An inequality compares mathematical quantities using an ordering, such as or for real numbers. A strict inequality excludes equality; a non-strict inequality allows it. For example, is strict, while is true because equality is allowed. Adding the same real number preserves either comparison; multiplying by a negative real number reverses its direction.
A linear inequality compares a linear functional of a vector of unknowns with a constant, for example or over the real numbers. For , its solution set is a closed or open half-space, respectively. The inequality allows all points on one side of the line , including the line. Multiplication by a negative real number reverses the inequality, so is equivalently .
A system of linear inequalities requires several linear inequalities to hold simultaneously. The notation compares coordinates componentwise, giving one inequality per row of the matrix . Its solution set is the intersection of the corresponding half-spaces. For non-strict inequalities it is a linear polyhedron, which may be empty. For example, , , are inconsistent: adding their equivalent forms , , gives . A Farkas certificate for linear inequalities generalizes this cancellation.
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In mathematics, an inequality is a relation that shows the relative size or order of two values. It indicates that one value is greater than, less than, greater than or equal to, or less than or equal to another value. Inequalities are an essential part of various mathematical concepts and applications, including algebra, calculus, and optimization. There are several types of inequalities, often denoted by specific symbols: 1. **Less than (<)**: Indicates that one quantity is smaller than another.