Linear Inequalities

Linear inequalities in general dimensions

In Rⁿ linear inequalities are the expressions that may be written in the form

${\displaystyle f({\bar {x}})<b}$ or ${\displaystyle f({\bar {x}})\leq b,}$

where f is a linear form (also called a linear functional), ${\displaystyle {\bar {x}}=(x_{1},x_{2},\ldots ,x_{n})}$ and b a constant real number.
More concretely, this may be written out as

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<b}$

or

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq b.}$

Here ${\displaystyle x_{1},x_{2},...,x_{n}}$ are called the unknowns, and ${\displaystyle a_{1},a_{2},...,a_{n}}$ are called the coefficients.
Alternatively, these may be written as

${\displaystyle g(x)<0\,}$ or ${\displaystyle g(x)\leq 0,}$

where g is an affine function.^[4]
That is

${\displaystyle a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<0}$

or

${\displaystyle a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq 0.}$

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

A system of linear inequalities is a set of linear inequalities in the same variables:

${\displaystyle {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;\leq \;&&&b_{1}\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;\leq \;&&&b_{2}\\\vdots \;\;\;&&&&\vdots \;\;\;&&&&\vdots \;\;\;&&&&&\;\vdots \\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;\leq \;&&&b_{m}\\\end{alignedat}}}$

Here ${\displaystyle x_{1},\ x_{2},...,x_{n}}$ are the unknowns, ${\displaystyle a_{11},\ a_{12},...,\ a_{mn}}$ are the coefficients of the system, and ${\displaystyle b_{1},\ b_{2},...,b_{m}}$ are the constant terms.
This can be concisely written as the matrix inequality

${\displaystyle Ax\leq b,}$

where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants.
In the above systems both strict and non-strict inequalities may be used.

• Not all systems of linear inequalities have solutions.