* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download integers and introduction to algebra
Survey
Document related concepts
Mathematics of radio engineering wikipedia , lookup
Infinitesimal wikipedia , lookup
Positional notation wikipedia , lookup
Factorization wikipedia , lookup
Abuse of notation wikipedia , lookup
Location arithmetic wikipedia , lookup
Large numbers wikipedia , lookup
Collatz conjecture wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
P-adic number wikipedia , lookup
Elementary mathematics wikipedia , lookup
Transcript
hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 119 chapter 2 INTEGERS AND INTRODUCTION INTRODUCTION TO ALGEBRA © 2010 McGraw-Hill Companies 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 2.10 The Addition Property of Equality page 192 2 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 134F in 1913 in Death Valley, California. • The record low temperature for the U.S. is –80F 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 63F during the day to –21F 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 10 feet and 5 feet. We examine these and several other climaterelated applications in Sections 2.1, 2.2, 2.3, and 2.7. 119 hut06236_ch02_A.qxd 9/22/08 Name 9:55 AM Page 120 Pretest Chapter 2 Section Date 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. ANSWERS [2.1] 1. See exercise 1. 6, 8, 4, 2, 10 2. 4, 2, 1, 0, 1, 5 3. Max: 7; Min: 5 4. 5 Represent the integers on the number line shown. 8 6 4 2 0 2 4 6 8 10 2. Place the following data set in ascending order: 5, 2, 4, 0, 1, 1. 5. 6 3. Determine the maximum and minimum of the following data set: 4, 1, 5, 6. 6 7, 3, 2. 7. 6 Evaluate: 8. 6 4. ⏐5⏐ 5. ⏐6⏐ 6. ⏐11 5⏐ 7. ⏐11⏐⏐5⏐ 8. ⏐4 5⏐⏐6 3⏐ 9. 16 10. 23 11. x8 12. w — 17x 13. No 14. Yes 11. 8 less than x 15. 10 12. the quotient when w is divided by the product of x and 17 16. 1 Identify which are expressions and which are not. 17. 5 18. 3 19. 19 20. 12 21. 0 22. 21 23. 7 24. 3 25. 55 26. 7 27. 8w 2t 28. a2 4a 3 Find the opposite of each integer. 9. 16 Write each of the phrases using symbols. 13. 7x 5 11 [2.2 to 2.5] [2.7] 14. 3x 2(x 1) Perform the indicated operations. 15. 7 (3) 16. 8 (9) 17. (3) (2) 18. 8 11 19. 8 11 20. 9 (3) 21. 6 (6) 22. (7)(3) 23. 27 6 3 Evaluate each expression. 24. 5 42 3 6 25. (45 3 5) 52 26. If x 2, y 7, and w 4, evaluate the expression [2.8] Combine like terms. 27. 5w 2t 3w 2t 120 x2y . w 28. 4a2 3a 5 7a 2 5a2 © 2010 McGraw-Hill Companies [2.6] 10. 23 hut06236_ch02_A.qxd 9/22/08 9:55 AM 2.1 Page 121 Introduction to Integers 2.1 OBJECTIVES 1. 2. 3. 4. 5. Represent integers on a number line Place a set of integers in ascending order Determine the extreme values of a data set Find the opposite of a given integer 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 © 2010 McGraw-Hill Companies On a given day the temperature in Chicago might be 10F below zero (10F). 110 100 90 8 7 6 5 4 3 2 1 0 –10 –20 121 hut06236_ch02_A.qxd 122 9/22/08 CHAPTER 2 9:55 AM Page 122 INTEGERS AND INTRODUCTION TO ALGEBRA An account could show a gain of $100 (100) or a loss of $100 (100). These numbers suggest the need to extend the whole numbers to include both positive numbers (like 100) 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 20 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 Zero is neither positive nor negative call zero the origin. 3 2 1 0 Negative numbers 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 2 3 Positive numbers 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, 0, 1, 2, 3, . . .} Example 1 Representing Integers on the Number Line Represent the integers on the number line shown. 3, 12, 8, 15, 7 12 7 3 15 10 5 8 0 5 15 10 15 © 2010 McGraw-Hill Companies NOTE Braces { and } are used 1 hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 123 INTRODUCTION TO INTEGERS SECTION 2.1 123 CHECK YOURSELF 1 Represent the integers on a number line. 1, 9, 4, 11, 8, 20 15 10 5 0 5 10 15 20 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. 4 3 2 1 0 1 2 3 4 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, 20, 13 From smallest to largest, the numbers are 20, 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 © 2010 McGraw-Hill Companies 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, 20, 13 20 is the minimum and 18 is the maximum. 124 9/22/08 CHAPTER 2 9:55 AM Page 124 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 2 Which of the real numbers, (a) 145, (b) 28, (c) 0.35, and (d) , are also integers? 3 (a) 145 is an integer. (b) 28 is an integer. (c) 0.35 is not an integer. 2 (d) is not an integer. 3 CHECK YOURSELF 4 Which of the real numbers are also integers? 23 1,054 0.23 0 500 4 5 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 (b) 9 The opposite of 5 is 5. The opposite of 9 is 9. CHECK YOURSELF 5 Find the opposite of each number. (b) 12 (a) 17 Definition: Absolute Value The absolute value of a number represents the distance of the point named by the number from the origin on the number line. © 2010 McGraw-Hill Companies hut06236_ch02_A.qxd hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 125 INTRODUCTION TO INTEGERS SECTION 2.1 125 Because we think of distance as a positive quantity (or as zero), the absolute value of a number is never negative. 5 units 5 5 units 0 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 Read “the absolute value of 5.” 5 5 Read “the absolute 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 © 2010 McGraw-Hill Companies (a) ⏐10⏐⏐10⏐ 10 10 20 (b) ⏐8 3⏐⏐5⏐ 5 Absolute value bars, like parentheses, serve as a set of grouping symbols, so do the operation inside first. (c) ⏐8⏐⏐3⏐ 8 3 5 Evaluate the absolute values, then subtract. CHECK YOURSELF 7 Evaluate. (a) ⏐9⏐⏐4⏐ (b) ⏐9 4⏐ (c) ⏐9⏐⏐4⏐ 126 9/22/08 CHAPTER 2 9:55 AM Page 126 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 are negative numbers. of zero (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 the origin. from CHECK YOURSELF ANSWERS 1. 119 20 15 10 5 1 0 4 8 5 20 10 15 20 2. (a) 13, 8, 3, 2, 12, 15 (b) 9, 3, 3, 6, 8 3. (a) minimum is 13; maximum is 15; (b) minimum is 9; maximum is 8 4. 23, 1054, 0, and 500 5. (a) 17; (b) 12 6. (a) 8; (b) 8; (c) 8. 7. (a) 13; (b) 5; (c) 5 © 2010 McGraw-Hill Companies hut06236_ch02_A.qxd hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 127 Boost your GRADE at ALEKS.com! Exercises 2.1 Represent each quantity with an integer. • Practice Problems • Self-Tests • NetTutor 1. An altitude of 400 ft above sea level • e-Professors • Videos Name 2. An altitude of 80 ft below sea level Section Date 3. A loss of $200 ANSWERS 4. A profit of $400 1. 400 5. A decrease in population of 25,000 2. 80 6. An increase in population of 12,500 3. 200 [Objective 1] Represent the integers on the number lines shown. 7. 5, 15, 18, 8, 3 15 20 8 4. 400 35 10 0 5. 25,000 18 10 6. 12,500 20 7. See exercise 8. 18, 4, 5, 13, 9 18 20 5 10 4 9 13 0 10 8. See exercise 20 9. 5, 175, 234 Which numbers in the sets are integers? 9. 2 5, , 175, 234, 0.64 9 10. 3 45, 0.35, , 700, 26 5 [Objective 2] Place each group of numbers in ascending order. 10. 45, 700, 26 11. 7, 5, 1, 0, 2, 3, 8 12. 8, 2, 1, 0, 1, 6, 7 13. 11, 6, 2, 1, 4, 5, 9 18, 15, 11, 5, 11. 3, 5, 2, 0, 7, 1, 8 12. 2, 7, 1, 8, 6, 1, 0 13. 9, 2, 11, 4, 6, 1, 5 14. 23, 18, 5, 11, 15, 14, 20 15. 16. 12, 13, 14, 14, 15, 15 16. 14, 15 14. 14, 20, 23 7, 6, 3, 3, 6, 7 15, 14, 13, 12, © 2010 McGraw-Hill Companies 15. 6, 7, 7, 6, 3, 3 [Objective 3] For each group of numbers, determine the maximum and minimum values. 17. 5, 6, 0, 10, 3, 15, 1, 8 18. 9, 1, 3, 11, 4, 2, 5, 2 19. 21, 15, 0, 7, 9, 16, 3, 11 20. 22, 0, 22, 31, 18, 5, 3 21. 3, 0, 1, 2, 5, 4, 1 22. 2, 7, 3, 5, 10, 5 17. Max: 15; Min: 6 18. Max: 11; Min: 4 19. Max: 21; Min: 15 20. Max: 22; Min: 31 21. Max: 5; Min: 2 22. Max: 7; Min: 10 SECTION 2.1 127 hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 128 ANSWERS 23. 15 24. 18 [Objective 4] Find the opposite of each number. 25. 11 23. 15 24. 18 26. 34 25. 11 26. 34 27. 19 28. 5 29. 7 30. 54 27. 19 28. 5 29. 7 30. 54 31. 17 32. 28 33. 10 34. 7 35. 3 36. 5 37. 8 38. 13 39. 5 40. 7 41. 18 42. 22 43. 0 44. 0 45. 7 46. 8 47. 7 48. 8 49. 11 50. 11 Label each statement as true or false. 51. 1 52. 5 53. All whole numbers are integers. 53. True 54. True 128 SECTION 2.1 31. ⏐17⏐ 32. ⏐28⏐ 33. ⏐10⏐ 34. ⏐7⏐ 35. ⏐3⏐ 36. ⏐5⏐ 37. ⏐8⏐ 38. ⏐13⏐ 39. ⏐2⏐⏐3⏐ 40. ⏐4⏐⏐3⏐ 41. ⏐9⏐⏐9⏐ 42. ⏐11⏐⏐11⏐ 43. ⏐4⏐⏐4⏐ 44. ⏐5⏐⏐5⏐ 45. ⏐15⏐⏐8⏐ 46. ⏐11⏐⏐3⏐ 47. ⏐15 8⏐ 48. ⏐11 3⏐ 49. ⏐9⏐⏐2⏐ 50. ⏐7⏐⏐4⏐ 51. ⏐8⏐⏐7⏐ 52. ⏐9⏐⏐4⏐ 54. All nonzero integers are real numbers. © 2010 McGraw-Hill Companies [Objective 5] Evaluate. hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 129 ANSWERS 55. All integers are whole numbers. 56. All real numbers are integers. 57. All negative integers are whole numbers. 55. False 56. False 57. False 58. True 59. ⏐6⏐ 2 8 60. ⏐8⏐⏐3⏐ 11 61. 6 ⏐2⏐ 8 62. 8 ⏐(3)⏐ 11 63. 5 cm 64. 2.5 cm 65. 50 dollars 66. 200 dollars 67. 10 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. 59. (6) 2 8 60. (8) (3) 11 61. 6 (2) 8 62. 8 (3) 11 Represent each quantity with a real number. 63. 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 $50 from a checking account. 66. Business and Finance The deposit of $200 in a savings account. 67. Science and Medicine The temperature change pictured. © 2010 McGraw-Hill Companies 2 60°F 50°F – – – – 1:00 P.M. 2:00 P.M. SECTION 2.1 129 hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 130 ANSWERS 68. 10° 69. 20° 70. 30° 71. 90,000,000 72. 60,000,000 68. Science and Medicine The temperature change indicated. 2 90 80 70 60 50 40 30 20 10 0 –10 – –30 –40 90 80 40 30 20 10 –20°F –30°F – 1:00 P.M. 69. Science and Medicine 2:00 P.M. The temperature change indicated. 2 90 80 70 60 50 40 30 20 10 0 –10 – –30 –40 90 80 40 30 20 10 0°F –30°F – 1:00 P.M. 70. Science and Medicine 2:00 P.M. The temperature change indicated. 2 90 80 40 30 20 10 F –30°F – 1:00 P.M. 2:00 P.M. A country exported $90,000,000 more than it imported, creating a positive trade balance. 71. Business and Finance A country exported $60,000,000 less than it imported, creating a negative trade balance. 72. Business and Finance 130 SECTION 2.1 © 2010 McGraw-Hill Companies 90 80 70 60 50 40 30 20 10 0 –10 – –30 –40 hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 131 ANSWERS For each group of numbers given in exercises 73 to 76, answer questions (a) to (d): (a) (b) (c) (d) Which number is smallest? Which number lies farthest from the origin? Which number has the largest absolute value? Which number has the smallest absolute value? 73. 6, 3, 8, 7, 2 74. 8, 3, 5, 4, 9 75. 2, 6, 1, 0, 2, 5 76. 9, 0, 2, 3, 6 73. 6; 8; 8; 2 74. 8; 9; 9; 3 75. 2; 6; 6; 0 76. 9; 9; 9; 0 77. 77. Simplify each of the following: (7) ((7)) (((7))) Based on your answers, generalize your results. Answers 3. 200 1. 400 or (400) 15 7. 10 35 0 5. 25,000 18 10 20 9. 5, 175, 234 7, 5, 1, 0, 2, 3, 8 13. 11, 6, 2, 1, 4, 5, 9 7, 6, 3, 3, 6, 7 17. Max: 15; Min: 6 19. Max: 21; Min: 15 Max: 5; Min: 2 23. 15 25. 11 27. 19 29. 7 17 33. 10 35. 3 37. 8 39. 5 41. 18 43. 0 7 47. 7 49. 11 51. 1 53. True 55. False 57. False ⏐6⏐ 2 8 61. 6 ⏐2⏐ 8 63. 5 cm 65. 50 dollars 10 69. 20 71. 90,000,000 73. 6; 8; 8; 2 2; 6; 6; 0 77. © 2010 McGraw-Hill Companies 11. 15. 21. 31. 45. 59. 67. 75. 20 8 SECTION 2.1 131 hut06236_ch02_A.qxd 9/22/08 9:55 AM 2.2 Page 132 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: 347 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: 3 4 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 Adding Integers (a) Add 3 4. 4 3 0 3 7 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). 4 7 3 3 0 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) 7 132 © 2010 McGraw-Hill Companies 347 hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 133 ADDITION OF INTEGERS SECTION 2.2 133 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: NOTE This means that the sum of two positive integers is positive and the sum of two negative integers is negative. Adding Integers Case 1: Same Sign 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). 6 3 3 0 3 First move 3 units to the right of the origin. Then move 6 units to the left. 3 (6) 3 (b) Add 4 7. 7 4 © 2010 McGraw-Hill Companies 4 0 3 This time move 4 units to the left of the origin as the first step. Then move 7 units to the right. 4 7 3 CHECK YOURSELF 2 Add. (a) 7 (5) (b) 4 (8) (c) 1 16 (d) 7 3 hut06236_ch02_A.qxd 134 9/22/08 CHAPTER 2 9:55 AM Page 134 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: RECALL We first encountered absolute values in Section 2.1. Adding Integers Case 2: Different Signs 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 (19 7 12). The sum of 7 and 19 has the sign () of the integer with the larger absolute value, 19. (b) 13 7 6 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 NOTE The opposite of a number is also called the additive inverse of that number. NOTE 3 and 3 are opposites. (b) 7 (8) (d) 7 (8) 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 3 0 3 The additive inverse property states that the sum of any number and its opposite is 0. Property: Additive Inverse Property NOTE Here a represents the For any number a, there exists a number a such that opposite of the number a. If a is positive, a is negative. If a is negative, a is positive. a (a) (a) a 0 The sum of any number and its opposite, or additive inverse, is 0. © 2010 McGraw-Hill Companies The opposite of 9 is 9. The opposite of 15 is 15. hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 135 ADDITION OF INTEGERS SECTION 2.2 135 Example 4 Adding Integers NOTE Later, we will show that 0 0; therefore, the opposite of 0 is 0. (a) 9 (9) 0 (b) 15 15 0 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 $250 in her checking account. She writes a check for $120 and makes a deposit of $90. 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 250 (120) 90 This expression can now be evaluated. 250 (120) 90 130 90 220 © 2010 McGraw-Hill Companies The resulting balance is $220. 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? 136 9/22/08 CHAPTER 2 9:55 AM Page 136 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 (b) To add two numbers with different signs, numbers. their absolute values. (c) The sum of any number and its opposite, or additive (d) The opposite of zero is , is 0. . CHECK YOURSELF ANSWERS 1. (a) 9; (b) 10; (c) 20; (d) 8 2. (a) 2; (b) 4; (c) 15; (d) 4 3. (a) 9; (b) 15; (c) 7; (d) 1 4. (a) 0; (b) 0 5. 12 17; the temperature was 5F. © 2010 McGraw-Hill Companies hut06236_ch02_A.qxd hut06236_ch02_A.qxd 9/22/08 2.2 9:55 AM Page 137 Boost your GRADE at ALEKS.com! Exercises [Objectives 1 and 2] Add. 1. 3 6 • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos 2. 5 9 Name 3. 11 5 4. 8 7 Section Date ANSWERS 5. (2) (3) 6. (1) (9) 7. 9 (3) 8. 10 (4) 1. 9 2. 14 3. 16 9. 9 0 10. 15 0 4. 15 5. 5 11. 7 (7) 12. 12 (12) 13. 7 (9) (5) 6 14. (4) 6 (3) 0 15. 7 (3) 5 (11) 16. 6 (13) 16 6. 10 7. 6 9. 9 10. 15 [Objective 3] In exercises 17 to 22, restate the problem using an expression involving integers and then answer the question. 11. 0 12. 0 Amir has $100 in his checking account. He writes a check for $23 and makes a deposit of $51. What is his new balance? 13. 1 14. 1 15. 2 16. 3 17. $128 18. $225 19. 120 lb/in.2 17. Business and Finance Olga has $250 in her checking account. She deposits $52 and then writes a check for $77. What is her new balance? 18. Business and Finance © 2010 McGraw-Hill Companies 8. 6 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? 19. Mechanical Engineering SECTION 2.2 137 hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 138 ANSWERS 20. 15 qt 21. 4°F 22. 3°F 23. 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? 20. Mechanical Engineering The lowest one-day temperature in Helena, Montana, was 21F at night. The temperature increased by 25 degrees by noon. What was the temperature at noon? 21. Science and Medicine 2 At 7 A.M., the temperature was 15F. By 1 P.M., the temperature had increased by 18 degrees. What was the temperature at 1 P.M.? 22. Science and Medicine 2 23. 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 3. 16 5. 5 7. 6 9. 9 11. 0 17. $128 19. 120 lb/in.2 21. 4F 23. 13. 1 © 2010 McGraw-Hill Companies 1. 9 15. 2 138 SECTION 2.2 hut06236_ch02_A.qxd 9/22/08 9:55 AM 2.3 Page 139 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 853 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: 8 5 8 (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 © 2010 McGraw-Hill Companies Change the subtraction symbol () to an addition symbol (). (a) 15 7 15 (7) 8 Replace 7 with its opposite, 7. (b) 9 12 9 (12) 3 (c) 6 7 6 (7) 13 (d) Subtract 5 from 2. We write the statement as 2 5 and proceed as before: 2 5 2 (5) 7 139 140 9/22/08 CHAPTER 2 9:55 AM Page 140 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 Change the subtraction to addition. (a) 5 (2) 5 (2) 5 2 7 Replace 2 with its opposite, 2 or 2. (b) 7 (8) 7 (8) 7 8 15 (c) 9 (5) 9 5 4 (d) Subtract 4 from 5. We write 5 (4) 5 4 1 CHECK YOURSELF 2 Subtract. (a) 8 (2) (c) 7 (2) OBJECTIVE 2 (b) 3 (10) (d) 7 (7) 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) 285 47 47 379 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. © 2010 McGraw-Hill Companies hut06236_ch02_A.qxd hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 141 SUBTRACTION OF INTEGERS SECTION 2.3 141 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 Enter each of the following into your calculator. (a) 24 24 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. (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) 256 297 Your display should read 41. (b) 312 (569) 312 569 Your display should read 881. CHECK YOURSELF 5 © 2010 McGraw-Hill Companies Find the sum for each pair of integers. (a) 368 547 Example 6 Subtracting Integers Find the difference for 356 (469). 356 469 Your display should read 113. (b) 596 (834) 142 9/22/08 CHAPTER 2 9:55 AM Page 142 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) 10; (b) 13; (c) 5; (d) 14 3. 97 (9) 9 115 yards 4. (a) 36; (b) 6 5. (a) 179; (b) 1,430 6. (a) 398; (b) 157 © 2010 McGraw-Hill Companies hut06236_ch02_A.qxd hut06236_ch02_A.qxd 2.3 9/22/08 9:56 AM Page 143 Boost your GRADE at ALEKS.com! Exercises [Objective 1] Subtract. 1. 21 13 • Practice Problems • Self-Tests • NetTutor 2. 36 22 • e-Professors • Videos 3. 82 45 Name 4. 103 56 5. 8 10 6. 14 19 7. 24 45 8. 136 352 9. 5 3 10. 15 8 11. 9 14 12. 8 12 13. 5 (11) 14. 7 (5) 15. 7 (12) 16. 3 (10) 17. 36 (24) 18. 28 (11) 19. 19 (27) 20. 11 (16) Section ANSWERS 1. 8 2. 14 3. 37 4. 47 [Objective 2] For exercises 21 to 23, write an integer expression that describes the situation. Then answer the question. The temperature at noon on a June day was 82F. It fell by 12 degrees in the next 4 h. What was the temperature at 4:00 P.M.? 21. Science and Medicine 2 Jason’s checking account shows a balance of $853. He has discovered that a deposit of $70 was accidently recorded as a check for $70. What is the corrected balance? 22. 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? 23. Business and Finance How long ago was the year 1250 B.C.E.? What year was 3,300 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. 24. Technology © 2010 McGraw-Hill Companies Date Inca culture in Peru—1400 A.D. The Ahmes Papyrus, a mathematical text from Egypt—1650 B.C.E. Babylonian arithmetic develops the use of a zero symbol—300 B.C.E. First Olympic Games—776 B.C.E. Pythagoras of Greece dies—500 B.C.E. Mayans in Central America independently develop use of zero—500 A.D. The Chou Pei, a mathematics classic from China—1000 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—850 A.D. Development of calculus in Europe—1670 A.D. Rise of abstract algebra—1860 A.D. Growing importance of probability and development of statistics—1902 A.D. 5. 2 6. 5 7. 21 8. 216 9. 8 10. 23 11. 23 12. 20 13. 16 14. 12 15. 19 16. 13 17. 12 18. 17 19. 8 20. 5 21. 82° 12° 70°F 853 (70) 70 22. $993 is the balance 947 86 (86) 23. $775 is the balance 24. SECTION 2.3 143 hut06236_ch02_A.qxd 9/22/08 9:56 AM Page 144 ANSWERS 25. Complete the following statement: “3 (7) is the same as ____ because . . .” 25. Write a problem that might be answered by doing this subtraction. 26. 26. Explain the difference between the two phrases: “a number subtracted from 5” 27. 84°F 28. 76°F 29. 15 30. 9 31. 917 32. 1,477 33. 780 34. 779 35. 72 36. 342 37. 690 38. 535 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? The greatest one-day temperature drop in the United States happened on Christmas Eve, 1924, in Montana. The temperature went from 63F during the day to 21F at night. What was the total temperature drop? 27. Science and Medicine 2 A similar one-day temperature drop happened in Alaska. The temperature went from 47F during the day to 29F at night. What was the total temperature drop? 28. Science and Medicine 2 The tide at the mouth of the Delaware River tends to vary between a maximum of 10 ft and a minimum of 5 ft. What is the difference in feet between the high tide and the low tide? 29. Science and Medicine 2 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? 30. Science and Medicine 2 Calculator Exercises 31. 789 (128) 32. 910 (567) 33. 349 (431) 34. 412 (367) 35. 47 (25) 36. 123 (219) 37. 234 (456) 38. 412 (123) Answers 1. 8 3. 37 5. 2 7. 21 9. 8 11. 23 13. 16 15. 19 17. 12 19. 8 21. 82° 12° 70F 23. 947 86 (86); $775 is the balance 25. 27. 84°F 29. 15 144 SECTION 2.3 31. 917 33. 780 35. 72 37. 690 © 2010 McGraw-Hill Companies Use your calculator to perform the following operations. hut06236_ch02_A.qxd 9/22/08 9:56 AM 2.4 Page 145 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, 3 4 4 4 4 12 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) 30 © 2010 McGraw-Hill Companies The product is negative. (b) (10)(10) 100 (c) (8)(12) 96 CHECK YOURSELF 1 Multiply. (a) (7)(5) (b) (12)(9) (c) (15)(8) 145 hut06236_ch02_A.qxd 146 9/22/08 CHAPTER 2 9:56 AM Page 146 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 (1)(2) 2 NOTE This number is decreasing by 1. Do you see that the product is increasing by 2 each time as you go down? (0)(2) 0 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) 40 The product of two negative numbers (same sign, ) is positive. CHECK YOURSELF 2 Multiply. (a) 10 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 Multiplying Integers by One and Zero Find each product. (a) (1)(7) 7 (b) (15)(1) 15 (c) (7)(0) 0 (d) 0 # 12 0 © 2010 McGraw-Hill Companies NOTE (1)(2) is the opposite hut06236_ch02_A.qxd 9/22/08 9:56 AM Page 147 MULTIPLICATION OF INTEGERS SECTION 2.4 147 CHECK YOURSELF 3 Multiply. (a) (10)(1) (b) (0)(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) 32 (3 3) 9 Note that the negative is not squared. CHECK YOURSELF 4 Evaluate each expression. (a) (4)2 (b) (4)3 (c) 42 In Example 5 we will apply the order of operations. Example 5 Using Order of Operations with Integers © 2010 McGraw-Hill Companies Evaluate each expression. (a) 7(9 12) 7(3) 21 Evaluate inside the parentheses first. (b) (8)(7) 40 56 40 16 Multiply first, then subtract. (c) (5)2 3 (5)(5) 3 25 3 22 Evaluate the power first. (d) 52 3 25 3 28 Note that 52 25. The power applies only to the 5. Note that (5)2 (5)(5) 25 148 9/22/08 CHAPTER 2 9:56 AM Page 148 INTEGERS AND INTRODUCTION TO ALGEBRA CHECK YOURSELF 5 Evaluate each expression. (a) 8(9 7) (c) (4)2 (4) (b) (3)(5) 7 (d) 42 (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 32, the is not squared. (d) The rules for order of operations were learned in Section . CHECK YOURSELF ANSWERS 1. (a) 35; (b) 108; (c) 120 2. (a) 120; (b) 72 3. (a) 10; (b) 0 4. (a) 16; (b) 64; (c) 16 5. (a) 16; (b) 22; (c) 20; (d) 12 © 2010 McGraw-Hill Companies hut06236_ch02_A.qxd hut06236_ch02_A.qxd 9/22/08 2.4 9:56 AM Page 149 Boost your GRADE at ALEKS.com! Exercises [Objective 1] Multiply. 1. 4 10 • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos 2. 3 14 Name 3. (5)(12) 4. (10)(2) Section 5. (8)(9) 6. (12)(3) ANSWERS 7. (8)(7) 8. (9)(8) Date 1. 40 2. 42 9. (5)(12) 11. (0)(18) 10. (7)(3) 12. (17)(0) 3. 60 4. 20 5. 72 13. (15)(0) 14. (0)(25) 6. 36 7. 56 [Objective 2] Do the indicated operations. Remember the rules for the order of operations. © 2010 McGraw-Hill Companies 15. 5(7 2) 16. 7(8 5) 17. 2(5 8) 18. 6(14 16) 19. 3(9 7) 20. 6(12 9) 21. 3(2 5) 22. 2(7 3) 23. (2)(3) 5 24. (6)(8) 27 25. 4(7) 5 26. (3)(9) 11 27. (5)(2) 12 28. (7)(3) 25 29. (3)(7) 20 30. (2)(6) 8 31. 4 (3)(6) 32. 5 (2)(3) 33. 7 (4)(2) 34. 9 (2)(7) 35. (7)2 17 36. (6)2 20 8. 72 9. 60 10. 21 11. 0 12. 0 13. 0 14. 0 15. 25 16. 21 17. 6 18. 12 19. 6 20. 18 21. 21 22. 20 23. 11 24. 75 25. 33 26. 16 27. 2 28. 4 29. 1 30. 4 31. 22 32. 11 33. 1 34. 5 35. 32 36. 16 SECTION 2.4 149 hut06236_ch02_A.qxd 9/22/08 9:56 AM Page 150 ANSWERS 37. 43 37. (5)2 18 38. (2)2 10 39. 62 4 40. 52 3 41. (4)2 (2)(5) 42. (3)3 (8)(2) 43. (8)2 52 44. (6)2 42 45. (6)2 (3)2 46. (8)2 (4)2 47. 82 52 48. 62 32 50. 92 (6)2 38. 14 39. 40 40. 28 41. 6 42. 43 43. 39 44. 20 45. 27 46. 48 49. 82 (5)2 47. 89 51. Business and Finance 48. 45 49. 89 50. 117 51. $17,086 52. $540 53. 22F 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, 201 units of product C, 377 units of product D, and 43 units of product E. What was their profit or loss that month? 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. 52. Statistics The temperature is 6F at 5:00 in the evening. If the temperature drops 2 degrees every hour, what is the temperature at 1:00 A.M.? 53. Science and Medicine 2 1. 40 3. 60 5. 72 15. 25 17. 6 19. 6 27. 2 29. 1 31. 22 39. 40 41. 6 43. 39 51. $17,086 53. 22F 150 SECTION 2.4 7. 56 9. 60 11. 0 13. 0 21. 21 23. 11 25. 33 33. 1 35. 32 37. 43 45. 27 47. 89 49. 89 © 2010 McGraw-Hill Companies Answers hut06236_ch02_A.qxd 9/22/08 9:56 AM 2.5 Page 151 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, 15 3 5 because 15 5 # 3 24 4 6 because 24 (6)(4) 30 6 5 because 30 (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. © 2010 McGraw-Hill Companies (a) (b) (c) (d) Positive Positive Negative Negative Negative Positive Positive Negative 28 4 7 Positive 36 9 4 Positive 42 6 7 Negative 75 25 3 Negative 151 152 9/22/08 CHAPTER 2 9:56 AM Page 152 INTEGERS AND INTRODUCTION TO ALGEBRA CHECK YOURSELF 1 Divide. 55 11 48 (c) 8 80 20 144 (d) 12 (a) (b) As discussed in Section 1.6, we must be very careful when 0 is involved in a division problem. Remember that 0 divided by any nonzero number is just 0. This rule can be extended to include integers, so that 0 0 7 because 0 (7)(0) However, if zero is the divisor, we have a special problem. Consider 9 ? 0 This means that 9 0 ?. Can 0 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 0 is not 0 allowed. We say that division by 0 is undefined. Example 2 Dividing Integers Divide, if possible. (a) 7 is undefined. 0 (b) 9 is undefined. 0 (c) 0 0 5 (d) 0 0 8 0 is called an indeterminate form. You will learn more about this 0 in later mathematics classes. Note: The expression © 2010 McGraw-Hill Companies hut06236_ch02_A.qxd hut06236_ch02_A.qxd 9/22/08 9:56 AM Page 153 DIVISION OF INTEGERS SECTION 2.5 153 CHECK YOURSELF 2 Divide, if possible. (a) 0 3 (b) 5 0 (c) 7 0 (d) 0 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) 42 14 3 3 Multiply in the numerator, then divide. (b) 3 (12) 9 3 3 3 Add in the numerator, then divide. (c) 4 (2)(6) 4 (12) 6 2 6 2 Multiply in the numerator. Then add in the numerator and subtract in the denominator. 16 2 8 Divide as the last step. CHECK YOURSELF 3 Evaluate each expression. (a) 4 (8) 6 (b) 3 (2)(6) 5 (c) (2)(4) (6)(5) (2)(11) © 2010 McGraw-Hill Companies 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) 457 734 Your display should read 335,438. hut06236_ch02_A.qxd 154 9/22/08 CHAPTER 2 9:56 AM Page 154 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. 384 16 384 16 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 ( 3 ) yx 6 that the negative is attached to the 3 before it is raised to a power. or, on some calculators ( () 3 ) ^ 6 Enter Either way, your display should read 729. CHECK YOURSELF 6 Evaluate. (2)9 © 2010 McGraw-Hill Companies NOTE The parentheses ensure hut06236_ch02_A.qxd 9/22/08 9:56 AM Page 155 DIVISION OF INTEGERS SECTION 2.5 155 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 © 2010 McGraw-Hill Companies 1. (a) 5; (b) 4; (c) 6; (d) 12 2. (a) 0; (b) undefined; (c) undefined; (d) 0 3. (a) 2; (b) 3; (c) 1 4. (a) 3,276; (b) 3,408 5. 32.5 6. 512 hut06236_ch02_A.qxd 9/22/08 Boost your GRADE at ALEKS.com! • Practice Problems • Self-Tests • NetTutor Page 156 2.5 • e-Professors • Videos Name Section 9:56 AM Exercises [Objective 1] Divide. 1. 20 4 2. 70 14 3. 48 6 4. 24 8 5. 50 5 6. 32 8 7. 52 4 8. 56 7 9. 75 3 10. 60 15 11. 0 8 12. 125 25 13. 9 1 14. 10 0 15. 96 8 16. 20 2 17. 18 0 18. 0 8 19. 17 1 20. 27 1 21. 144 16 22. 150 6 Date ANSWERS 1. 5 2. 5 3. 8 4. 3 5. 10 6. 4 7. 13 8. 8 9. 25 10. 4 11. 0 12. 5 14. Undefined 13. 16. 9 10 15. 12 17. Undefined 18. 0 19. 17 20. 21. 9 22. 25 23. 9 24. 15 25. 4 26. 4 27. 2 28. 9 29. 8 30. [Objective 2] Perform the indicated operations. 2 33. Undefined 34. Undefined 32. 156 SECTION 2.5 23. (6)(3) 2 24. (9)(5) 3 25. (8)(2) 4 26. (7)(8) 14 27. 24 4 8 28. 36 7 3 29. 12 12 3 30. 14 4 6 31. 55 19 12 6 32. 11 7 14 8 33. 75 22 34. 10 6 44 3 3 © 2010 McGraw-Hill Companies 31. 27 hut06236_ch02_A.qxd 9/22/08 9:56 AM Page 157 ANSWERS For exercises 35 to 37, use integers to write an expression that represents the situation. Then answer the question. 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? 35. 125 9 9 $44 36. 42 — 14 weeks 3 35. Business and Finance 36. Social Science A woman lost 42 lb. If she lost 3 lb each week, how long has she been dieting? Suppose that you and your two brothers bought equal shares of an investment for a total of $20,000 and sold it later for $16,232. How much did each person lose? 37. 20,000 16,232 —— $1,256 3 38. 675 39. 936 40. 1,736 41. 952 42. 1,349 43. 2 44. 625 45. 1,024 37. Business and Finance Calculator Exercises Use your calculator to multiply and divide. 38. (15) (45) 39. (78) (12) 40. (56) (31) 41. (34) (28) 42. (71) (19) 43. 44. (5)4 45. (4)5 28 14 © 2010 McGraw-Hill Companies Answers 1. 5 15. 12 27. 2 3. 8 5. 10 7. 13 9. 25 17. Undefined 19. 17 21. 9 29. 8 31. 2 33. Undefined 20,000 16,232 $1,256 3 45. 1,024 37. 39. 936 11. 0 13. 9 23. 9 25. 4 35. 125 9 9 $44 41. 952 43. 2 SECTION 2.5 157