Does anybody know about articles about the importance of "inequalities" in Algebra?
As any local maximal point is bigger than all points to the left and to the right, inequalities are all that matters in all subjects where optimisation is relevant. I work within game theory, and the only parts of algebra that I (almost) ever use is how to solve inequalities.
I solve inequalities daily. Without solving them one can only guess at a quantity, or as many engineers will tell you "without data, it's just another opinion."
Inequalities are very important in geometry as well. Sharp inequalities provide best estimate of geometric quantities.
Triangle inequality basically conveys the simple notion that the shortest path between two given points is the straight line joining them. This apparently harmless inequality however has far reaching consequences in the most general axiomatic definition of the "norm" of a vector space, which makes it important for vector analysis.
Similarly triangle inequality enters into the defining rules of any "metric space" not only vector spaces.
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