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:. It shows the data which is not equal in graph form. A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign. 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 (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", "y") 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 linear inequalities are the expressions that may be written in the form where "f" is a linear form (also called a  linear functional), formula_4 and "b" a constant real number. More concretely, this may be written out as or Here formula_7 are called the unknowns, and formula_8 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 formula_14 are the unknowns, formula_15 are the coefficients of the system, and formula_16 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. Variables can be eliminated from systems of linear inequalities using Fourier–Motzkin elimination. 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 (possibly unbounded, e.g., a half-space, a slab between two parallel half-spaces or a polyhedral cone). It may also be empty or a convex polyhedron of lower dimension confined to an  affine subspace of the "n"-dimensional space R. Linear programming. A linear programming problem seeks to optimize (find a maximum or minimum value) a function (called the objective function) subject to a number of constraints on the variables which, in general, are linear inequalities. The list of constraints is a system of linear inequalities. Generalization. The above definition requires well-defined operations of addition, multiplication and comparison; therefore, the notion of a linear inequality may be extended to ordered rings, and in particular to ordered fields.