Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

1.1 Some Basic of Algebra ■ Algebraic Expressions ■ Equations and Inequalities ■ Sets of Numbers

Slide 1- 3 Copyright © 2012 Pearson Education, Inc. Terminology A letter that can be any one of various numbers is called a variable. If a letter always represents a particular number that never changes, it is called a constant.

Slide 1- 4 Copyright © 2012 Pearson Education, Inc. Algebraic Expressions An algebraic expression consists of variables, numbers, and operation signs. Examples: When an equal sign is placed between two expressions, an equation is formed.

Slide 1- 5 Copyright © 2012 Pearson Education, Inc. Exponential Notation The expression a n, in which n is a counting number, means n factors. In a n, a is called the base and n is called the exponent, or power. When no exponent appears, it is assumed to be 1. Thus a 1 = a.

Slide 1- 6 Copyright © 2012 Pearson Education, Inc. Example The base of a triangle is 10 feet and the height is 3.1 feet. Find the area of the triangle. Solution 10·3.1 = 15.5 square feet h b

Slide 1- 7 Copyright © 2012 Pearson Education, Inc. Rules for Order of Operations 1. Simplify within any grouping symbols, such as parentheses and brackets. 2. Simplify all exponential expressions. 3. Perform all multiplication and division, as either occurs, working from left to right. 4. Perform all addition and subtraction, as either occurs, working from left to right.

Slide 1- 8 Copyright © 2012 Pearson Education, Inc. Example Evaluate the expression Solution 2(x + 3) 2 – 12 x 2 Substituting Simplifying 5 2 and 2 2 Multiplying and Dividing Subtracting = 2(2 + 3) 2 – Working within parentheses

Slide 1- 9 Copyright © 2012 Pearson Education, Inc. Example Evaluate the expression Solution 4x 2 + 2xy – z = 4· · 3 · 2 – 8 = – 8 = 40 = 4·9 + 2·3·2 – 8 Substituting Simplifying 3 2 Multiplying Adding and Subtracting

Slide Copyright © 2012 Pearson Education, Inc. Example Determine whether 8 is a solution of x + 12 = 21. Since the left-hand and right-hand sides differ, 8 is not a solution. 20  21 False | 21 Solution x + 12 = 21 Equations and Inequalities

Slide Copyright © 2012 Pearson Education, Inc. Sets of Numbers Natural Numbers, Whole Numbers, and Integers Natural Numbers (Counting Numbers) Those numbers used for counting: {1, 2, 3,…} Whole Numbers The set of natural numbers with 0 included: {0, 1, 2, 3,…} Integers The set of all whole numbers and their opposites: {…,-3, -2, -1, 0, 1, 2, 3,…} The dots are called ellipses and indicate that the pattern continues without end.

Slide Copyright © 2012 Pearson Education, Inc. Set Notation Roster notation: {2, 4, 6, 8} Set-builder notation: {x | x is an even number between 1 and 9} “The set of all xsuch thatx is an even number between 1 and 9”

Slide Copyright © 2012 Pearson Education, Inc. Rational Numbers Numbers that can be expressed as an integer divided by a nonzero integer are called rational numbers:

Slide Copyright © 2012 Pearson Education, Inc. Real Numbers Numbers that are either rational or irrational are called real numbers. The set of all real numbers is often represented as. Numbers like are said to be irrational. Decimal notation for irrational numbers neither terminates nor repeats.

Slide Copyright © 2012 Pearson Education, Inc.

Slide Copyright © 2012 Pearson Education, Inc. Example Which numbers in the following list are a) whole numbers? b) integers? c) rational numbers? d) irrational numbers? e) real numbers? Solution Whole numbers: 0, 72, and 12/2 Integers: ‒ 26, 0, 72, and 12/2 Rational: ‒ 26, ‒ 8/5, 0, 5.5, 72, and 12/2 Irrational: Rea: All the numbers are real numbers.