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In [[mathematics]], an '''inequation''' is a statement that an [[inequality (mathematics)|inequality]] holds between two values.<ref>{{cite book |title=The A to Z of Mathematics: A Basic Guide |author=Thomas H. Sidebotham |page=252 |publisher=John Wiley and Sons |year=2002 |isbn=0-471-15045-2}}</ref> It is usually written in the form of a pair of [[expression (mathematics)|expression]]s denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are:
:<math>a < b,\,</math>
:<math>x+y+z \leq 1,\,</math>
:<math>n > 1,\,</math>
:<math>x \neq 0.\,</math>
Some authors apply the term only to inequations in which the inequality relation is, specifically, not-equal-to (≠).<ref>{{MathWorld|title=Inequation|urlname=Inequation}}</ref>

==Chains of inequations==
A shorthand notation is used for the [[conjunction (logic)|conjunction]] of several inequations involving common expressions, by chaining them together. For example, the chain
:<math>0 \leq a < b \leq 1\,</math>
is shorthand for
:<math>0 \leq a~\mathrm{and}~a< b~\mathrm{and}~b\leq 1.\,</math>

==Solving inequations==

[[File:Linear Programming Feasible Region.svg|thumb|Solution set for example inequations]]
Similar to [[equation solving]], '''inequation solving''' means finding what values (numbers, functions, sets, etc.) 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, mathematical expressions. 
A '''solution''' of the equation is an assignment of expressions to the ''unknowns'' that satisfies the inequation(s); in other words, expressions such that, when they are substituted for the unknowns, the inequations become true propositions.
Often, an additional '''objective''' expression is given that is to be minimized by an ''optimal'' solution.

For example, <math>0 \leq x_1 \leq 690 - 1.5 \cdot x_2 \;\land\; 0 \leq x_2 \leq 530 - x_1 \;\land\; x_1 \leq 640 - 0.75 \cdot x_2</math> is a conjunction of inequations, partly written as chains; the set of its solutions is shown in blue in the picture (the red, green, and orange line corresponding to the 1st, 2nd, and 3rd conjunct, respectively).
See [[Linear_programming#Example]] for a larger 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 supports solving algorithms for particular classes of inequalities (and other relations) as a basic language feature, see [[constraint logic programming]].

== Special ==
<math>\sqrt{{f(x)}} < g(x)</math><br><math>\Leftrightarrow</math><math>\begin{cases}
 f(x) \ge 0 \\
 g(x) > 0\\
 f(x) < \left [ g(x) \right ]^2 \quad
\end{cases}</math>

==See also==
{{Wiktionary}}
* [[Equation]]
* [[Equals sign]]
* [[Inequality (mathematics)]]
* [[Relational operator]]

==References==
{{reflist}}

[[Category:Elementary algebra]]