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 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:
a x + b y < c and a x + b y ≥ c , {\displaystyle ax+by<c{\text{ and }}ax+by\geq c,}
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 (x0, y0) 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 (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 Rn linear inequalities are the expressions that may be written in the form
f ( x ¯ ) < b {\displaystyle f({\bar {x}})<b} or f ( x ¯ ) ≤ b , {\displaystyle f({\bar {x}})\leq b,}
where f is a linear form (also called a linear functional), x ¯ = ( x 1 , x 2 , … , x n ) {\displaystyle {\bar {x}}=(x_{1},x_{2},\ldots ,x_{n})} and b a constant real number. More concretely, this may be written out as
a 1 x 1 + a 2 x 2 + ⋯ + a n x n < b {\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<b}
or
a 1 x 1 + a 2 x 2 + ⋯ + a n x n ≤ b . {\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq b.}
Here x 1 , x 2 , . . . , x n {\displaystyle x_{1},x_{2},...,x_{n}} are called the unknowns, and a 1 , a 2 , . . . , a n {\displaystyle a_{1},a_{2},...,a_{n}} are called the coefficients. Alternatively, these may be written as
g ( x ) < 0 {\displaystyle g(x)<0\,} or g ( x ) ≤ 0 , {\displaystyle g(x)\leq 0,}
where g is an affine function. That is
a 0 + a 1 x 1 + a 2 x 2 + ⋯ + a n x n < 0 {\displaystyle 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 + ⋯ + a n x n ≤ 0. {\displaystyle a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq 0.}
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