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    <p>It's late enough that there are probably a number of slick reductions, but this works...</p> <pre> ineq={...}; pivotAt[set_, j_] := Select[set, And[ Not[FreeQ[#, x[u_] /; u &lt;= j]], FreeQ[#, x[u_] /; u > j] ] &] triangularize[set_] := Module[{left, i, new}, left = set; Reap[ For[i = 0, i &lt;= 21, i++, new = pivotAt[left, i]; Sow[new]; left = Complement[left, new]; ]][[2, 1]] ] Module[{ tri, workingIntervals, partials, increment, i }, tri = triangularize[ineq]; workingIntervals[set_] := set /. { t_ &lt;= c_ :> {t, Interval[{-\[Infinity], Max[c]}]}, t_ == c_ :> {t, Interval[{Min[c], Max[c]}]}, t_ >= c_ :> {t, Interval[{Max[c], \[Infinity]}]}}; partials = {}; increment[slice_] := Rule[#[[1, 1]], IntervalIntersection @@ #[[All, 2]]] &[ workingIntervals[slice /. partials ] ]; For[i = 1, i &lt;= Length[tri], i++, partials = Join[partials, {increment[tri[[i]]]}]; ]; partials ] </pre> <p>It's permissive in that correlations between variables ("this high means that low") are not accounted.</p> <p>-- EDIT --</p> <p>The result of the above is, of course</p> <pre> {x[0] -> Interval[{3, 3}], x[1] -> Interval[{1, 3}], x[2] -> Interval[{1, 3}], x[3] -> Interval[{1, 3}], x[4] -> Interval[{1, 3}], x[5] -> Interval[{1, 6}], x[6] -> Interval[{1, 6}], x[7] -> Interval[{1, 6}], x[8] -> Interval[{1, 6}], x[9] -> Interval[{1, 6}], x[10] -> Interval[{1, 6}], x[11] -> Interval[{1, 6}], x[12] -> Interval[{1, 6}], x[13] -> Interval[{1, 6}], x[14] -> Interval[{1, 10}], x[15] -> Interval[{1, 10}], x[16] -> Interval[{1, 10}], x[17] -> Interval[{1, 16}], x[18] -> Interval[{1, 16}], x[19] -> Interval[{1, 16}], x[20] -> Interval[{1, 16}], x[21] -> Interval[{1, 1}]} </pre>
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