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Inequation

Mathematical statement that two values are not equal

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated August 6, 2026

In mathematics, an inequation is a statement relating two values, that is either an strict inequality ("greater than", <, or "less than", >) or a relation "not equal to" (≠). 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:

  • a<b
  • x+y+z<1
  • n>1
  • x\neq 0

Sometimes, the term "inequation" is restricted to the relaton "not equal to", or is enlarged for including the relations "less than or equal to", ≤, and "greater than or equal to", ≥.

01Chains of inequations

A shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example, the chain

0\leq a<b\leq 1

is shorthand for

0\leq a~~\mathrm {and} ~~a<b~~\mathrm {and} ~~b\leq 1

which also implies that 0<b and a<1.

In rare cases, chains without such implications about distant terms are used. For example i\neq 0\neq j is shorthand for i\neq 0~~\mathrm {and} ~~0\neq j, which does not imply i\neq j. Similarly, a<b>c is shorthand for a<b~~\mathrm {and} ~~b>c, which does not imply any order of a and c.

Solution set (portrayed as feasible region) for a sample list of inequations
Solution set (portrayed as feasible region) for a sample list of inequations

02Solving 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, expressions. A solution of the inequation 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, make the inequations true propositions. Often, an additional objective expression (i.e., an optimization equation) is given, that is to be minimized or maximized by an optimal solution.

For example,

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}

is a conjunction of inequations, partly written as chains (where \land can be read as "and"); 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). 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 (and other relations) as a basic language feature. For more, see constraint logic programming.

03Combinations of meanings

Usually because of the properties of certain functions (like square roots), some inequations are equivalent to a combination of multiple others. For example, the inequation \textstyle {\sqrt {f(x)}}<g(x) is logically equivalent to the following three inequations combined:

  1. f(x)\geq 0
  2. g(x)>0
  3. f(x)<\left(g(x)\right)^{2}
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Sources and credits

This article is adapted from the Wikipedia article “Inequation”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Apartness relation

In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. An apartness relation is often written as # {\displaystyle \#} to distinguish from the negation of equality, which is weaker.

Equation

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables while an égalité does not require variables, but in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.

Equals sign

The equals sign or equal sign (American English), also known as the equality sign, is the mathematical symbol =, which is used to indicate equality. In an equation, it is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value.