# Algebraic Expressions, Equations, and Inequalities

## Overview

### Description

Expressions, equations, and inequalities are used to describe relationships between quantities. The associative, commutative, and identity properties of addition and multiplication are used to simplify algebraic expressions. Algebraic equations and inequalities represent mathematical and practical relationships with operations, constants, and variables. A value or set of values that makes an equation or inequality true is a solution. While an equation often has a single solution, the solutions of inequalities may be represented as intervals of values. These can be represented using set-builder or interval notation. Some equations and inequalities do not have solutions.

### At A Glance

• The associative, commutative, and identity properties of addition can be used to simplify expressions with real numbers. The additive identity is zero, and the additive inverse, or opposite, of a real number $a$ is $-a$.
• The associative, commutative, and identity properties of multiplication can be used to simplify expressions with real numbers. The multiplicative identity is 1, and the multiplicative inverse, or reciprocal, of a real number $a$ is $\frac{1}{a}$.
• The distributive property of multiplication over addition can be used to simplify expressions with real numbers.
• Expressions can be simplified by combining the properties of operations with the order of operations.
• Algebraic expressions are used to represent verbal descriptions of a situation.
• To evaluate an algebraic expression, substitute the given values of the variables in the expression, and simplify.
• Algebraic equations can represent verbal statements of situations.
• Equations have many applications that aid in calculating values that are used in the real world.
• To check a solution of an equation, substitute the value of the variables in the equation and determine whether the resulting statement is true.
• Algebraic inequalities are used to represent verbal descriptions of problems.
• An inequality can have infinitely many solutions. The set of solutions of an inequality can be written using set-builder or interval notation.
• To check a solution of an inequality, substitute the value of the variables in the inequality, and determine whether the resulting statement is true.