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When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4 x < 2 x + 1 ≤ 3 x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction.
Reducing and re-arranging the coefficients by adding multiples of as necessary, we can assume < (in fact, this is the unique such satisfying the equation and inequalities). Similarly we take u , v {\displaystyle u,v} satisfying N − k = u a + v b {\displaystyle N-k=ua+vb} and 0 ≤ u < b {\displaystyle 0\leq u<b} .
Two-dimensional linear inequalities are expressions in two variables of the form: + < +, where the inequalities may either be strict or not. The solution set of such an inequality can be graphically represented by a half-plane (all the points on one "side" of a fixed line) in the Euclidean plane. [2]
Bernstein inequalities (probability theory) Boole's inequality; Borell–TIS inequality; BRS-inequality; Burkholder's inequality; Burkholder–Davis–Gundy inequalities; Cantelli's inequality; Chebyshev's inequality; Chernoff's inequality; Chung–Erdős inequality; Concentration inequality; Cramér–Rao inequality; Doob's martingale inequality
Computer support in solving inequations is described in constraint programming; in particular, the simplex algorithm finds optimal solutions of linear inequations. [6] The programming language Prolog III also supports solving algorithms for particular classes of inequalities (and other relations) as a basic language feature.
Hardy's inequality is an inequality in mathematics, named after G. H. Hardy.. Its discrete version states that if ,,, … is a sequence of non-negative real numbers, then for every real number p > 1 one has
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