INTEGERS AND ALGEBRA. chapter CHAPTER 2 OUTLINE INTRODUCTION

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1 hut6236_ch2_a.qxd 9/22/8 9:55 AM Page 119 chapter 2 INTEGERS AND INTRODUCTION INTRODUCTION TO ALGEBRA CHAPTER 2 OUTLINE Section 2.1 Introduction to Integers page 121 Section 2.2 Addition of Integers page 132 Section 2.3 Subtraction of Integers page 139 Section 2.4 Multiplication of Integers page 145 Section 2.5 Division of Integers page 151 Section 2.6 Introduction to Algebra: Variables and Expressions page 158 Section 2.7 Evaluating Algebraic Expressions page 167 Section 2.8 Simplifying Algebraic Expressions page 177 Section 2.9 Introduction to Linear Equations page 186 Section 2.1 The Addition Property of Equality page Of all the sciences, meteorology may be both the least precise and the most talked about. Meteorologists study the weather and climate. Together with geologists, they provide us with many predictions that play a part in our everyday decisions. Among the things we might look for are temperatures, rainfall, water level, and tide level. On a day with unusual weather, we often become curious about the record for that day or even the all-time record. Here are a few of those records: The record high temperature for the United States is 134 F in 1913 in Death Valley, California. The record low temperature for the U.S. is 8 F in 1971 in Prospect Creek Camp, Alaska. The greatest one day temperature drop in the U.S. happened on Christmas Eve, 1924, in Montana. The temperature went from 63 F during the day to 21 F at night. The use of negative numbers for temperatures below zero is common for such numbers, but there are many other applications. For example, the tide range in Delaware varies between 1 feet and 5 feet. We examine these and several other climaterelated applications in Sections 2.1, 2.2, 2.3, and

2 Name Section ANSWERS Date 1. See exercise 2. 4, 2, 1,, 1, 5 3. Max: 7; Min: x w 17x 13. No 14. Yes Pretest Chapter 2 This pretest will provide a preview of the types of exercises you will encounter in each section of this chapter. The answers for these exercises can be found in the back of the text. If you are working on your own or are ahead of the class, this pretest can help you identify the sections in which you should focus more of your time. [2.1] Represent the integers on the number line shown. 1. 6, 8, 4, 2, 1 2. Place the following data set in ascending order: 5, 2, 4,, 1, Determine the maximum and minimum of the following data set: 4, 1, 5, 7, 3, 2. Evaluate: Find the opposite of each integer [2.6] Write each of the phrases using symbols less than x the quotient when w is divided by the product of x and Identify which are expressions and which are not x x 2(x 1) [2.2 to 2.5] Perform the indicated operations ( 3) ( 9) 17. ( 3) ( 2) ( 3) ( 6) 22. ( 7)( 3) w 2 t 28. a 2 4a 3 [2.7] Evaluate each expression (45 3 5) If x 2, y 7, and w 4, evaluate the expression [2.8] Combine like terms. x 2 y w w 2 t 3w 2 t 28. 4a 2 3a 5 7a 2 5a 2 12

3 2.1 Introduction to Integers 2.1 OBJECTIVES 1. Represent integers on a number line 2. Place a set of integers in ascending order 3. Determine the extreme values of a data set 4. Find the opposite of a given integer 5. Evaluate expressions involving absolute value When numbers are used to represent physical quantities (altitudes, temperatures, and amounts of money are examples), it may be necessary to distinguish between positive and negative quantities. The symbols and are used for this purpose. For instance, the altitude of Mount Whitney is 14,495 ft above sea level ( 14,495 ft). 14,495 ft Mount Whitney The altitude of Death Valley is 282 ft below sea level ( 282 ft). 282 ft Death Valley On a given day the temperature in Chicago might be 1 F below zero ( 1 F)

4 122 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA An account could show a gain of $1 ( 1) or a loss of $1 ( 1). These numbers suggest the need to extend the whole numbers to include both positive numbers (like 1) and negative numbers (like 282). To represent the negative numbers, we extend the number line to the left of zero and name equally spaced points. Numbers corresponding to points to the right of zero are positive numbers. They are written with a positive ( ) sign or with no sign at all. 6 and 9 are positive numbers Numbers corresponding to points to the left of zero are negative numbers. They are always written with a negative ( ) sign. 3 and 2 are negative numbers Read negative 3. The positive and negative numbers, as well as zero, are called the real numbers. Here is the number line extended to include positive and negative numbers, and zero. NOTE On the number line, we call zero the origin. Zero is neither positive nor negative Negative numbers Positive numbers NOTE Braces { and } are used to hold a collection of numbers. We call the collection a set. The dots are called ellipses and indicate that the pattern continues. OBJECTIVE 1 The numbers used to name the points shown on the number line are called the integers. The integers consist of the natural numbers, their negatives, and the number zero. We can represent the set of integers by {..., 3, 2, 1,, 1, 2, 3,...} Example 1 Represent the integers on the number line shown. 3, 12, 8, 15, Representing Integers on the Number Line

5 INTRODUCTION TO INTEGERS SECTION CHECK YOURSELF 1 Represent the integers on a number line. 1, 9, 4, 11, 8, The set of numbers on the number line is ordered. The numbers get smaller moving to the left on the number line and larger moving to the right When a group of numbers is written from smallest to largest, the numbers are said to be in ascending order. OBJECTIVE 2 Example 2 Ordering Integers Place each group of numbers in ascending order. (a) 9, 5, 8, 3, 7 From smallest to largest, the numbers are 8, 5, 3, 7, 9 Note that this is the order in which the numbers appear on a number line as we move from left to right. (b) 3, 2, 18, 2, 13 From smallest to largest, the numbers are 2, 13, 2, 3, 18 CHECK YOURSELF 2 Place each group of numbers in ascending order. (a) 12, 13, 15, 2, 8, 3 (b) 3, 6, 9, 3, 8 The least and greatest numbers in a group are called the extreme values. The least number is called the minimum, and the greatest number is called the maximum. OBJECTIVE 3 Example 3 Labeling Extreme Values For each group of numbers, determine the minimum and maximum values. (a) 9, 5, 8, 3, 7 From our previous ordering of these numbers, we see that 8, the least number, is the minimum, and 9, the greatest number, is the maximum. (b) 3, 2, 18, 2, 13 2 is the minimum and 18 is the maximum.

6 124 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA CHECK YOURSELF 3 For each group of numbers, determine the minimum and maximum values. (a) 12, 13, 15, 2, 8, 3 (b) 3, 6, 9, 3, 8 Each point on the number line corresponds to a real number.there are more points on the number line than integers. The real numbers include decimals, fractions, and other numbers. Example 4 Identifying Real Numbers that Are Integers Which of the real numbers, (a) 145, (b) 28, (c).35, and (d) 2, are also integers? 3 (a) 145 is an integer. (b) 28 is an integer. (c).35 is not an integer. (d) 2 3 is not an integer. CHECK YOURSELF 4 Which of the real numbers are also integers? 23 1, Sometimes we refer to the negative of a number as its opposite. For a nonzero number, this corresponds to a point the same distance from the origin as the given number, but on the other side of zero. Example 5 illustrates this. OBJECTIVE 4 Example 5 Find the Opposite of Each Number (a) 5 The opposite of 5 is 5. (b) 9 The opposite of 9 is 9. CHECK YOURSELF 5 Find the opposite of each number. (a) 17 Definition: Absolute Value (b) 12 The absolute value of a number represents the distance of the point named by the number from the origin on the number line.

7 INTRODUCTION TO INTEGERS SECTION Because we think of distance as a positive quantity (or as zero), the absolute value of a number is never negative. 5 units 5 units 5 5 The absolute value of 5 is 5. The absolute value of 5 is also 5. As a consequence of the definition, the absolute value of a positive number or zero is itself. The absolute value of a negative number is its opposite. In symbols, we write 5 5 and 5 5 Read the absolute Read the absolute value of 5. value of negative 5. The absolute value of a number does not depend on whether the number is to the right or to the left of the origin, but on its distance from the origin. OBJECTIVE 5 Example 6 Simplifying Absolute Value Expressions (a) 7 7 (b) 7 7 (c) 7 7 This is the negative, or opposite, of the absolute value of negative 7. CHECK YOURSELF 6 Evaluate. (a) 8 (b) 8 (c) 8 To determine the order of operation for an expression that includes absolute values, note that the absolute value bars are treated as a grouping symbol. Example 7 Adding or Subtracting Absolute Values (a) (b) Absolute value bars, like parentheses, serve as a set of grouping symbols, so do the operation inside first. (c) Evaluate the absolute values, then subtract. CHECK YOURSELF 7 Evaluate. (a) 9 4 (b) 9 4 (c) 9 4

8 126 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA READING YOUR TEXT The following fill-in-the-blank exercises are designed to assure that you understand the key vocabulary used in this section. Each sentence comes directly from the section. You will find the correct answers in Appendix C. Section 2.1 (a) When numbers are used to represent physical quantities, it may be necessary to distinguish between positive and quantities. (b) Numbers that correspond to points to the of zero are negative numbers. (c) When a set of numbers is written from smallest to largest, the numbers are said to be in order. (d) The absolute value of a number depends on its from the origin. CHECK YOURSELF ANSWERS (a) 13, 8, 3, 2, 12, 15 (b) 9, 3, 3, 6, (a) minimum is 13; maximum is 15; (b) minimum is 9; maximum is , 154,, and 5 5. (a) 17; (b) (a) 8; (b) 8; (c) (a) 13; (b) 5; (c) 5

9 2.1 Exercises Boost your GRADE at ALEKS.com! Represent each quantity with an integer. 1. An altitude of 4 ft above sea level Practice Problems Self-Tests NetTutor e-professors Videos 2. An altitude of 8 ft below sea level 3. A loss of $2 4. A profit of $4 5. A decrease in population of 25, 6. An increase in population of 12,5 Name Section ANSWERS Date [Objective 1] Represent the integers on the number lines shown. 7. 5, 15, 18, 8, , 6. 12, , 4, 5, 13, See exercise 8. See exercise Which numbers in the sets are integers? 9. 5, 2, 175, 234, ,.35, 3, 7, , 175, , 7, , 5, 1,, 2, 3, 8 [Objective 2] Place each group of numbers in ascending order , 2, 1,, 1, 6, , 6, 2, 1, 4, 5, , 5, 2,, 7, 1, , 7, 1, 8, 6, 1, , 15, 11, 5, 14, 2, , 2, 11, 4, 6, 1, , 18, 5, 11, 15, 14, , 6, 3, 3, 6, , 7, 7, 6, 3, , 13, 14, 14, 15, , 14, 13, 12, 14, 15 [Objective 3] For each group of numbers, determine the maximum and minimum values , 6,, 1, 3, 15, 1, , 1, 3, 11, 4, 2, 5, , 15,, 7, 9, 16, 3, ,, 22, 31, 18, 5, ,, 1, 2, 5, 4, , 7, 3, 5, 1, Max: 15; Min: Max: 11; Min: Max: 21; Min: Max: 22; Min: Max: 5; Min: Max: 7; Min: 1 SECTION

10 ANSWERS [Objective 4] Find the opposite of each number [Objective 5] Evaluate True 54. True Label each statement as true or false. 53. All whole numbers are integers. 54. All nonzero integers are real numbers. 128 SECTION 2.1

11 ANSWERS 55. All integers are whole numbers. 56. All real numbers are integers. 57. All negative integers are whole numbers. 58. Zero is neither positive nor negative. Place absolute value bars in the proper location on the left side of the expression so that the equation is true. 55. False 56. False 57. False 58. True ( 3) cm 59. ( 6) ( 8) ( 3) ( 2) ( 3) 11 Represent each quantity with a real number cm dollars dollars Science and Medicine The erosion of 5 centimeters (cm) of topsoil from an Iowa cornfield. 64. Science and Medicine The formation of 2.5 cm of new topsoil on the African savanna. 65. Business and Finance The withdrawal of $5 from a checking account. 66. Business and Finance The deposit of $2 in a savings account. 67. Science and Medicine The temperature change pictured. 2 6 F 1: P.M. 5 F 2: P.M. SECTION

12 ANSWERS Science and Medicine The temperature change indicated ,, 72. 6,, F F 1: P.M. 2: P.M. 69. Science and Medicine The temperature change indicated F F 1: P.M. 2: P.M. 7. Science and Medicine The temperature change indicated F 1: P.M F 2: P.M. 71. Business and Finance A country exported $9,, more than it imported, creating a positive trade balance. 72. Business and Finance A country exported $6,, less than it imported, creating a negative trade balance. 13 SECTION 2.1

13 ANSWERS For each group of numbers given in exercises 73 to 76, answer questions (a) to (d): (a) Which number is smallest? (b) Which number lies farthest from the origin? (c) Which number has the largest absolute value? (d) Which number has the smallest absolute value? 73. 6; 8; 8; ; 9; 9; ; 6; 6; 76. 9; 9; 9; 73. 6, 3, 8, 7, , 3, 5, 4, , 6, 1,, 2, ,, 2, 3, Simplify each of the following: ( 7) ( ( 7)) ( ( ( 7))) Based on your answers, generalize your results. Answers 1. 4 or ( 4) , , 175, , 5, 1,, 2, 3, , 6, 2, 1, 4, 5, , 6, 3, 3, 6, Max: 15; Min: Max: 21; Min: Max: 5; Min: True 55. False 57. False cm dollars ,, 73. 6; 8; 8; ; 6; 6; 77. SECTION

14 2.2 Addition of Integers 2.2 OBJECTIVES 1. Add two integers with the same sign 2. Add two integers with opposite signs 3. Solve applications involving integers In Section 2.1 we introduced the idea of negative numbers. Here we examine the four arithmetic operations (addition, subtraction, multiplication, and division) and see how those operations are performed when integers are involved. We start by considering addition. An application may help. We will represent a gain of money as a positive number and a loss as a negative number. If you gain $3 and then gain $4, the result is a gain of $7: If you lose $3 and then lose $4, the result is a loss of $7: 3 ( 4) 7 If you gain $3 and then lose $4, the result is a loss of $1: 3 ( 4) 1 If you lose $3 and then gain $4, the result is a gain of $1: The number line can be used to illustrate the addition of integers. Starting at the origin, we move to the right for positive integers and to the left for negative integers. OBJECTIVE 1 Example 1 (a) Add 3 4. Adding Integers Start at the origin and move 3 units to the right. Then move 4 more units to the right to find the sum. From the number line, we see that the sum is (b) Add ( 3) ( 4). 7 4 Start at the origin and move 3 units to the left. Then move 4 more units to the left to find the sum. From the number line, we see that the sum is ( 3) ( 4)

15 ADDITION OF INTEGERS SECTION CHECK YOURSELF 1 Add. (a) ( 4) ( 5) (c) ( 5) ( 15) (b) ( 3) ( 7) (d) ( 5) ( 3) You have probably noticed a helpful pattern in the previous example. This pattern will allow you to do the work mentally without having to use the number line. Look at the following rule. Property: Adding Integers Case 1: Same Sign NOTE This means that the sum of two positive integers is positive and the sum of two negative integers is negative. If two integers have the same sign, add their absolute values. Give the result the sign of the original integers. We can use the number line to illustrate the addition of two integers. This time the integers will have different signs. OBJECTIVE 2 Example 2 Adding Integers (a) Add 3 ( 6) First move 3 units to the right of the origin. Then move 6 units to the left. 3 ( 6) 3 (b) Add This time move 4 units to the left of the origin as the first step. Then move 7 units to the right CHECK YOURSELF 2 Add. (a) 7 ( 5) (b) 4 ( 8) (c) 1 16 (d) 7 3

16 134 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA You have no doubt noticed that, in adding a positive integer and a negative integer, sometimes the sum is positive and sometimes it is negative. This depends on which of the integers has the larger absolute value. This leads us to the second part of our addition rule. Property: Adding Integers Case 2: Different Signs RECALL We first encountered absolute values in Section 2.1. If two integers have different signs, subtract their absolute values, the smaller from the larger. Give the result the sign of the integer with the larger absolute value. Example 3 Adding Integers (a) 7 ( 19) 12 Because the two integers have different signs, subtract the absolute values ( ). The sum of 7 and 19 has the sign ( ) of the integer with the larger absolute value, 19. (b) Subtract the absolute values (13 7 6). The sum of 13 and 7 has the sign ( ) of the integer with the larger absolute value, 13. CHECK YOURSELF 3 Add mentally. (a) 5 ( 14) (c) 8 15 (b) 7 ( 8) (d) 7 ( 8) NOTE The opposite of a number is also called the additive inverse of that number. In Section 1.2, we discussed the commutative, associative, and additive identity properties. There is another property of addition that we should mention. Recall that every number has an opposite. It corresponds to a point the same distance from the origin as the given number but in the opposite direction. 3 3 NOTE 3 and 3 are opposites. 3 3 The opposite of 9 is 9. The opposite of 15 is 15. NOTE Here a represents the opposite of the number a. If a is positive, a is negative. If a is negative, a is positive. The additive inverse property states that the sum of any number and its opposite is. Property: Additive Inverse Property For any number a, there exists a number a such that a ( a) ( a) a The sum of any number and its opposite, or additive inverse, is.

17 ADDITION OF INTEGERS SECTION Example 4 Adding Integers NOTE Later, we will show that ; therefore, the opposite of is. (a) 9 ( 9) (b) CHECK YOURSELF 4 Add. (a) ( 17) 17 (b) 12 ( 12) When solving an application of integer arithmetic, the first step is to translate the phrase or statement using integers. Example 5 illustrates this step. OBJECTIVE 3 Example 5 An Application of the Addition of Integers Shanique has $25 in her checking account. She writes a check for $12 and makes a deposit of $9. What is the resulting balance? First, translate the phrase using integers. Such problems will usually include something that is represented by negative integers and something that is represented by positive integers. In this case, a check can be represented as a negative integer and a deposit as a positive integer. We have 25 ( 12) 9 This expression can now be evaluated. 25 ( 12) The resulting balance is $22. CHECK YOURSELF 5 Translate the problem into an integer expression and then answer the question. When Kirin awoke, the temperature was twelve degrees below zero, Fahrenheit. Over the next six hours, the temperature increased by seventeen degrees. What was the temperature at that time?

18 136 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA READING YOUR TEXT The following fill-in-the-blank exercises are designed to assure that you understand the key vocabulary used in this section. Each sentence comes directly from the section. You will find the correct answers in Appendix C. Section 2.2 (a) In Section 2.1, we introduced the idea of numbers. (b) To add two numbers with different signs, their absolute values. (c) The sum of any number and its opposite, or additive, is. (d) The opposite of zero is. CHECK YOURSELF ANSWERS 1. (a) 9; (b) 1; (c) 2; (d) 8 2. (a) 2; (b) 4; (c) 15; (d) 4 3. (a) 9; (b) 15; (c) 7; (d) 1 4. (a) ; (b) ; the temperature was 5 F.

19 2.2 Exercises Boost your GRADE at ALEKS.com! [Objectives 1 and 2] Add ( 2) ( 3) 6. ( 1) ( 9) Practice Problems Self-Tests NetTutor Name Section ANSWERS 1. 9 e-professors Videos Date 7. 9 ( 3) 8. 1 ( 4) ( 7) ( 12) ( 9) ( 5) ( 4) 6 ( 3) ( 3) 5 ( 11) ( 13) 16 [Objective 3] In exercises 17 to 22, restate the problem using an expression involving integers and then answer the question. 17. Business and Finance Amir has $1 in his checking account. He writes a check for $23 and makes a deposit of $51. What is his new balance? Business and Finance Olga has $25 in her checking account. She deposits $52 and then writes a check for $77. What is her new balance? Mechanical Engineering A pneumatic actuator is operated by a pressurized air reservoir. At the beginning of an operator s shift, the pressure in the reservoir was 126 pounds per square inch (lb/in. 2 ). At the end of each hour, the operator recorded the reservoir s change in pressure. The values recorded (in lb/in. 2 ) were a drop of 12, a drop of 7, a rise of 32, a drop of 17, a drop of 15, a rise of 31, a drop of 4, and a drop of 14. What was the pressure in the tank at the end of the shift? 17. $ $ lb/in. 2 SECTION

20 ANSWERS qt F F 2. Mechanical Engineering A diesel engine for an industrial shredder has an 18-quart (qt) oil capacity. When a maintenance technician checked the oil, it was 7 qt low. Later that day, she added 4 qt to the engine. What was the oil level after the 4 qt were added? Science and Medicine The lowest one-day temperature in Helena, Montana, was 21 F at night. The temperature increased by 25 degrees by noon. What was the temperature at noon? Science and Medicine At 7 A.M., the temperature was 15 F. By 1 P.M., the temperature had increased by 18 degrees. What was the temperature at 1 P.M.? In this chapter, it is stated that every number has an opposite. The opposite of 9 is 9. This corresponds to the idea of an opposite in English. In English, an opposite is often expressed by a prefix, for example, un- or ir-. (a) Write the opposite of these words: unmentionable, uninteresting, irredeemable, irregular, uncomfortable. (b) What is the meaning of these expressions: not uninteresting, not irredeemable, not irregular, not unmentionable? (c) Think of other prefixes that negate or change the meaning of a word to its opposite. Make a list of words formed with these prefixes and write a sentence with three of the words you found. Make a sentence with two words and phrases from parts (a) and (b). What is the value of [ ( 5)]? What is the value of ( 6)? How does this relate to the given examples? Write a short description about this relationship. Answers $ lb/in F SECTION 2.2

21 2.3 Subtraction of Integers 2.3 OBJECTIVES 1. Find the difference of two integers 2. Solve applications involving the subtraction of integers To begin our discussion of subtraction when integers are involved, we can look back at a problem using natural numbers. We know that From our work in adding integers, we know that it is also true that 8 ( 5) 3 Comparing these equations, we see that the results are the same. This leads us to an important pattern. Any subtraction problem can be written as a problem in addition. Subtracting 5 is the same as adding the opposite of 5, or 5. We can write this fact as follows: ( 5) 3 This leads us to the following rule for subtracting integers. Property: Subtracting Integers 1. To rewrite the subtraction problem as an addition problem: a. Change the subtraction operation to addition. b. Replace the integer being subtracted with its opposite. 2. Add the resulting integers as before. In symbols, a b a ( b) Example 1 illustrates the use of this property while subtracting. OBJECTIVE 1 Example 1 Subtracting Integers (a) ( 7) Change the subtraction symbol ( ) to an addition symbol ( ). Replace 7 with its opposite, 7. 8 (b) ( 12) 3 (c) ( 7) 13 (d) Subtract 5 from 2. We write the statement as 2 5 and proceed as before: ( 5) 7 139

22 14 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA CHECK YOURSELF 1 Subtract. (a) 18 7 (b) 5 13 (c) 7 9 (d) 2 7 The subtraction rule is used in the same way when the integer being subtracted is negative. Change the subtraction to addition. Replace the negative integer being subtracted with its opposite, which is positive. Example 2 illustrates this principle. Example 2 Subtracting Integers (a) 5 ( 2) 5 ( 2) (b) 7 ( 8) 7 ( 8) (c) 9 ( 5) (d) Subtract 4 from 5. We write 5 ( 4) Change the subtraction to addition. Replace 2 with its opposite, 2 or 2. CHECK YOURSELF 2 Subtract. (a) 8 ( 2) (c) 7 ( 2) (b) 3 ( 1) (d) 7 ( 7) OBJECTIVE 2 Example 3 An Application of the Subtraction of Integers Susanna s checking account shows a balance of $285. She has discovered that a deposit for $47 was accidently recorded as a check for $47. Write an integer expression that represents the correction on the balance. Then find the corrected balance. 285 ( 47) (47) Subtract the check and then add the deposit. 285 ( 47) (47) The corrected balance is $379. CHECK YOURSELF 3 It appears that Marshal, a running back, gained 97 yards in the last game. A closer inspection of the statistics revealed that a 9-yard gain had been recorded as a 9-yard loss. Write an integer expression that represents the corrected yards gained and then find that number.

23 SUBTRACTION OF INTEGERS SECTION Using Your Calculator to Add and Subtract Integers Your scientific (or graphing) calculator has a key that makes a number negative. This key is different from the subtraction key. The negative key is marked as either or ( ). With a scientific calculator, this key is pressed after the number you wish to make negative is entered. All of the instructions in this section assume that you have a scientific calculator. Example 4 Entering a Negative Integer into the Calculator NOTE The 12 changes between positive and negative in the display. The final display is 12, because there are an even number of negative signs in front of the 12. Enter each of the following into your calculator. (a) (b) ( ( ( 12))) 12 CHECK YOURSELF 4 Enter each number into your calculator. (a) 36 (b) ( ( 6)) Example 5 Adding Integers Find the sum for each pair of integers. (a) 256 ( 297) Your display should read 41. (b) 312 ( 569) Your display should read 881. CHECK YOURSELF 5 Find the sum for each pair of integers. (a) Example 6 Subtracting Integers Find the difference for 356 ( 469) Your display should read 113. (b) 596 ( 834)

24 142 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA CHECK YOURSELF 6 Find each difference. (a) 349 ( 49) (b) 294 ( 137) READING YOUR TEXT The following fill-in-the-blank exercises are designed to assure that you understand the key vocabulary used in this section. Each sentence comes directly from the section. You will find the correct answers in Appendix C. Section 2.3 (a) Any subtraction problem can be written as a problem in. (b) To rewrite a subtraction problem as an addition problem, change the subtraction operation to addition and replace the integer being subtracted with its. (c) The opposite of 2 is. (d) The calculator key that makes a number negative is different from the key. CHECK YOURSELF ANSWERS 1. (a) 11; (b) 8; (c) 16; (d) 9 2. (a) 1; (b) 13; (c) 5; (d) ( 9) yards 4. (a) 36; (b) 6 5. (a) 179; (b) 1,43 6. (a) 398; (b) 157

25 2.3 Exercises Boost your GRADE at ALEKS.com! [Objective 1] Subtract ( 11) ( 5) ( 12) ( 1) ( 24) ( 11) ( 27) ( 16) [Objective 2] For exercises 21 to 23, write an integer expression that describes the situation. Then answer the question. 21. Science and Medicine The temperature at noon on a June day was 82 F. It fell by 12 degrees in the next 4 h. What was the temperature at 4: P.M.? 22. Business and Finance Jason s checking account shows a balance of $853. He has discovered that a deposit of $7 was accidently recorded as a check for $7. What is the corrected balance? 23. Business and Finance Ylena s checking account shows a balance of $947. She has discovered that a check for $86 was recorded as a deposit of $86. What is the corrected balance? 24. Technology How long ago was the year 125 B.C.E.? What year was 3,3 years ago? Make a number line and locate the following events, cultures, and objects on it. How long ago was each item in the list? Which two events are the closest to each other? You may want to learn more about some of the cultures in the list and the mathematics and science developed by each culture. 2 Practice Problems Self-Tests NetTutor Name Section ANSWERS e-professors Videos Date Inca culture in Peru 14 A.D. The Ahmes Papyrus, a mathematical text from Egypt 165 B.C.E. Babylonian arithmetic develops the use of a zero symbol 3 B.C.E. First Olympic Games 776 B.C.E. Pythagoras of Greece dies 5 B.C.E. Mayans in Central America independently develop use of zero 5 A.D. The Chou Pei, a mathematics classic from China 1 B.C.E. The Aryabhatiya, a mathematics work from India 499 A.D. Trigonometry arrives in Europe via the Arabs and India 1464 A.D. Arabs receive algebra from Greek, Hindu, and Babylonian sources and develop it into a new systematic form 85 A.D. Development of calculus in Europe 167 A.D. Rise of abstract algebra 186 A.D. Growing importance of probability and development of statistics 192 A.D F ( 7) 7 $993 is the balance ( 86) $775 is the balance SECTION

26 ANSWERS F F , Complete the following statement: 3 ( 7) is the same as because... Write a problem that might be answered by doing this subtraction. 26. Explain the difference between the two phrases: a number subtracted from 5 and a number less than 5. Use algebra and English to explain the meaning of these phrases. Write other ways to express subtraction in English. Which ones are confusing? 27. Science and Medicine The greatest one-day temperature drop in the United States happened on Christmas Eve, 1924, in Montana. The temperature went from 63 F during the day to 21 F at night. What was the total temperature drop? 28. Science and Medicine A similar one-day temperature drop happened in Alaska. The temperature went from 47 F during the day to 29 F at night. What was the total temperature drop? Science and Medicine The tide at the mouth of the Delaware River tends to vary between a maximum of 1 ft and a minimum of 5 ft. What is the difference in feet between the high tide and the low tide? 3. Science and Medicine The tide at the mouth of the Sacramento River tends to vary between a maximum of 7 ft and a minimum of 2 ft. What is the difference in feet between the high tide and the low tide? Calculator Exercises Use your calculator to perform the following operations ( 128) ( 567) ( 431) ( 367) ( 25) ( 219) ( 456) ( 123) Answers F ( 86); $775 is the balance F SECTION 2.3

27 2.4 Multiplication of Integers 2.4 OBJECTIVES 1. Find the product of two or more integers 2. Use the order of operations with integers When you first considered multiplication in arithmetic, it was thought of as repeated addition. Now we look at what our work with the addition of integers can tell us about multiplication when integers are involved. For example, We interpret multiplication as repeated addition to find the product, 12. Now, consider the product (3)( 4): (3)( 4) ( 4) ( 4) ( 4) 12 Looking at this product suggests the first portion of our rule for multiplying integers. The product of a positive integer and a negative integer is negative. Property: Multiplying Integers Case 1: Different Signs The product of two integers with different signs is negative. To use this rule in multiplying two integers with different signs, multiply their absolute values and attach a negative sign. OBJECTIVE 1 Example 1 Multiplying Integers Multiply. (a) (5)( 6) 3 (b) (c) The product is negative. ( 1)(1) 1 (8)( 12) 96 CHECK YOURSELF 1 Multiply. (a) ( 7)(5) (b) ( 12)(9) (c) ( 15)(8) 145

28 146 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA The product of two negative integers is harder to visualize. The following pattern may help you see how we can determine the sign of the product. (3)( 2) 6 (2)( 2) 4 NOTE This number is decreasing by 1. (1)( 2) 2 ()( 2) Do you see that the product is increasing by 2 each time as you go down? NOTE ( 1)( 2) is the opposite of 2. ( 1)( 2) 2 What should the product ( 2)( 2) be? Continuing the pattern shown, we see that ( 2)( 2) 4 This suggests that the product of two negative integers is positive, which is the case. We can extend our multiplication rule. Property: Multiplying Integers Case 2: Same Sign The product of two integers with the same sign is positive. Example 2 Multiplying Integers Multiply. (a) 9 # 7 63 The product of two positive numbers (same sign, ) is positive. (b) ( 8)( 5) 4 The product of two negative numbers (same sign, ) is positive. CHECK YOURSELF 2 Multiply. (a) 1 12 (b) ( 8)( 9) The multiplicative identity property and multiplicative property of zero studied in Section 1.5 can be applied to integers, as illustrated in Example 3. Example 3 Find each product. (a) (1)( 7) 7 (b) (15)(1) 15 (c) ( 7)() (d) # 12 Multiplying Integers by One and Zero

29 MULTIPLICATION OF INTEGERS SECTION CHECK YOURSELF 3 Multiply. (a) ( 1)(1) (b) ()( 17) We can now extend the rules for the order of operations learned in Section 1.8 to simplify expressions containing integers. First, we will work with integers raised to a power. OBJECTIVE 2 Example 4 Integers with Exponents Evaluate each expression. NOTE In part (b) of Example 4, we have a negative integer raised to a power. In part (c), only the 3 is raised to a power. We have the opposite of 3 squared. (a) ( 3) 2 ( 3)( 3) 9 (b) ( 3) 3 ( 3)( 3)( 3) 27 (c) 3 2 (3 3) 9 Note that the negative is not squared. CHECK YOURSELF 4 Evaluate each expression. (a) ( 4) 2 (b) ( 4) 3 (c) 4 2 In Example 5 we will apply the order of operations. Example 5 Using Order of Operations with Integers Evaluate each expression. (a) 7( 9 12) 7(3) 21 Evaluate inside the parentheses first. (b) ( 8)( 7) (c) ( 5) 2 3 ( 5)( 5) Multiply first, then subtract. Evaluate the power first. (d) Note that The power applies only to the Note that ( 5) 2 ( 5)( 5) 25

30 148 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA CHECK YOURSELF 5 Evaluate each expression. (a) 8( 9 7) (b) ( 3)( 5) 7 (c) ( 4) 2 ( 4) (d) 4 2 ( 4) READING YOUR TEXT The following fill-in-the-blank exercises are designed to assure that you understand the key vocabulary used in this section. Each sentence comes directly from the section. You will find the correct answers in Appendix C. Section 2.4 (a) The product of two integers with different signs is. (b) The product of two integers with the same sign is. (c) Given the expression 3 2, the is not squared. (d) The rules for order of operations were learned in Section. CHECK YOURSELF ANSWERS 1. (a) 35; (b) 18; (c) (a) 12; (b) (a) 1; (b) 4. (a) 16; (b) 64; (c) (a) 16; (b) 22; (c) 2; (d) 12

31 2.4 Exercises Boost your GRADE at ALEKS.com! [Objective 1] Multiply (5)( 12) 4. (1)( 2) Practice Problems Self-Tests NetTutor Name Section e-professors Videos Date 5. ( 8)(9) 6. ( 12)(3) ANSWERS 7. ( 8)( 7) 8. ( 9)( 8) 9. ( 5)( 12) 1. ( 7)( 3) 11. ()( 18) 12. ( 17)() 13. (15)() 14. ()(25) [Objective 2] Do the indicated operations. Remember the rules for the order of operations (7 2) 16. 7(8 5) 17. 2(5 8) 18. 6(14 16) (9 7) 2. 6(12 9) 21. 3( 2 5) 22. 2( 7 3) 23. ( 2)(3) ( 6)(8) ( 7) ( 3)( 9) ( 5)( 2) ( 7)( 3) (3)( 7) 2 3. (2)( 6) ( 3)(6) ( 2)(3) ( 4)( 2) ( 2)( 7) 35. ( 7) ( 6) SECTION

32 ANSWERS $17, $ F 37. ( 5) ( 2) ( 4) 2 ( 2)( 5) 42. ( 3) 3 ( 8)( 2) 43. ( 8) ( 6) ( 6) 2 ( 3) ( 8) 2 ( 4) ( 5) ( 6) Business and Finance Stores occasionally sell products at a loss in order to draw in customers or to reward good customers. The theory is that customers will buy other products along with the discounted item and the store will ultimately profit. Beguhn Industries sells five different products. The company makes $18 on each product-a item sold, loses $4 on product-b items, earns $11 on product C, makes $38 on product D, and loses $15 on product E. One month, Beguhn Industries sold 127 units of product A, 273 units of product B, 21 units of product C, 377 units of product D, and 43 units of product E. What was their profit or loss that month? 52. Statistics In Atlantic City, Nick played the slot machines for 12 h. He lost $45 an hour. Use integers to represent the change in Nick s financial status at the end of the 12 h. 53. Science and Medicine The temperature is 6 F at 5: in the evening. If the temperature drops 2 degrees every hour, what is the temperature at 1: A.M.? 2 Answers $17, F 15 SECTION 2.4

33 2.5 Division of Integers 2.5 OBJECTIVES 1. Find the quotient of two integers 2. Use the order of operations with integers You know from your work in arithmetic that multiplication and division are related operations. We can use that fact, and our work of Section 2.4, to determine rules for the division of integers. Every division problem can be stated as an equivalent multiplication problem. For instance, because because because 15 5 # 3 24 (6)( 4) 3 ( 5)(6) These examples illustrate that because the two operations are related, the rule of signs that we stated in Section 2.4 for multiplication is also true for division. Property: Dividing Integers 1. The quotient of two integers with different signs is negative. 2. The quotient of two integers with the same sign is positive. Again, the rule is easy to use. To divide two integers, divide their absolute values. Then attach the proper sign according to the rule. OBJECTIVE 1 Example 1 Dividing Integers Divide. (a) Positive 28 4 Positive Positive 7 (b) Negative 36 9 Negative 4 Positive (c) Negative 42 6 Positive 7 Negative (d) Positive Negative 3 Negative 151

34 152 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA CHECK YOURSELF 1 Divide. (a) (c) (b) (d) As discussed in Section 1.6, we must be very careful when is involved in a division problem. Remember that divided by any nonzero number is just. This rule can be extended to include integers, so that 7 because ( 7)() However, if zero is the divisor, we have a special problem. Consider 9? This means that 9?. Can times a number ever be 9? No, so there is no solution. 9 Because cannot be replaced by any number, we agree that division by is not allowed. We say that division by is undefined. Example 2 Dividing Integers Divide, if possible. (a) 7 is undefined. (b) 9 is undefined. (c) 5 (d) 8 Note: The expression is called an indeterminate form. You will learn more about this in later mathematics classes.

35 DIVISION OF INTEGERS SECTION CHECK YOURSELF 2 Divide, if possible. 5 7 (a) (b) (c) (d) 3 9 The fraction bar, like parentheses and the absolute value bars, serves as a grouping symbol. This means that all operations in the numerator and denominator should be performed separately. Then the division is done as the last step. Example 3 illustrates this property. OBJECTIVE 2 Example 3 Using Order of Operations Evaluate each expression. (a) ( 6)( 7) Multiply in the numerator, then divide. (b) 3 ( 12) Add in the numerator, then divide. (c) 4 (2)( 6) ( 12) Multiply in the numerator. Then add in the numerator and subtract in the denominator. Divide as the last step. CHECK YOURSELF 3 Evaluate each expression. 4 ( 8) 3 (2)( 6) (a) (b) (c) 6 5 ( 2)( 4) ( 6)( 5) ( 2)(11) Using Your Calculator to Multiply and Divide Integers Finding the product of two integers using a calculator is relatively straightforward. Example 4 Multiplying Integers Find the product. (457) ( 734) Your display should read 335,438.

36 154 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA CHECK YOURSELF 4 Find the products. (a) (36) ( 91) (b) ( 12) ( 284) Finding the quotient of integers is also straightforward. Example 5 Dividing Integers Find the quotient Your display should read 24. CHECK YOURSELF 5 Find the quotient. ( 7,865) ( 242) We can also use the calculator to raise an integer to a power. Example 6 Raising a Number to a Power Evaluate. ( 3) 6 NOTE The parentheses ensure that the negative is attached to the 3 before it is raised to a power. ( 3 ) y x 6 or, on some calculators ( ( ) 3 ) ^ 6 Enter Either way, your display should read 729. CHECK YOURSELF 6 Evaluate. ( 2) 9

37 DIVISION OF INTEGERS SECTION READING YOUR TEXT The following fill-in-the-blank exercises are designed to assure that you understand the key vocabulary used in this section. Each sentence comes directly from the section. You will find the correct answers in Appendix C. Section 2.5 (a) The quotient of two integers with different signs is. (b) The quotient of two integers with the same sign is. (c) Division by is not allowed. (d) The fraction bar serves as a symbol. CHECK YOURSELF ANSWERS 1. (a) 5; (b) 4; (c) 6; (d) (a) ; (b) undefined; (c) undefined; (d) 3. (a) 2; (b) 3; (c) 1 4. (a) 3,276; (b) 3,

38 Boost your GRADE at ALEKS.com! 2.5 Exercises Practice Problems Self-Tests NetTutor Name e-professors Videos [Objective 1] Divide Section ANSWERS Date Undefined Undefined [Objective 2] Perform the indicated operations. ( 6)( 3) ( 9)(5) ( 8)(2) Undefined 34. Undefined (7)( 8) SECTION 2.5

39 ANSWERS For exercises 35 to 37, use integers to write an expression that represents the situation. Then answer the question. 35. Business and Finance Patrick worked all day mowing lawns and was paid $9 per hour. If he had $125 at the end of a 9-h day, how much did he have before he started working? $ weeks 2, 16,232 $1, , , Social Science A woman lost 42 lb. If she lost 3 lb each week, how long has she been dieting? 37. Business and Finance Suppose that you and your two brothers bought equal shares of an investment for a total of $2, and sold it later for $16,232. How much did each person lose? ,24 Calculator Exercises Use your calculator to multiply and divide. 38. (15) ( 45) 39. (78) ( 12) 4. ( 56) (31) 41. ( 34) ( 28) 42. ( 71) ( 19) ( 5) ( 4) 5 Answers Undefined Undefined $44 2, 16, $1, ,24 SECTION