# Systems of Linear Inequalities by A. S. Solodovnikov

By A. S. Solodovnikov

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Extra info for Systems of Linear Inequalities

Sample text

They form a feasible set (they are found by substituting x? , x~ all compatible values of the unknown X3 (they are found by substituting x? and x~ into the system (5(11 - )))) and so on. , (s

Consider an arbitrary system of linear inequalities. For notational convenience consider for the time being that the number of unknowns is three, although everything stated above equally applies to systems in any number of unknowns. So we are given the system (Itx+hly+ctz+dl ~O (l2 X + h 2 J' + C2 Z + £1 2 ~ 0 (1) a,»: + hillY + CII/Z + diu ~ 0 Multiply both sides of the inequality of (I) by a nonnegative number k 1 , both sides of the second inequality by a nonnegative number k 2 and so 011. and then add the obtained inequalities together to come as a result to the inequality (kta!

K, The set (A 1. A 2 ,· ... A 1'), the convex hull of the system of points AI' A 2 , .. ). But in that case. according to Lemma 1, the set