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:

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

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 (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.

In Rⁿ linear inequalities are the expressions that may be written in the form

where f is a linear form (also called a linear functional), ${\displaystyle {\bar {x}}=(x_{1},x_{2},\ldots ,x_{n})}$ and b a constant real number.

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