# Linear inequality

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

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

- ,

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 (*x*_{0}, *y*_{0}) which is not on the line and observe whether or not the inequality is satisfied.

For example, to draw the solution set of *x* + 3*y* < 9, one first draws the line with equation *x* + 3*y* = 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 **R**^{n} linear inequalities are the expressions that may be written in the form

- or ,

where *f* is a linear form (also called a *linear functional*), and *b* a constant real number.

More concretely, this may be written out as

- ,

or

- .

Here are called the unknowns, and are called the coefficients.

Alternatively, these may be written as

- or ,

where *g* is an affine function.

That is

- ,

or

- .

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:

Here are the unknowns, are the coefficients of the system, and are the constant terms.

This can be concisely written as the matrix inequality:

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.