# Inequality

## Examples

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 x within the set of real numbers. Associating each of the real numbers with a point on a line (called the real number line) provides a way of picturing this location relative to all the other real numbers.

In addition, this same expression defines a solution set, or subset of the set of real numbers, namely all values of x for which the expression is true. More generally, an expression in two variables, such as y > 5x + 6, defines a solution set (or relation) whose members are ordered pairs of real numbers. Associating each ordered pair of real numbers with points in a plane (called the Cartesian coordinate system) it is possible to picture the solution set as being that portion of the plane that makes the expression true.

## Resources

### Books

Bittinger, Marvin L., and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Davison, David M., Marsha Landau, Leah McCracken, and Linda Thompson. Prentice Hall Pre-Algebra. Needham, MA: Prentice Hall, 1992.

McKeague, Charles P. Intermediate Algebra. Fort Worth, TX: Saunders College Publishing, 1995.

## KEY TERMS

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Ordering relation

—An ordering relation is a rule for comparing the members of a set in a way that provides a method for placing each member in a specific order relative to the other members of the set. The integers, and the alphabet are examples of ordered sets.

Relation

—A relation between two sets X and Y is a subset of all possible ordered pairs (x,y) for which there exists a specific relationship between each x and y.

Set

—A set is a collection of things called members or elements of the set. In mathematics, the members of a set will often be numbers.

Solution set

—The solution set of an inequality is that subset of an ordered set which makes the inequality a true statement.