Algebraic Expressions, Mathematical Models, and Real Numbers

Objectives

• Evaluate algebraic expressions.

• Use mathematical models.

• Find the intersection and union of two sets.

• Recognize subsets of the real numbers.

• Use inequality symbols.

• Evaluate absolute value and use it to express distance.

• Identify properties of real numbers.

• Simplify algebraic expressions.

Algebraic Expressions

Variables and Expressions

If a letter is used to represent various numbers, it is called a variable. A combination of variables and numbers using the operations of addition, subtraction, multiplication, division, powers, or roots is called an algebraic expression. Evaluating an algebraic expression means to find the value of the expression for a given value of the variable.

• Exponential Expression: An expression of the form is called an exponential expression.

Example: Evaluate for .

Example: Evaluate for .

Formulas and Mathematical Models

Equations and Mathematical Modeling

An equation is formed when an equal sign is placed between two algebraic expressions. A formula is an equation that uses variables to express a relationship between two or more quantities. The process of finding formulas to describe real-world phenomena is called mathematical modeling. Such formulas, together with the meaning assigned to the variables, are called mathematical models.

Example: The formula models the average cost of tuition and fees, , for public U.S. colleges for the school year ending years after 2000. Use the formula to project the average cost for 2015 ().

Checkpoint: Use the formula above to find the average cost for 2010 ().

Sets and Set Operations

Definitions and Notation

• A set is a collection of objects whose contents can be clearly determined. The objects are called elements.

• Braces { } are used to represent a set.

• The roster method lists elements explicitly, e.g., {1, 2, 3, 4, 5, ...}.

• An ellipsis (...) indicates the set continues indefinitely.

• The empty set (or null set) has no elements and is denoted by .

Set-Builder Notation

In set-builder notation, elements are described but not listed. For example:

• This is equivalent to the roster method: {1, 2, 3, 4, 5}

Intersection and Union of Sets

• Intersection: The intersection of sets and , written , is the set of elements common to both sets.

• Union: The union of sets and , written , is the set of elements that are in , , or both.

Example:

Example:

The Set of Real Numbers

Subsets of Real Numbers

• Natural Numbers:

• Whole Numbers:

• Integers:

• Rational Numbers: Numbers that can be written as , where and are integers and .

• Irrational Numbers: Numbers that cannot be expressed as a quotient of integers (e.g., , ).

All these sets are subsets of the set of real numbers.

Recognizing Subsets

Example: Given the set :

• Rational numbers:

• Irrational numbers:

The Real Number Line

The real number line is a graph used to represent the set of real numbers. The origin is labeled 0. Numbers to the right are positive; numbers to the left are negative. Real numbers increase from left to right.

Inequality Symbols

• : is less than

• : is less than or equal to

• : is greater than

• : is greater than or equal to

These symbols always point to the lesser of the two numbers when the statement is true. Comparing real numbers is called ordering the real numbers.

Absolute Value and Distance

Definition of Absolute Value

The absolute value of a real number , denoted , is the distance from 0 to on the number line. The distance is always nonnegative.

Definition:

Examples: ,

Distance Between Two Points

If and are any two points on the real number line, the distance between and is given by:

• or

Example: The distance between and $5|-4 - 5| = |-9| = 9$.

Example: The distance between $3-5|3 - (-5)| = |8| = 8$.

Properties of the Real Numbers

• Commutative Property of Addition:

• Commutative Property of Multiplication:

• Associative Property of Addition:

• Associative Property of Multiplication:

• Distributive Property:

• Identity Property of Addition:

• Identity Property of Multiplication:

• Inverse Property of Addition:

• Inverse Property of Multiplication: (for )

Simplifying Algebraic Expressions

To simplify an algebraic expression, combine like terms. Like terms have exactly the same variable factors. An expression is simplified when parentheses have been removed and like terms combined.

Example: Simplify

Solution:

Example: Simplify

Solution:

Summary Table: Subsets of Real Numbers

Additional info: This guide covers the foundational concepts from Chapter P (Fundamental Concepts of Algebra) in College Algebra, including algebraic expressions, mathematical modeling, set theory, real numbers, inequalities, absolute value, and properties of real numbers.