
(An equation (1, 1, 1) x - 10 == 0,
 An inequality (-1, 1, 0) x + 0 >= 0,
 An inequality (-1, -2, 0) x + 10 >= 0)

Since everything is linear your solution set is a polyhedron. For simplicity, lets take it closed (use <= instead of <). Then:

sage: P = Polyhedron(eqns=[(-10,1,1,1)], ieqs=[(0,-1,1,0), (0,0,-1,1)])
sage: P.Hrepresentation()
(An equation (1, 1, 1) x - 10 == 0,
 An inequality (-1, 1, 0) x + 0 >= 0,
 An inequality (-1, -2, 0) x + 10 >= 0)

sage: P = Polyhedron(eqns=[(-10,1,1,1)], ieqs=[(0,-1,1,0), (0,0,-1,1), (0,1,0,0)])
sage: P.Hrepresentation()
(An equation (1, 1, 1) x - 10 == 0,
 An inequality (-1, 1, 0) x + 0 >= 0,
 An inequality (-1, -2, 0) x + 10 >= 0,
 An inequality (1, 0, 0) x + 0 >= 0)
sage: P.integral_points()
((0, 0, 10),
 (0, 1, 9),
 (0, 2, 8),
 (0, 3, 7),
 (0, 4, 6),
 (0, 5, 5),
 (1, 1, 8),
 (1, 2, 7),
 (1, 3, 6),
 (1, 4, 5),
 (2, 2, 6),
 (2, 3, 5),
 (2, 4, 4),
 (3, 3, 4))