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The graph of y = ln x. Any monotonically increasing function, by its definition, [9] may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the domain of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality ...
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]
Consider the sum = = = (). The two sequences are non-increasing, therefore a j − a k and b j − b k have the same sign for any j, k.Hence S ≥ 0.. Opening the brackets, we deduce:
Three examples of the triangle inequality for triangles with sides of lengths x, y, z.The top example shows a case where z is much less than the sum x + y of the other two sides, and the bottom example shows a case where the side z is only slightly less than x + y.
In graph theory, isoperimetric inequalities are at the heart of the study of expander graphs, which are sparse graphs that have strong connectivity properties. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory , design of robust computer networks , and the theory of ...
A Minnesota couple has reportedly been sentenced to four years after they locked their children in cages for "their safety." Benjamin and Christina Cotton from Red Wing, were sentenced by a ...