Inequation


In mathematics, an inequation is a statement that either an inequality or a relation "not equal to" holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between the two sides, indicating the specific inequality relation. Some examples of inequations are:
In some cases, the term "inequation" has a more restricted definition, reserved only for statements whose inequality relation is "not equal to".

Chains of inequations

A shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example, the chain
is shorthand for
which also implies that and.
In rare cases, chains without such implications about distant terms are used.
For example is shorthand for, which does not imply Similarly, is shorthand for, which does not imply any order of and.

Solving inequations

Similar to equation solving, inequation solving means finding what values fulfill a condition stated in the form of an inequation or a conjunction of several inequations. These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled. To be precise, what is sought are often not necessarily actual values, but, more in general, expressions. A solution of the inequation is an assignment of expressions to the unknowns that satisfies the inequation; in other words, expressions such that, when they are substituted for the unknowns, make the inequations true propositions.
Often, an additional objective expression is given, that is to be minimized or maximized by an optimal solution.
For example,
is a conjunction of inequations, partly written as chains ; the set of its solutions is shown in blue in the picture. For a larger example. see Linear programming#Example.
Computer support in solving inequations is described in constraint programming; in particular, the simplex algorithm finds optimal solutions of linear inequations. The programming language Prolog III also supports solving algorithms for particular classes of inequalities as a basic language feature. For more, see constraint logic programming.

Combinations of meanings

Usually because of the properties of certain functions, some inequations are equivalent to a combination of multiple others. For example, the inequation is logically equivalent to the following three inequations combined:
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