- Linear inequality
In mathematics a linear inequality is an
inequality which involves alinear function .Formal definitions
When operating in terms of
real number s, 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"×"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.
Linear inequalities in terms of other mathematical objects
The above definition requires well-defined operations of
addition ,multiplication andcomparison , therefore the notion of a linear inequality may be extended toordered ring s, in, particular, toordered field s.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 case s this convex set if aconvex polyhedron (possibly unbounded, e.g., a half-space, a slab between two parallel half-spaces or apolyhedral cone ). It may also be empty or a convex polyhedron of lower dimension confined to anaffine 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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