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Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
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 .
In mathematics, an inequation is a statement that an inequality holds between two values. [1] [2] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are:
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. [2]
Here the feasible set is the set of pairs (x, y) in which the value of x is at least 1 and at most 10 and the value of y is at least 5 and at most 12. The feasible set of the problem is separate from the objective function , which states the criterion to be optimized and which in the above example is x 2 + y 4 . {\displaystyle x^{2}+y^{4}.}
The solution to the isoperimetric problem is usually expressed in the form of an inequality that relates the length L of a closed curve and the area A of the planar region that it encloses. The isoperimetric inequality states that , and that the equality holds if and only if the curve is a circle.
In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997, [1] to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu ...
Whiteley, J.N. (1969). "On Newton's Inequality for Real Polynomials". The American Mathematical Monthly. 76 (8). The American Mathematical Monthly, Vol. 76, No. 8: 905– 909. doi:10.2307/2317943. JSTOR 2317943. Niculescu, Constantin (2000). "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics. 1 (2 ...