# Linear inequality

Posted By on November 5, 2014

In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:

• < is less than
• > is greater than
• ≤ is less than or equal to
• ≥ is greater than or equal to
• ≠ is not equal to

A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.

### Two-dimensional linear inequalities

Graph of linear inequality:
x + 3y < 9

Two-dimensional linear inequalities are expressions in two variables of the form:

$ax + by < c text{ and } ax + by geq c$,

where the inequalities may either be strict or not. The solution set of such an inequality can be graphically represented by a half-plane (all the points on one “side” of a fixed line) in the Euclidean plane. The line that determines the half-planes (ax + by = c) is not included in the solution set when the inequality is strict. A simple procedure to determine which half-plane is in the solution set is to calculate the value of ax + by at a point (x0, y0) which is not on the line and observe whether or not the inequality is satisfied.

For example, to draw the solution set of x + 3y < 9, one first draws the line with equation x + 3y = 9 as a dotted line, to indicate that the line is not included in the solution set since the inequality is strict. Then, pick a convenient point not on the line, such as (0,0). Since 0 + 3(0) = 0 < 9, this point is in the solution set, so the half-plane containing this point (the half-plane “below” the line) is the solution set of this linear inequality.

### Linear inequalities in general dimensions

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

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

where f is a linear form (also called a linear functional), $bar{x} = (x_1,x_2,ldots,x_n)$ and b a constant real number.

More concretely, this may be written out as

$a_1 x_1 + a_2 x_2 + cdots + a_n x_n < b$,

or

$a_1 x_1 + a_2 x_2 + cdots + a_n x_n leq b$.

Here $x_1, x_2,...,x_n$ are called the unknowns, and $a_{1}, a_{2},..., a_{n}$ are called the coefficients.

Alternatively, these may be written as

$g(x) < 0 ,$ or $g(x) leq 0$,

where g is an affine function.

That is

$a_0 + a_1 x_1 + a_2 x_2 + cdots + a_n x_n < 0$,

or

$a_0 + a_1 x_1 + a_2 x_2 + cdots + a_n x_n leq 0$.

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:

begin{alignat}{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{alignat}

Here $x_1, x_2,...,x_n$ are the unknowns, $a_{11}, a_{12},..., a_{mn}$ are the coefficients of the system, and $b_1, b_2,...,b_m$ are the constant terms.

This can be concisely written as the matrix inequality:

$Ax leq b$

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.

#### Posted by Akash Kurup

Founder and C.E.O, World4Engineers Educationist and Entrepreneur by passion. Orator and blogger by hobby

Website: http://world4engineers.com