Linear inequality

Linear inequality

In mathematics a linear inequality is an inequality which involves a linear function.

Formal definitions

When operating in terms of real numbers, linear inequalities are the ones written in the forms

: f(x) < b ext{ or }f(x) leq b,

where f(x) is a linear functional in real numbers and "b" is a constant real number. Alternatively, these may be viewed as

: g(x) < 0 ext{ or }g(x) leq 0,

where g(x) is an affine function.

The above are commonly written out as: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.

Sometimes they may be written out in the forms

: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,dots,x_n are called the unknowns, a_{1}, a_{2},dots, a_{n} are called the coefficients, and b is the constant term.

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

A system of linear inequalities is a set of linear inequalities in the same variables:

:egin{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,dots,x_n are the unknowns, a_{11}, a_{12},dots, a_{mn} are the coefficients of the system, and b_1, b_2,dots,b_m are the constant terms.

This can be concisely written as the matrix inequality:

:Ax leq b

where "A" is an "m"&times;"n" matrix, "x" is an "n"&times;1 column vector of variables, and "b" is an "m"&times;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.

Linear inequalities in terms of other mathematical objects

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, in, particular, to ordered fields.

Linear inequalities in real numbers

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-planes defined by individual inequalities. It is a convex set, since the half-planes are convex sets, and the intersection of a set of convex sets is also convex. In the non-degenerate cases this convex set if 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"n".

Sets of linear inequalities (called constraints) are used in the definition of linear programming.

References


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