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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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 . Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities ...
This can be concisely written as the matrix inequality , where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. [citation needed] In the above systems both strict and non-strict inequalities may be used. Not all systems of linear inequalities have solutions.
A particularly suggestive and easy-to-remember version is the following: if a set of linear inequalities has no solution, then a contradiction can be produced from it by linear combination with nonnegative coefficients.
Solution set (portrayed as feasible region) for a sample list of inequations. Similar to equation solving, inequation solving means finding what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an inequation or a conjunction of several inequations.
Therefore, the solution = is extraneous and not valid, and the original equation has no solution. For this specific example, it could be recognized that (for the value x = − 2 {\displaystyle x=-2} ), the operation of multiplying by ( x − 2 ) ( x + 2 ) {\displaystyle (x-2)(x+2)} would be a multiplication by zero.
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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