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:
- < less than
- > greater than
- ≤ less than or equal to
- ≥ greater than or equal to
- ≠ not equal to
Linear inequalities of real numbers
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 in the Euclidean plane. The line that determines the half-planes 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 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. Since 0 + 3 = 0 < 9, this point is in the solution set, so the half-plane containing this point 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 formwhere f is a linear form, 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
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 of constants, x is an n×1 column vector of variables, b is an m×1 column vector of constants and the inequality relation is understood row-by-row.
In the above systems both strict and non-strict inequalities may be used.
- Not all systems of linear inequalities have solutions.
Applications
Polyhedra
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. It may also be empty or a convex polyhedron of lower dimension confined to an affine subspace of the n-dimensional space Rn.