Resolving a misconception in combining inequalities

This article addresses a common misconception about combining inequalities. Inequality is used in algebra to show magnitude comparison, as well as in set theory and mathematical programming to represent a set of values satisfying inequality. It is a widespread misconception that two inequalities can be combined under any circumstances. In this article, we prove that when inequality represents a set of values, combining inequalities would fail to maintain the set they represent and would result in a larger set. Therefore, even though it is valid when inequality is used to represent relation of magnitude comparison, combining inequalities is not valid when inequality is used to represent a set of values. Six examples are diagnosed in applications of solving simultaneous equations and constructing constraints in linear and integer programming, and pitfalls due to invalidly combining inequalities are indicated in those examples.