[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/linear-inequality-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/linear-inequality-wikipedia\/","headline":"Linear inequality – Wikipedia","name":"Linear inequality – Wikipedia","description":"before-content-x4 In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of","datePublished":"2016-12-09","dateModified":"2016-12-09","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/d\/d7\/Linearineq1.svg\/220px-Linearineq1.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/d\/d7\/Linearineq1.svg\/220px-Linearineq1.svg.png","height":"220","width":"220"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/linear-inequality-wikipedia\/","about":["Wiki"],"wordCount":5012,"articleBody":" (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:[1]< less than> greater than\u2264 less than or equal to\u2265 greater than or equal to\u2260 not equal to (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.Table of Contents (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Linear inequalities of real numbers[edit]Two-dimensional linear inequalities[edit]Linear inequalities in general dimensions[edit]Systems of linear inequalities[edit]Applications[edit]Polyhedra[edit]Linear programming[edit]Generalization[edit]References[edit]Sources[edit]External links[edit]Linear inequalities of real numbers[edit]Two-dimensional linear inequalities[edit] Graph of linear inequality:x + 3y < 9Two-dimensional linear inequalities are expressions in two variables of the form: (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4ax+by)b,{displaystyle f({bar {x}})leq b,}where f is a linear form (also called a linear functional), x\u00af=(x1,x2,\u2026,xn){displaystyle {bar {x}}=(x_{1},x_{2},ldots ,x_{n})} and b a constant real number.More concretely, this may be written out asa1x1+a2x2+\u22ef+anxnb.{displaystyle a_{1}x_{1}+a_{2}x_{2}+cdots +a_{n}x_{n}leq b.}Here x1,x2,...,xn{displaystyle x_{1},x_{2},…,x_{n}} are called the unknowns, and a1,a2,...,an{displaystyle a_{1},a_{2},…,a_{n}} are called the coefficients.Alternatively, these may be written asg(x)+anxn0.{displaystyle a_{0}+a_{1}x_{1}+a_{2}x_{2}+cdots +a_{n}x_{n}leq 0.}Note that any inequality containing a “greater than” or a “greater than or equal” sign can be rewritten with a “less than” or “less than or equal” sign, so there is no need to define linear inequalities using those signs.Systems of linear inequalities[edit]A system of linear inequalities is a set of linear inequalities in the same variables:a11x1+a12x2+\u22ef+a1nxn\u2264b1a21x1+a22x2+\u22ef+a2nxn\u2264b2\u22ee\u22ee\u22ee\u22eeam1x1+am2x2+\u22ef+amnxn\u2264bm{displaystyle {begin{alignedat}{7}a_{11}x_{1}&&;+;&&a_{12}x_{2}&&;+cdots +;&&a_{1n}x_{n}&&;leq ;&&&b_{1}\\a_{21}x_{1}&&;+;&&a_{22}x_{2}&&;+cdots +;&&a_{2n}x_{n}&&;leq ;&&&b_{2}\\vdots ;;;&&&&vdots ;;;&&&&vdots ;;;&&&&&;vdots \\a_{m1}x_{1}&&;+;&&a_{m2}x_{2}&&;+cdots +;&&a_{mn}x_{n}&&;leq ;&&&b_{m}\\end{alignedat}}}Here x1,\u00a0x2,...,xn{displaystyle x_{1}, x_{2},…,x_{n}} are the unknowns, a11,\u00a0a12,...,\u00a0amn{displaystyle a_{11}, a_{12},…, a_{mn}} are the coefficients of the system, and b1,\u00a0b2,...,bm{displaystyle b_{1}, b_{2},…,b_{m}} are the constant terms.This can be concisely written as the matrix inequalityAx\u2264b,{displaystyle Axleq b,}where A is an m\u00d7n matrix, x is an n\u00d71 column vector of variables, and b is an m\u00d71 column vector of constants.[citation needed]In the above systems both strict and non-strict inequalities may be used.Not all systems of linear inequalities have solutions.Variables can be eliminated from systems of linear inequalities using Fourier\u2013Motzkin elimination.[5]Applications[edit]Polyhedra[edit]The set of solutions of a real linear inequality constitutes a half-space of the ‘n’-dimensional real space, one of the two defined by the corresponding linear equation.The set of solutions of a system of linear inequalities corresponds to the intersection of the half-spaces defined by individual inequalities. It is a convex set, since the half-spaces are convex sets, and the intersection of a set of convex sets is also convex. In the non-degenerate cases this convex set is a convex polyhedron (possibly unbounded, e.g., a half-space, a slab between two parallel half-spaces or a polyhedral cone). It may also be empty or a convex polyhedron of lower dimension confined to an affine subspace of the n-dimensional space Rn.Linear programming[edit]A linear programming problem seeks to optimize (find a maximum or minimum value) a function (called the objective function) subject to a number of constraints on the variables which, in general, are linear inequalities.[6] The list of constraints is a system of linear inequalities.Generalization[edit]The above definition requires well-defined operations of addition, multiplication and comparison; therefore, the notion of a linear inequality may be extended to ordered rings, and in particular to ordered fields.References[edit]^ Miller & Heeren 1986, p. 355^ Technically, for this statement to be correct both a and b can not simultaneously be zero. In that situation, the solution set is either empty or the entire plane.^ Angel & Porter 1989, p. 310^ In the 2-dimensional case, both linear forms and affine functions are historically called linear functions because their graphs are lines. In other dimensions, neither type of function has a graph which is a line, so the generalization of linear function in two dimensions to higher dimensions is done by means of algebraic properties and this causes the split into two types of functions. However, the difference between affine functions and linear forms is just the addition of a constant.^ G\u00e4rtner, Bernd; Matou\u0161ek, Ji\u0159\u00ed (2006). Understanding and Using Linear Programming. Berlin: Springer. ISBN\u00a03-540-30697-8.^ Angel & Porter 1989, p. 373Sources[edit]Angel, Allen R.; Porter, Stuart R. (1989), A Survey of Mathematics with Applications (3rd\u00a0ed.), Addison-Wesley, ISBN\u00a00-201-13696-1Miller, Charles D.; Heeren, Vern E. (1986), Mathematical Ideas (5th\u00a0ed.), Scott, Foresman, ISBN\u00a00-673-18276-2External links[edit] (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/linear-inequality-wikipedia\/#breadcrumbitem","name":"Linear inequality – Wikipedia"}}]}]