Editing Inequation

{{short description|Mathematical statement that two values are not equal}}
{{Redirect-distinguish|≠|‡|ǂ}}
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In [[mathematics]], an '''inequation''' is a statement that either an [[inequality (mathematics)|''inequality'']] (relations "greater than" and "less than", < and >) or a relation "'''not equal to'''" (≠) 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><ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/Inequation.html|title=Inequation|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</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 the [[Sides of an equation|two sides]], 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>

In some cases, the term "inequation" has a more restricted definition, reserved only for statements whose inequality relation is "not equal to" (or "distinct").<ref name=":1" /><ref>{{Cite web|url=http://bestmaths.net/online/index.php/year-levels/year-9/year-9-topics/equations-and-inequations/|title=BestMaths|website=bestmaths.net|access-date=2019-12-03}}</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>
which also implies that <math>0 < b</math> and <math>a < 1</math>.

In rare cases, chains without such implications about distant terms are used.
For example <math>i \neq 0 \neq j</math> is shorthand for <math>i \neq 0 ~ ~ \mathrm{and} ~ ~ 0 \neq j</math>, which does not imply <math>i \neq j.</math>{{citation needed|date=December 2019}} Similarly, <math>a < b > c</math> is shorthand for <math>a < b ~ ~ \mathrm{and} ~ ~ b > c</math>, which does not imply any order of <math>a</math> and <math>c</math>.<ref>{{cite book | author1=Brian A. Davey | author2=Hilary Ann Priestley | author2-link=Hilary Priestley | title=Introduction to Lattices and Order |title-link= Introduction to Lattices and Order |publisher=Cambridge University Press | series=Cambridge Mathematical Textbooks | isbn=0-521-36766-2 | lccn=89009753 | year=1990 |at= definition of a [[fence (mathematics)|fence]] in exercise 1.11, p.23}}</ref>

==Solving inequations==

[[File:Linear Programming Feasible Region.svg|thumb|Solution set (portrayed as [[feasible region]]) for a sample list of 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.<ref>{{Cite web|url=https://www.purplemath.com/modules/linprog.htm|title=Linear Programming: Introduction|last=Stapel|first=Elizabeth|website=Purplemath|access-date=2019-12-03}}</ref>

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 (where <math>\land</math> 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.<ref>{{Cite web|url=https://www.britannica.com/science/optimization|title=Optimization - The simplex method|website=Encyclopedia Britannica|language=en|access-date=2019-12-03}}</ref> 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]].

== Combinations 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 <math>\textstyle \sqrt{{f(x)}} < g(x)</math> is logically equivalent to the following three inequations combined:

# <math> f(x) \ge 0</math>
# <math> g(x) > 0</math>
# <math>  f(x) < \left(g(x)\right)^2</math>

== See also ==
{{Wiktionary}}
* [[Apartness relation]] &mdash; a form of inequality in constructive mathematics
* [[Equation]]
* [[Equals sign]]
* [[Inequality (mathematics)]]
* [[Relational operator]]

==References==
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[[Category:Elementary algebra]]
[[Category:Mathematical terminology]]