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Section 1.1 Numbers and Their Properties

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OBJECTIVES A Write a set of numbers using roster or set–builder notation.

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OBJECTIVES B Write a rational number as a decimal.

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OBJECTIVES C Classify a number as natural, whole, integer, rational, irrational, or real.

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OBJECTIVES D Find the additive inverse of a number.

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OBJECTIVES E Find the absolute value of a number.

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OBJECTIVES F Given two numbers, use the correct notation to indicate equality or which is larger.

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DEFINITION NATURAL NUMBERS The set of numbers used for counting.

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DEFINITION WHOLE NUMBERS The set of natural numbers and zero.

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DEFINITION INTEGERS The set of whole numbers and their opposites(negatives).

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**DEFINITION RATIONAL NUMBERS**

All numbers that can be written as the ratio of two integers.

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**DEFINITION IRRATIONAL NUMBERS**

Numbers that cannot be written as ratios of two integers.

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**DEFINITION REAL NUMBERS**

Numbers that are either rational or irrational:

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DEFINITION ADDITIVE INVERSE The additive inverse(opposite) of a is –a.

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**DEFINITION ABSOLUTE VALUE**

The distance between a and 0 on the real-number line

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CAUTION The absolute value is always positive or zero.

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**DEFINITION TRICHOTOMY LAW**

If given any two real numbers, only one of three things is true: a is equal to b, denoted by a = b, or a is less than b, denoted by a < b, or a is greater than b, denoted by a >b.

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**Chapter 1 The Real Numbers Section 1.1A**

Practice Test Exercise #1

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**Use roster notation to list the natural numbers between 5 and 9.**

The set of natural numbers between 5 and 9 is {6, 7,8} Note 5 and 9 are not included

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**Chapter 1 The Real Numbers Section 1.1B**

Practice Test Exercise #2

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**Chapter 1 The Real Numbers Section 1.1C**

Practice Test Exercise #3

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**Classify the given number by making a check mark () in the appropriate row(s).**

Natural number Whole number Integer Rational number Irrational number Real number

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**Chapter 1 The Real Numbers Section 1.1D**

Practice Test Exercise #4

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**Chapter 1 The Real Numbers Section 1.1E**

Practice Test Exercise #5

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Find:

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**Chapter 1 The Real Numbers 1.1F**

Practice Test Exercise #6

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**Fill in the blank with <, >, or = to make the resulting statement true:**

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Section 1.2 Operations and Properties of Real Numbers

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OBJECTIVES A Add, subtract, multiply, and divide signed numbers.

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OBJECTIVES B Identify uses of the properties of the real numbers.

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**PROCEDURE TO ADD TWO NUMBERS WITH THE SAME SIGN:**

Add their absolute values and give the sum the common sign.

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**PROCEDURE TO ADD TWO NUMBERS WITH DIFFERENT SIGNS:**

Find the absolute value. Subtract the smaller from the greater number. Use the sign of the number with the greater absolute value.

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DEFINITION ADDITIVE IDENTITY For any real number a:

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DEFINITION SUBTRACTION OF SIGNED NUMBERS If a and b are real numbers:

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DEFINITION ADDITIVE INVERSE For any real number a:

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PROCEDURE SIGNIFY MULTIPLICATION

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**PROCEDURE MULTIPLYING NUMBERS WITH OPPOSITE SIGNS**

To multiply a positive number by a negative number, multiply their absolute values and make the product negative.

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**DEFINITION SIGNS OF MULTIPLICATION PRODUCTS Same signs: Positive(+)**

Different signs: Negative(–)

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DEFINITION IDENTITY FOR MULTIPLICATION For any real number a:

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DEFINITION MULTIPLICATION OF FRACTIONS

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**DEFINITION DIVISION OF REAL NUMBERS**

If a and b are real numbers and b is not zero:

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**DEFINITION SIGNS OF A FRACTION**

For any real number a and nonzero real number b, there are two cases of signs:

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DEFINITION ZERO IN DIVISION For a ≠ 0:

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CAUTION

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**DEFINITION MULTIPLICATIVE INVERSE (RECIPROCAL)**

Every nonzero real number a has a reciprocal such that:

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DEFINITION DIVISION OF FRACTIONS

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**Chapter 1 The Real Numbers**

Section 1.2A Practice Test Exercise #7

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Find.

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**Chapter 1 The Real Numbers**

Section 1.2A Practice Test Exercise #8

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Find.

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**Chapter 1 The Real Numbers**

Section 1.2A Practice Test Exercise #9

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Find. Least common denominator = 8. Now add numerators.

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Find. Least common denominator = 12.

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Now add numerators.

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**Chapter 1 The Real Numbers**

Section 1.2A Practice Test Exercise #10

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Find.

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**Chapter 1 The Real Numbers**

Section 1.2A Practice Test Exercise #11

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Find. 1

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Find. 1 3 1 4

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**Chapter 1 The Real Numbers**

Section 1.2B Practice Test Exercise #12

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**Name the property illustrated in the statement.**

Commutative Property of Addition Associative Property of Addition

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**Chapter 1 The Real Numbers**

Section 1.2B Practice Test Exercise #13

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**Name the property illustrated in the statement.**

Inverse Property of Multiplication. Inverse Property of Addition.

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Section 1.3 Properties of Exponents

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OBJECTIVES A Evaluate expressions containing natural numbers as exponents.

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OBJECTIVES B Write an expression containing negative exponents as a fraction.

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OBJECTIVES C Multiply and divide expressions containing exponents.

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OBJECTIVES D Raise a power to a power.

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OBJECTIVES E Raise a quotient to a power.

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OBJECTIVES F Convert between ordinary decimal notation and scientific notation, and use scientific notation in computations.

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**DEFINITION EXPONENT AND BASE**

If a is a real number and n is a natural number:

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**Chapter 1 The Real Numbers**

Section 1.3A Practice Test Exercise #14

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Evaluate.

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**Chapter 1 The Real Numbers**

Section 1.3B Practice Test Exercise #15

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**Write without negative exponents.**

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**Chapter 1 The Real Numbers**

Section 1.3C Practice Test Exercise #16

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**Perform the indicated operation and simplify.**

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**Perform the indicated operation and simplify.**

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**Perform the indicated operation and simplify.**

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**Chapter 1 The Real Numbers**

Section 1.3D, E Practice Test Exercise #17

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Simplify.

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Simplify.

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**Chapter 1 The Real Numbers**

Section 1.3F Practice Test Exercise #18

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**The exponent of 10, (–3), means move the decimal point 3 places to the left.**

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**Chapter 1 The Real Numbers**

Section 1.3F Practice Test Exercise #19

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**The exponent of 10, (5), means move the decimal point 5 places to the right.**

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**Chapter 1 The Real Numbers**

Section 1.3F Practice Test Exercise #20

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**Perform the calculation and write your answer in scientific notation.**

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**Perform the calculation and write your answer in scientific notation.**

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Section 1.4 Algebraic Expressions and The Order of Operations

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OBJECTIVES A Evaluate numerical expressions with grouping symbols.

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**OBJECTIVES Evaluate expressions using the correct order of operations.**

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OBJECTIVES C Evaluate algebraic expressions.

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OBJECTIVES D Use the distributive property to simplify expressions.

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OBJECTIVES E Simplify expressions by combining like terms.

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OBJECTIVES F Simplify expressions by removing grouping symbols and combining like terms.

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**PROCEDURE ORDER OF OPERATIONS Do the operations in the (). P**

Evaluate exponential expressions. Perform multiplications and divisions from left to right. Perform additions and subtractions from left to right. P E (MD) (AS)

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**Identity for Multiplication**

PROCEDURE Identity for Multiplication For any real number a:

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PROCEDURE Additive Inverse For any real number a:

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**Additive Inverse of a Sum**

PROCEDURE Additive Inverse of a Sum

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**Additive Inverse of a Difference**

PROCEDURE Additive Inverse of a Difference

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DEFINITION LIKE TERMS Constant terms or terms with exactly the same variable factors are called similar or like terms.

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**Chapter 1 The Real Numbers**

Section 1.4A Practice Test Exercise #21

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Evaluate. 11 1

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**Chapter 1 The Real Numbers**

Section 1.4B Practice Test Exercise #22

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Evaluate.

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**Chapter 1 The Real Numbers**

Section 1.4C Practice Test Exercise #23

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Evaluate. a.

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Evaluate. b.

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**Chapter 1 The Real Numbers**

Section 1.4D, E Practice Test Exercise #24

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Simplify.

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**Chapter 1 The Real Numbers**

Section 1.4F Practice Test Exercise #25

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Simplify.

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