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Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Numbers, Variables, and Expressions Natural Numbers and Whole Numbers Prime Numbers and Composite Numbers Variables, Algebraic Expressions, and Equations Translating Words to Expressions 1.1

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Natural Numbers and Whole Numbers The set of natural numbers are also known as the counting numbers. 1, 2, 3, 4, 5, 6,… Because there are infinitely many natural numbers, three dots are used to show that the list continues in the same pattern without end. The whole numbers can be expressed as 0, 1, 2, 3, 4, 5, … Slide 3 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Prime Numbers and Composite Numbers When two natural numbers are multiplied, the result is another natural number. The product of 6 and 7 is = 42 The numbers 6 and 7 are factors of 42. A prime number has only itself and 1 as factors. A natural number greater than 1 that is not prime is a composite number. Any composite number can be written as a product of prime numbers. Slide 4 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Prime Factorization The prime factorization of 120. Slide 5 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 6 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Classifying numbers as prime or composite Classify each number as prime or composite. If a number is composite, write it as a product of prime numbers. a. 37b. 3c. 45d. 300 a. 37 The only factors of 37 are 1 and itself. The number is prime. b. 3 The only factors of 3 are 1 and itself. The number is prime. c. 45 Composite because 9 and 5 are factors. Prime factorization: 3 2 5

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Slide 7 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Classifying numbers as prime or composite Classify each number as prime or composite. If a number is composite, write it as a product of prime numbers. a. 37b. 3c. 45d. 300 d. 300 Prime factorization

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Variables, Algebraic Expressions, and Equations Variables are often used in mathematics when tables of numbers are inadequate. A variable is a symbol, typically an italic letter used to represent an unknown quantity. An algebraic expression consists of numbers, variables, operation symbols, such as +,,, and, and grouping symbols, such as parentheses. An equation is a mathematical statement that two algebraic expressions are equal. A formula is a special type of equation that expresses a relationship between two or more quantities. Slide 8 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 9 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Evaluating algebraic expressions with one variable Evaluate each algebraic expression for x = 6. a. x + 4b. 4xc. 20 – x d. a. x = 10 b. 4x 4(6) = 24 c. 20 – x 20 – 6 = 14 d.

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Slide 10 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Evaluating algebraic expressions with two variables Solution Evaluate each algebraic expression for y = 3 and z = 9 a. 5yzb. z – y c. a. 5yz 5(3)(9) = 135 b. z – y 9 – 3 = 6 c.

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Slide 11 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Evaluating formulas Find the value of y for x = 20 and z = 5. a. y = x + 4b. y = 9xz a. y = x + 4 y = b. y = 9xz y = 9(20)(5) = 24= 900

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Translating Words to Expressions Slide 12 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 13 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Translating words to expressions Translate each phrase to an algebraic expression. a. Twice the cost of a book b. Ten less than a number c. The product of 8 and a number a. Twice the cost of a book b. Ten less than a number c. The product of 8 and a number 2c where c is the cost of the book n – 10 where n is the number 8n where n is the number

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Slide 14 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding the area of a rectangle The area A of a rectangle equals its length L times its width W. a. Write a formula that shows the relationship between these three quantities. b. Find the area of a yard that is 100 feet long and 75 feet wide. a. The word times indicates the length and width should be multiplied. The formula is A = LW. b. A = LW = (100)(75) = 7500 square feet

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Slide 15 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Fractions Basic Concepts Simplifying Fractions to Lowest Terms Multiplication and Division of Fractions Addition and Subtraction of Fractions Applications 1.2

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Basic Concepts The parts of a fraction are named as follows. Slide 17 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fraction bar Numerator Denominator

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Slide 18 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Identifying numerators and denominators Give the numerator and denominator of each fraction. a. b. c. a. The numerator is 8 and the denominator is 19. b. The numerator is mn, and the denominator is p. c. The numerator is c + d, and the denominator is f – 7.

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Simplifying Fractions to Lowest Terms When simplifying fractions, we usually factor out the greatest common factor (GCF) for the numerator and the denominator. The greatest common factor is the largest factor common to both the numerator and the denominator. Slide 19 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 20 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding the greatest common factor Find the greatest common factor (GCF) for each pair of numbers. a. 14, 21 b. 42, 90 a. Because 14 = 7 2 and 21 = 7 3, the number 7 is the largest factor that is common to both 14 and 21. Thus the GCF of 14 and 21 is 7.

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Slide 21 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE continued b. When working with larger numbers, one way to determine the greatest common factor is to find the prime factorization of each number. 42 = 6 7 = 2 3 7and 90 = 6 15 = The prime factorizations have one 2 and one 3 in common. Thus the GCF for 42 and 90 is 6.

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Slide 22 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Simplifying fractions to lowest terms Simplify each fraction to lowest terms. a. b. c. a. The GCF of 9 and 15 is 3. b.The GCF of 20 and 28 is 4. c.The GCF of 45 and 135 is 45.

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Multiplication of Fractions Slide 23 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 24 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Multiplying fractions Multiply each expression and simplify the result when appropriate. a. b. c. a. b. c.

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Slide 25 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding fractional parts Find each fractional part. a.One-third of one-fourth b.One half of three-fourths a. b.

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Division of Fractions Slide 26 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 27 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Dividing fractions Divide each expression. a. b. c. a. b. c.

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Slide 28 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Fractions with Like Denominators Slide 29 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 30 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Adding and subtracting fractions with common denominators Add or subtract as indicated. Simplify your answer to lowest terms when appropriate. a. b. a. b.

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Fractions with Unlike Denominators Slide 31 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 32 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Rewriting fractions with the LCD Rewrite each set of fractions using the LCD. a. b. a. The LCD is 24 b.The LCD is 40.

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Slide 33 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Adding and subtracting fractions with unlike denominators Add or subtract as indicated. Simplify your answer to lowest terms when appropriate. a. b. a. b.

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Slide 34 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Applying fractions to carpentry A pipe measures inches long and needs to be cut into three equal pieces. Find the length of each piece. Begin by writing as the improper fraction

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Slide 35 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Exponents and Order of Operations Natural Number Exponents Order of Operations Translating Words to Expressions 1.3

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Natural Number Exponents The area of a square equals the length of one of its sides times itself. If the square is 5 inches on a side, then its area is 5 5 = 5 2 = 25 square inches The expression 5 2 is an exponential expression with base 5 and exponent 2. Slide 37 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Base Exponent

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Slide 38 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 39 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Writing products in exponential notation Write each product as an exponential expression. a. b. c.

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Slide 40 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Evaluating exponential notation Evaluate each expression. a. b.

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Order of Operations Slide 41 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 42 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Evaluating arithmetic expressions Evaluate each expression by hand. a. 12 – 6 – 2 b. 12 – (6 – 2) c. a. 12 – 6 – 2 6 – 2 4 b. 12 – (6 – 2) 12 – 4 8 c.

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Slide 43 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Evaluating arithmetic expressions Evaluate each expression. a. b. c. a. 15 – 6 9 b.c.

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Slide 44 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Writing and evaluating expressions Write each expression and then evaluate it. a. Two to the fifth power plus three b. Twenty-four less two times four a. Two to the fifth power plus three b. Twenty-four less two times four

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Slide 45 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Real Numbers and the Number Line Signed Numbers Integers and Rational Numbers Square Roots Real and Irrational Numbers The Number Line Absolute Value Inequality 1.4

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Signed Numbers The opposite, or additive inverse, of a number a isa. Slide 47 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 48 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding opposites (or additive inverses) Find the opposite of each expression. a. 29b. c.d. (13) a. The opposite of 29 is 29. b. The opposite of is c. d. (13) = 13, so the opposite of (13) is 13.

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Slide 49 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding an additive inverse (or opposite) Find the additive inverse of –x, if x =. The additive inverse of x is x = because (x) = x by the double negative rule.

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Integers and Rational Numbers The integers include the natural numbers, zero, and the opposite of the natural numbers. …,2, 1, 0, 1, 2,… Slide 50 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A rational number is any number that can be expressed as the ratio of two integers, where q 0.

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Slide 51 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Classifying numbers Classify each number as one or more of the following: natural number, whole number, integer, or rational number. a. b. 9c. a.Because, the number is a natural number, whole number, integer, and rational number. b. The number 9 is an integer and rational number, but not a natural number or a whole number. c. The fraction is a rational number because it is the ratio of two integers. However it is not a natural number, a whole number, or an integer.

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Square Roots Square roots are frequently used in algebra. The number b is a square root of a number a if b b = a. Every positive number has one positive square root and one negative square root. Slide 52 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 53 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Calculating principal square roots Evaluate each square root. Approximate to three decimal places when appropriate. a. b. c. a. because 8 8 = 64 and 8 is nonnegative. b. because = 169 and 13 is nonnegative. c. is a number between 4 and 5. We can estimate the value of with a calculator.

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Real and Irrational Numbers Slide 54 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 55 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Classifying numbers Identify the natural numbers, whole numbers, integers, rational numbers, and irrational numbers in the following list. Natural numbers: Whole numbers: Integers: Rational numbers: Irrational numbers:

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Slide 56 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Plotting numbers on a number line Plot each real number on a number line. a. b. c. a. Plot a dot halfway between 2 and 3. b. Plot a dot between 2 and 3. c. Plot a dot halfway between 3 and 4.

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Absolute Value The absolute value of a real number equals its distance on the number line from the origin. Because distance is never negative, the absolute value of a real number is never negative. Slide 57 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 58 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding the absolute value of a real number Write the expression without the absolute value sign. a. b. c.d. a. because the distance between the origin and 9 is 9. b. because the distance is 0 between the origin and 0. c. d.

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Slide 59 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Ordering real numbers List the following numbers from least to greatest. Then plot these numbers on a number line.

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Slide 60 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Addition and Subtraction of Real Numbers Addition of Real Numbers Subtraction of Real Numbers Applications 1.5

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There are four arithmetic operations: addition, subtraction, multiplication, and division. Slide 62 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 63 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Addition of Real Numbers In an addition problem the two numbers added are called addends, and the answer is called the sum = 13 5 and 8 are the addends 13 is the sum The opposite (or additive inverse) of a real number a is a.

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Slide 64 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Adding Opposites Find the opposite of each number and calculate the sum of the number and its opposite. a. b.

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Slide 65 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Addition of Real Numbers

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Slide 66 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Adding real numbers Evaluate each expression. a. b. The numbers are both negative, add the absolute values. The sign would be negative as well.

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Subtraction of Real Numbers Slide 67 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 68 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Subtracting real numbers Find each difference by hand. a. 12 – 16 b. –6 – 2 a. 12 – (–16) 4 b. –6 – 2 –6 + (–2) 8

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Slide 69 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Adding and subtracting real numbers Evaluate each expression. a. b. a. b.

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Slide 70 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Balancing a checking account The initial balance in a checking account is $326. Find the final balance if the following represents a list of withdrawals and deposits: $20, $15, $200, and $150 Find the sum of the five numbers. The final balance is $341.

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Slide 71 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Multiplication and Division of Real Numbers Multiplication of Real Numbers Division of Real Numbers Applications 1.6

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Multiplication of Real Numbers Slide 73 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 74 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Multiplying real numbers Find each product by hand. a. 4 8b. c. d. b. The product is positive because both factors are positive. c. Since both factors are negative, the product is positive. d. a. The resulting product is negative because the factors have unlike signs. Thus 4 8 = 32.

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Slide 75 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 76 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Evaluating real numbers with exponents Evaluate each expression by hand. a. (6) 2 b. 6 2 b. This is the negation of an exponential expression with base 6. Evaluating the exponent before negative results in 6 2 = (6)(6) = 36. a. Because the exponent is outside of parentheses, the base of the exponential expression is 6. The expression is evaluated as (6) 2 = (6)(6) = 36.

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Division of Real Numbers Slide 77 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 78 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Dividing real numbers Evaluate each expression by hand. a. b. c. d. b. c. d. 4 ÷ 0 is undefined. The number 0 has no reciprocal. a.

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Slide 79 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Slide 80 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Converting fractions to decimals Convert the measurement to a decimal number. a. Begin by dividing 5 by Thus the mixed number is equivalent to

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Slide 81 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Converting decimals to fractions Convert each decimal number to a fraction in lowest terms. a. 0.32b b. The decimal equals eight hundred seventy-five thousandths. a. The decimal 0.32 equals thirty-two hundredths.

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Slide 82 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Application After surveying 125 pediatricians, 92 stated that they had admitted a patient to the childrens hospital in the last month for pneumonia. Write the fraction as a decimal. One method for writing the fraction as a decimal is to divide 92 by 125 using long division. An alternative method is to multiply the fractions by so the denominator becomes Then, write the numerator in the thousandths place in the decimal.

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Slide 83 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Properties of Real Numbers Commutative Properties Associative Properties Distributive Properties Identity and Inverse Properties Mental Calculations 1.7

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Slide 85 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Commutative Properties The commutative property for addition states that two numbers, a and b, can be added in any order and the result will be the same = The commutative property for multiplication states that two numbers, a and b, can be multiplied in any order and the result will be the same. 9 4 = 4 9

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Slide 86 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Applying the commutative properties Use the commutative properties to rewrite each expression. a. b. c.

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The associative property allows us to change how numbers are grouped. Slide 87 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Associative Properties

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Slide 88 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Applying the associative properties Use the associative property to rewrite each expression. a. b.

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Slide 89 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Identifying properties of real numbers State the property that each equation illustrates. a. b. Associative property of multiplication because the grouping of the numbers has been changed. Commutative property for addition because the order of the numbers has changed.

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Distributive Properties Slide 90 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The distributive properties are used frequently in algebra to simplify expressions. 7(3 + 8) = The 7 must be multiplied by both the 3 and the 8.

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Slide 91 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Applying the distributive properties Apply a distributive property to each expression. a. 4(x + 3)b. –8(b – 5)c. 12 – (a + 2) a. 4(x + 3) = 4 x = 4x + 12 b. –8(b – 5) c. 12 – (a + 2) = 12 + ( 1)(a + 2) = 8 b ( 8) 5 = 8b + 40 = 12 + ( 1) a + (–1) 2 = 12 a – 2 = 10 a

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Slide 92 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Inserting parentheses using the distributive property Use the distributive property to insert parentheses in the expression and then simplify the result. a. b. a. b.

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Slide 93 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Identifying properties of real numbers State the property or properties illustrated by each equation. a. b. Distributive property. Commutative and associative properties for addition.

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The identity property of 0 states that if 0 is added to any real number a, the result is a. The number 0 is called the additive identity = 3 The identity property of 1 states that if any number a is multiplied by 1, the result is a. The number 1 is called the multiplicative identity. 4 1 = 4 Slide 94 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Identity and Inverse Properties

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Slide 95 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Identity and Inverse Properties

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Slide 96 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Identifying identity and inverse properties State the property or properties illustrated by each equation. a. b. Identity property for 0. Additive inverse property and the identity property for 0.

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Slide 97 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Performing calculations mentally Use the properties of real numbers to calculate each expression mentally. a. b.

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Slide 98 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Simplifying and Writing Algebraic Expressions Terms Combining Like Terms Simplifying Expressions Writing Expressions 1.8

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Terms Slide 100 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A term is a number, a variable, or a product of numbers and variables raised to powers. Examples of terms include 4, z, 5x, and 6xy 2. The coefficient of a term is the number that appears in the term.

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Slide 101 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Identifying terms Determine whether each expression is a term. If it is a term, identify its coefficient. a. 97b. 17x c.4a – 6b d. 9y 2 b. The product of a number and a variable is a term. The coefficient is 17. c. The difference of two terms in not a term. d. The product of a number and a variable with an exponent is a term. Its coefficient is 9. a. A number is a term. The coefficient is 97.

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Slide 102 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Identifying like terms Determine whether the terms are like or unlike. a. 9x, 15x b. 16y 2, 1 c. 5a 3, 5b 3 d. 11, 8z b. The term 1 has no variable and the 16 has a variable of y 2. They are unlike terms. c. The variables are different, so they are unlike terms. d. The term 11 has no variable and the 8 has a variable of z. They are unlike terms. a. The variable in both terms is x, with the same power of 1, so they are like terms.

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Slide 103 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Combining like terms Combine terms in each expression, if possible. a. 2y + 7y b. 4x 2 – 6x b. They are unlike terms, so they can not be combined. a. Combine terms by applying the distributive property. 2y + 7y = (2 + 7)y = 5y

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Slide 104 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Simplifying expressions Simplify each expression. a z – 9 + 7z b. 9x – 2(x – 5) b. a z – 9 + 7z = 13 +(– 9) + z + 7z = 13 +(– 9) + (1+ 7)z = 4 + 8z 9x – 2(x – 5) = 9x + (– 2)x + (2)(– 5) = 9x – 2x + 10 = 7x + 10

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Slide 105 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Simplifying expressions Simplify each expression. a. 6x 2 – y + 9x 2 – 3y b. b. a. 6x 2 – y + 9x 2 – 3y = 6x 2 + 9x 2 + (–1y) + (–3y) = (6 + 9)x 2 + (–1+ (– 3))y = 15x 2 –4y

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Slide 106 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Writing and simplifying an expression A sidewalk has a constant width w and comprises several short sections with lengths 11, 4, and 18 feet. a.Write and simplify an expression that gives the number of square feet of sidewalk. b.Find the area of the sidewalk if its width is 3 feet. a. 11w + 4w + 18w = ( )w = 33w b. 33w = 33 3 = 99 square feet 11 ft 4 ft 18 ft w

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