epäyhtäläiset

Epäyhtäläiset are mathematical relations that express that one quantity is related to another by an inequality. In Finnish mathematics, they include both strict inequalities such as x > y and x < y, and non-strict inequalities such as x ≥ y and x ≤ y. Each epäyhtäläiset defines a set of admissible values for the variables involved.

Solving epäyhtäläiset follows rules similar to those for equations, but with attention to the direction of the inequality. Adding or subtracting the same value to both sides preserves the inequality; multiplying or dividing by a positive number also preserves direction, while multiplying or dividing by a negative number reverses it. For polynomial and other more complex inequalities, techniques such as factoring, sign analysis, and finding critical points help determine the solution set.

Graphically, epäyhtäläiset are represented as regions rather than single points. On a number line, a single-variable inequality is shown by shading the region that satisfies the relation. In two variables, linear and nonlinear epäyhtäläiset define half-planes or more general regions in the plane; intersection of multiple inequalities yields the feasible region.

Systems of epäyhtäläiset consist of multiple inequalities whose common solutions form a feasible area, often polygonal in linear cases. These systems are central to optimization, economics, and physics as a way to model constraints on variables.

Related concepts include equality (yhtälö) and systems of equations. The study of epäyhtäläiset spans linear, polynomial, rational, and transcendental cases, and employs algebraic, graphical, and calculus-based methods.