|
|
| Line 1: |
Line 1: |
| {{merge|Inequality (mathematics)|discuss=Talk:Inequation#Merger proposal|date=May 2012}}
| | She is recognized by the title of Myrtle Shryock. To collect coins is what his family members and him enjoy. California is our birth location. She is a librarian but she's usually wanted her personal company.<br><br>Here is my web page ... [http://Www.Alhuloul.com/?p=235037 at home std test] |
| {{refimprove|date=March 2010}}
| |
| | |
| 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]]
| |
She is recognized by the title of Myrtle Shryock. To collect coins is what his family members and him enjoy. California is our birth location. She is a librarian but she's usually wanted her personal company.
Here is my web page ... at home std test