# Inequality - Ordered Sets, Algebra Of Inequalities, Examples

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positive statement integers subset

In **mathematics**, an inequality is a statement about the relative order of members of a set. For instance, if S
**Figure 1.** Illustration by Hans & Cassidy. Courtesy of Gale Group.

is the set of positive **integers**, and the symbol < is taken to mean less than, then the statement 5 < 6 (read "5 is less than 6") is a true statement about the relative order of 5 and 6 within the set of positive integers. The comparison that is symbolized by < is said to define an ordering **relation** on the set of positive integers. An inequality is often used for defining a subset of an ordered set. The subset is also the solution set of the inequality.

## Additional Topics

A set is ordered if its members obey three simple rules. First, an ordering relation such as "less than" (<) must apply to every member of the set, that is, for any two members of the set, call them a and b, either a < b or b < a. Second, no member of the set can have more than one position within the ordering, in other words, a < a has no meaning. Third, …

Inequalities involving real numbers are particularly important. There are four types of inequalities, or ordering relations, that are important when dealing with real numbers. They are (together with their symbols) "less than" (<), "less than or equal to" (≤), "greater than" (>), and "greater than or equal to" (≥).…

As stated previously, an inequality can be a statement about the general location of a member within an ordered set, or it can be interpreted as defining a solution set or relation. For example, consider the compound expression 5 < x < 6 (read "5 is less than x, and x is less than 6") where x is a real number. This expression is a statement about the general location of…

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