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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: [1] < less than. > greater than. ≤ less than or equal to. ≥ greater than or equal to. ≠ not equal to.
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 .
Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the ...
Isoperimetric inequality. In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume , where is a unit sphere. The equality holds only when is a sphere in .
Kuznets ratio and Kuznets curve. The Kuznets ratio is a measurement of the ratio of income going to the highest-earning households (usually defined by the upper 20%) to income going to the lowest-earning households, [4] which is commonly measured by either the lowest 20% or lowest 40% of income. Comparing 20% to 20%, a completely even ...
Poincaré inequality. In mathematics, the Poincaré inequality[1] is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the ...
Cauchy–Schwarz inequality. The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) [1][2][3][4] is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.
QM-AM-GM-HM inequalities. In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that are positive real numbers.
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