Abstract:

In order to justify a solution of an equation/inequality, one has to view this equation/inequality as a statement concerning the unknowns that is given in terms of suitable functions. The next step is to present a sequence of equivalent statements concerning the unknowns - that we call conjunction-disjunction complexes - such that the last statement describes the solution set of the equation/inequality. To ensure that the intermediate statements are indeed equivalent (or, in the case of solving an equation, each such statement is, at least, a consequence of the previous one), one needs to follow rules of logic and keep track of the domains of the functions involved. This talk is devoted to a discussion of general principles of solving equations/inequalities based on the notion of function combined with utilization of simple rules of logic. We then apply these principles to different types of equations/inequalities such as involving rational, irrational, logarithmic and other functions and provide concrete examples where such equations/inequalities are solved. Understanding the approach suggested in the paper allows to avoid many common mistakes that occur when solving complex equations/inequalities and systems of equations/inequalities. Presented material shall be of interest to high school/college instructors who teach algebra/pre-calculus courses.

Keywords:

Equations, inequalities, functions, logic, conjunction-disjunction complexes.