Linear inequality

Posted By on November 5, 2014

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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 ,


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,


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:

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

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.

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Posted by Akash Kurup

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