Chapter 2 Integers and Introduction to Solving Equations

Transcription

1 Chapter Integers and Introduction to Solving Equations.1 Check Points 1. a. 500 b. 8.. a. 6> 7 because 6 is to the right of 7 on the number line. b. 8< 1 because 8 is to the left of 1 on the number line. c. 5 < because 5 is to the left of on the number line. d. 14 < 0 because 14 is to the left of 0 on the number line. 4. a. 8 8 because 8 is 8 units from 0. b. 6 6 because 6 is 6 units from 0. b. 8 8 because 8 is 8 units from 0 and the negative of 8 is a. The opposite of 14 is 14. b. The opposite of 17 is 17. c. The opposite of 0 is 0. to simplify. 6. a. We can use the double-negative rule, ( a) a, ( 5) 5 b. We cannot use the double-negative rule when one of the negatives is inside the absolute value bars c. We can use the double-negative rule, ( a) a, ( 0) 0 0 inside the absolute value bars. 7. a. Take aspirin daily: 5; Blood relative 95 or older: 10; Less than 6 to 8 hours sleep: 1; Less than 1 years education: 6 b. Number line: c. Less than 1 years education; Less than 6 to 8 hours sleep; Take aspirin daily; Blood relative 95 or older Copyright 017 Pearson Education, Inc. 61

2 Chapter Integers and Introduction to Solving Equations.1 Concept and Vocabulary Check 1....,,, 1,0,1,,,.... left. the distance from 0 to a 4. opposites 5. a.1 Eercise Set , ,000,000, < 7 because is to the left of 7 on the number line < 1 because 1 is to the left of 1 on the number line < because 1 is to the left of on the number line > 1 because 1 is to the right of 1 on the number line > 50 because 8 is to the right of 50 on the number line > 9 because 7 is to the right of 9 on the number line < 0 because 100 is to the left of 0 on the number line > 00 because 0 is to the right of 00 on the number line ,000 00, ,000,000 1,000, The opposite of 7 is The opposite of 8 is The opposite of 1 is 1.. The opposite of 15 is 15.. ( 70) ( 80) ( 1) ( 14) ( 1) ( 14) >, because 6 > 6 Copyright 017 Pearson Education, Inc.

3 Section.1 Introduction to Integers 4. 0 < 50, because 0 < <, because 6 < > 50, because 0 > ( 5) > 5, because 5 > ( 7) > 7, because 7 > has the greater absolute value because it is further from zero on the number line has the greater absolute value because it is further from zero on the number line ( ) ; Ulysses S. Grant ; Ronald Reagan 5. Cleveland 54. Reagan 55. Van Buren and Cleveland 56. Van Buren and Cleveland 57. Grant and Reagan 58. Grant, Kennedy, and Reagan 59. a. 60. a. b. Rhode Island, Georgia, Louisiana, Florida, Hawaii b. Wyoming, Wisconsin, Washington, West Virginia, Virginia 61. a. 4 F c. Yes, 4 and 4 are the same distance from 0 on opposite sides of 0 on a number line. 6. a. 9 F b. 9 F c. Yes, 9 and 9 are the same distance from 0 on opposite sides of 0 on a number line. 6. When the wind speed is 5 miles per hour and the air temperature is 5 F the temperature feels like 16 F. When the wind speed is 50 miles per hour and the air temperature is 10 F the temperature feels like 17 F. It feels colder when the wind speed is 50 miles per hour and the air temperature is 10 F. 64. When the wind speed is 60 miles per hour and the air temperature is 15 F the temperature feels like 11 F. When the wind speed is 15 miles per hour and the air temperature is 5 F the temperature feels like 1 F. It feels colder when the wind speed is 15 miles per hour and the air temperature is 5 F Answers will vary. 71. does not make sense; Eplanations will vary. Sample eplanation: The Titanic s resting place is higher because it is less feet below sea level. 7. does not make sense; Eplanations will vary. Sample eplanation: The lowest a class size could be is zero. 7. makes sense 74. does not make sense; Eplanations will vary. Sample eplanation: We cannot use the doublenegative rule when one of the negatives is inside the absolute value bars. 75. true 76. true 77. true b. 4 F Copyright 017 Pearson Education, Inc. 6

4 Chapter Integers and Introduction to Solving Equations 78. false; Changes to make the statement true will vary. A sample change is: If a > b and a and b are integers, then a can be a negative or positive integer depending upon the value of b. 79. ( 7) ( 600 ) 76 ( 600) (9 [ 5) + ] 6(4) [ + ] 4 [ + ] 6 [ ] associative property of addition ( 90) ( 10) 50. Check Points 1. a. A loss of $60 followed by a loss of $40 results in a loss of $ ( 40) 100 b. A gain of $60 followed by a loss of $40 results in a gain of $ ( 40). 4 + ( 7) Start at 4 and move 7 units to the left.. a. 1 + ( ) 4 Start at 1 and move units to the left. b. 5+ Start at 5 and move units to the right. 4. a. 7 + ( ) 9; Add the absolute values and use the common sign. b ( 46) 64; Add the absolute values and use the common sign. c ; Add the absolute values and use the common sign. 5. a ; Subtract the absolute values and use the sign of the number with the greater absolute value. b. 0 + ( ) 17; Subtract the absolute values and use the sign of the number with the greater absolute value. 6. a ; Subtract the absolute values and use the sign of the number with the greater absolute value. b. 7 + ( 95) 68; Subtract the absolute values and use the sign of the number with the greater absolute value. 7. a. 6 + ( 48) ; The addends have different signs, so subtract the absolute values and use the sign of the number with the greater absolute value. b. 5 + ( 10) 17; The addends have the same signs, so add the absolute values and use the common sign. c ; The addends have different signs, so subtract the absolute values and use the sign of the number with the greater absolute value. d ( 79) ; The sum of any integer and its opposite is ( 66) + 8 ( ) + ( ) + ( 66) 8 + ( 89) 7 64 Copyright 017 Pearson Education, Inc.

5 Section. Adding Integers 9. + ( 4) ( 5) + ( + 1+ ) + [( 4) + ( 5) ] 6 + ( 9) At the end of 5 months the water level was down feet.. Concept and Vocabulary Check 7. A loss of $4 followed by a gain of $5 results in a gain of $ A loss of $7 followed by a gain of $7 results in a gain of $ ( ) 4 1. a; right; left; a; sum. 0. negative ( ) 5 4. negative integer 5. positive integer ( ) positive integer 8. negative integer 1. + ( ) Eercise Set 1. A loss of $8 followed by a loss of $ results in a loss of $ ( ) 10. A loss of $10 followed by a loss of $6 results in a loss of $ ( 6) 16. A gain of $1 followed by a loss of $8 results in a gain of $ ( 8) 4 4. A gain of $15 followed by a loss of $10 results in a gain of $ ( 10) 5 5. A gain of $0 followed by a loss of $5 results in a loss of $ ( 5) 5 6. A gain of $0 followed by a loss of $6 results in a loss of $ ( 6) ( ) ( 5) ( 10) ( 6) ( 6) 5 Copyright 017 Pearson Education, Inc. 65

6 Chapter Integers and Introduction to Solving Equations ( 47) ( 6) ( 11) ( 76) ( 89) ( 91) ( 8) ( 47) ( 58) ( 7) + ( 5) 4+ ( 7) + ( 5) 4+ ( 1) ( ) + ( 8) 10 + ( ) + ( 8) 10 + ( 11) ( 10) + + ( ) ( 7+ ) + ( 10) + ( ) 9+ ( 1) ( 7) + + ( 6) ( 5+ ) + ( 7) + ( 6) 8+ ( 1) ( ) + 17 ( 1+ 17) + ( 19) + ( ) 0 + ( 5) ( 4) + 5 ( ) + ( 18) + ( 4) 40 + ( 5) ( 1) ( 8) ( ) + ( 1) + ( 8) 41+ ( 41) ( 16) ( ) ( ) + ( 16) + ( ) 49 + ( 49) ( 6) ( 74) Copyright 017 Pearson Education, Inc.

7 Section. Adding Integers [ 10] [ 8] Left side: [ ] Right side: [ ] 6+ + ( 1) > + 4+ ( 8) because 5> Left side: Right side: [ ] ( 8) + ( 6) 10 < ( 1) because 4 < The high temperature was 44 F The high temperature was 45 F The elevation of the person is 600 feet below sea level The elevation of the person is 10 feet above sea level The temperature at 4:00 P.M. was F ( 4) 6 The team had a total loss of 6 yards ( ) + ( 1) The location of the football at the end of the fourth play is at the 5-yard line ( ) + ( 1) + ( 4) + ( ) + ( + ( 1) + ( 4) ) The level of the reservoir is 18 feet. 69. a ( 60) 199 The deficit is $199 billion for 011. b ( 57) 1087 The deficit is $1087 billion for 01. This is better than 011 because 01 had less debt. c ( 1087) 86 The combined deficit is $86 billion. 70. a ( 518) 141 The deficit is $141 billion for 009. b ( 457) 194 The deficit is $194 billion for 010. This is better than 009 because 010 had less debt. c ( 194) 707 The combined deficit is $707 billion Answers will vary. 77. makes sense 78. makes sense 79. does not make sense; Eplanations will vary. Sample eplanation: The sum of two negative integers is a negative integer. 80. makes sense 81. true 8. false; Changes to make the statement true will vary. A sample change is: The sum of a positive integer and a negative integer can be positive or negative or zero. 8. true 84. false; Changes to make the statement true will vary. The absolute value of two negative integers is always a positive integer. 85. The sum is negative. The sum of two negative numbers is always a negative number. Copyright 017 Pearson Education, Inc. 67

8 Chapter Integers and Introduction to Solving Equations 86. The sum is negative. When finding the sum of numbers with different signs, use the sign of the number with the greater absolute value as the sign of the sum. Since a is further from 0 than c, we use a negative sign. 87. The sum is positive. When finding the sum of numbers with different signs, use the sign of the number with the greater absolute value as the sign of the sum. Since c is further from 0 than b, we use a positive sign. 88. Though the sum inside the absolute value is negative, the absolute value of this sum is positive Answers will vary y , true The number is a solution. 9. commutative property of addition ( 10) ( 1) ( 1) Check Points Note that for Check Points #1, first change all subtractions to additions of opposites. 1. a ( 11) 8 b. 4 ( 5) c. 7 ( ) a ( 87) 1 b. 19 ( 17) c. 164 ( 8) First change all subtractions to additions of opposites. Then add the positive numbers and negative numbers separately. 10 ( 1) 4 ( ) ( 4) + + ( 6) ( ) + ( 4) + ( 6) 5 + ( 10) 15 [ ] 4. The difference can be found by subtracting the elevation of the Marianas Trench from the elevation of Mount Everest ( 10,915) ,915 19,76 The difference in elevation is 19,76 meters. 5. a. ( 5) The difference in lifespan is 8 years. b. 1 ( 15) The difference in lifespan is 14 years. c. + ( 5) You shrink your lifespan by years. d. 5+ ( 5) There is no change to your lifespan.. Concept and Vocabulary Check 1. ( 14) ; ( 14) 5. ; ( 1 ); ( ). Eercise Set 1. a. 1 b ( 1). a. 10 b ( 10) 68 Copyright 017 Pearson Education, Inc.

9 Section. Subtracting Integers. a. 7 b. 5 ( 7) a. 8 b. ( 8) ( 8) ( ) ( 14) ( 15) 1 9. ( 0) ( 17) ( 18) ( 19) ( ) ( ) ( 17) ( 1) ( 45) ( 65) ( ) ( 6) 1. 1 ( 1) ( 15) ( 1) ( 15) ( 1) ( 15) ( 86) ( 95) ( 91) ( 419) ( 89) ( 98) ( 146) ( 6) ( 146) ( 6) ( 8) 1 + ( ) + 8 ( 1+ 8) + ( ) 1+ ( ) ( 7) 14 + ( ) + 7 ( ) + ( ) 1+ ( ) ( 8) + + ( 7) ( 9+ ) + ( 8) + ( 7) 1 + ( 15) ( ) + 5+ ( 1) ( 8+ 5) + ( ) + ( 1) 1 + ( 15) Copyright 017 Pearson Education, Inc. 69

10 Chapter Integers and Introduction to Solving Equations ( ) + + ( 10) ( 6) + ( ) + ( 10) ( 5) + 4+ ( 17) ( 9) + ( 5) + ( 17) ( 5) ( ) ( 10) + ( ) + ( 5 + 7) ( ) ( 11) ( 6) + ( 11) + ( + 8) ( 7) + ( 5) ( ) + ( 11) ( 5) ( 6) ( 1) ( 19) + ( 8) + 6 ( 1) [( 19) ( 8) ] [ 6 ( 1) ] ( 17) + ( 7) ( 48) + 17 ( 7) + ( 48) + ( ) ( 1) 17 + ( 4) ( 78) + 1 ( 4) + ( 78) + ( ) ( 8) + ( ) + ( ) ( 816) + ( ) + ( ) ( 17) ( 11) ( 7) ( 8) ( 4) ( 5) ( 40) ( 60) ( 0) ( 0) ( 100) ( 100) ( 40) ( 60) Copyright 017 Pearson Education, Inc.

11 Section. Subtracting Integers 65. Left side: 6 ( 18) Right side: 60 ( 48) ( 18) > 60 ( 48) because 8> Left side: ( 51) 77 Right side: ( 7) < 44 7 because 77 < y 1 ( ) ( 5) 14 + ( 5) y 15 ( ) ( 7) 18 + ( 7) ( 4) , true The number is a solution ( 7) , true The number is a solution. 71. ( 6 10) 6+ ( 10) ( 4) ( 4 1) 5 4+ ( 1) 5 ( 8) Elevation of Mount McKinley elevation of Death Valley,0 ( 8),60 The difference in elevation between the two geographic locations is 0,60 feet. 74. Elevation of Mount Kilimanjaro elevation of Qattara Depression 19, 1 ( 46) 19, 757 The difference in elevation between the two geographic locations is 19,757 feet. 75. ( 19) The difference is 1 F ( 1) The difference is 18 F ( ) 19 + F warmer ( 19) F warmer ( 100) 71 The combined deficit is $71 billion ( 141) + ( 100) 874 The combined deficit is $874 billion ( 100) The difference is $148 billion ( 141) The difference is $1541 billion Answers will vary. 86. makes sense 87. makes sense 88. makes sense 89. makes sense 90. true 91. false; Changes to make the statement true will vary true 9. true A sample change is: 94. positive 95. negative Copyright 017 Pearson Education, Inc. 71

12 Chapter Integers and Introduction to Solving Equations 96. negative 97. positive y ( 8) + y y From least to greatest is: y ; y ; + y Answers will vary seventy-si thousand, three hundred five ( ) ( ) + ( ) + ( ) + ( ) ( ) ( ) + ( ) + ( ) ( ) 6 1( ) 0( ) 1( ) ( ) 6 ( ) 9 4( ) 1.4 Check Points 1. a. 8( 5) 40 The product of two integers with different signs is negative. b. ( 6) 1 The product of two integers with different signs is negative.. a. ( 7)( 10) 70 The product of two integers with same signs is positive. b. ( 4)( 9) 6 The product of two integers with same signs is positive.. a. 15( 19) 85 The product of two integers with different signs is negative. b. ( 6)( 04) 14 The product of two integers with same signs is positive. c. ( 04)( 0) The product of any number and zero is zero. d The product of two integers with different signs is negative. 4. a. ( )()( 1)(4) 4 When multiplying an even number of negative integers the product is positive. b. ( 1)( )()( 1)(5) 0 When multiplying an odd number of negative integers the product is negative. 5. a. 6. a b c d e f The quotient of two integers with 5 different signs is negative. b. ( 0) ( 10) The quotient of two integers with same signs is positive. c The quotient of two integers with different signs is negative. d Any nonzero integer divided into zero is zero. 7 Copyright 017 Pearson Education, Inc.

13 Section.4 Multiplying and Dividing Integers 7. a. Because there are 7 days in a week, we can find the number of deaths in a week by multiplying the number of daily deaths by 7. ( 156, 000) 7 1, 09, 000 Each week, there are 1,09,000 deaths in the world. b. Because there are 7 days in a week, we can find the number of births in a week by multiplying the number of daily births by 7. 84,000 7,688,000 Each week, there are,688,000 births in the world. c. We find the increase in world population each week by combining the number of deaths and the number of births. ( 1,09,000) +,688,000 1,596,000 Each week, the population increases by 1,596, Concept and Vocabulary Check 1. negative. positive. positive 4. negative negative 7. positive undefined.4 Eercise Set 1. 5( 9) ( 7) 70. ( 8)( ) 4 4. ( 9)( 5) ( 19)( 1) ( 11)( 1) ( 19) 10. 0( 11) 11. 1( 1) ( 14) ( 6)( 07) ( 6)( 08) ( 07)( 0) 16. ( 08)( 0) 17. ( 5)( )() ( 6)( )(10) ( 4)( )( 1)(6) 7 0. ( )( 7)( 1)() 4 1. ( )( 4)( 1) 4. ( )( 5)( 1) 0. ( )( )( ) 7 4. ( 4)( 4)( 4) ( )( 1)() ( 5)( )() () ( )( )( )( )( ) 8. ( )( )( )( )( ) ( ) ( 8)( 4)(0)( 17)( 6) 0. ( 9)( 1)( 18)(0)( ) Copyright 017 Pearson Education, Inc. 7

14 Chapter Integers and Introduction to Solving Equations ( 60) ( 80) ( 180) ( 0) ( 10) ( 0) ( 10) ( 10) 59. ( 10) 0 undefined 60. ( 10) 0 undefined ( 74) ( 971) ( 7)( 4) ( 96)( 5) ( 15) ( 1) Copyright 017 Pearson Education, Inc.

15 Section.4 Multiplying and Dividing Integers Left side: Right side: 18 ( ) 9 85 < 18 ( ) because 40 < Left side: Right side: 60 ( 5) 1 15 < 60 ( 5) because 45 < y 6 ( )( ) y 5 15 ( )( ) ( ) , false The number is not a solution ( 7 ) , false The number is not a solution ( ) 18 The decrease is ( 4) 8 The decrease is The total loss is $ The total loss is $ a. 16( 5) 80 You are fined 80 cents. b. 10( 8) 80 You are owed 80 cents. c. Neither owes money to the other because the values are opposites. 90. a. 1( 5) 60 You are fined 60 cents b. 16( 8) 18 You are owed 18 cents. c. ( 60) We owe you 68 cents., Each person losses $ , Each person losses $500. ( 11) + ( 0) + ( 57) + ( 44) + ( 8) On average, women with college degrees get paid $6 less per week than their male counterparts during this five-year period On average, women with college degrees get paid $41 less per week than their male counterparts during this two-year period Answers will vary. Copyright 017 Pearson Education, Inc. 75

16 Chapter Integers and Introduction to Solving Equations 101. does not make sense; Eplanations will vary. Sample eplanation: The sign rules for multiplication and division are the same. 10. does not make sense; Eplanations will vary. Sample eplanation: When I multiply more than two integers, I determine the sign of the product by counting the number of negative factors ( 11 8) 6+ [ ] [ ] [ ] makes sense 104. makes sense 105. false; Changes to make the statement true will vary. A sample change is: The addition of two negative numbers results in a negative answer. While, the multiplication of two negative integers results in a positive answer true 107. false; Changes to make the statement true will vary , true 109. A sample change is: ( 5) 6( 5) ( 0) 15+ ( 5) 5 0, false The number is not a solution , Answers will vary ( ) ( ) ( ) Mid-Chapter Check Point Chapter ( 1) ( 16) ( 11) ( 15) ( 15) + ( 4 + 7) [ 5] ( 0) ( 4) 5 8. ( 5)( )( 1) ( 19) ( 11) ( 5) ( 5) + ( ) [ 1] > Copyright 017 Pearson Education, Inc.

17 Section.5 Order of Operations with Integers; Mathematical Models ( 1) ,400 ( 760) 15, ,160 The difference is 16,160 feet ( 8) ( 7) + ( 1) ( 5+ + ) + ( 8) + ( 7) + ( 1) 10 + [ 16] 6 The temperature at the final measurement is a. 1 ( 9) b. It was 4 F colder in Wisconsin The mean was 11 F. c The temperature was F..5 Check Points 1. There are no grouping symbols or eponents. We start by performing the division. + ( 5) 5 + ( 5). There are no grouping symbols or eponents. We begin by evaluating the eponential epression. 0 10( ) 10( 7) ( 7) 54. Because grouping symbols appear, we perform the operations in parentheses first. 0 ( 7 ) 0 ( 14) 0 ( 11) There are no grouping symbols or eponents. We begin by evaluating the eponential epression ( ) ( 9) 7 1 5( 9) ( 45) + ( 7) 1+ ( 5) 1 5. Begin by performing the operation within the innermost grouping symbol, the parentheses, first ( 6) 8 10( ) [ 40] 8 + ( 40) [ ] Fraction bars are grouping symbols that separate epressions into two parts. Simplify above and below the fraction bar separately Copyright 017 Pearson Education, Inc. 77

18 Chapter Integers and Introduction to Solving Equations 7. Because absolute value donates grouping, we perform the operations inside the absolute value bars first Begin by substituting 5 for each occurrence of in the algebraic epression. Then use the order of operations to evaluate the epression ( 5) 9( 5) 4 5 9( 5) 4 5 ( 45) ( 1) 5 45 ( 1) Begin by substituting the values for each variable in the algebraic epression. Then use the order of operations to evaluate the epression. b 4ac ( 5) 6 ( 40) a. To determine whether 9 is a solution to the equation, substitute 9 for all occurrences of the variable and evaluate each side of the equation ( 6) , true The number is a solution. 78 Copyright 017 Pearson Education, Inc.

19 Section.5 Order of Operations with Integers; Mathematical Models b. To determine whether is a solution to the equation, substitute for all occurrences of the variable and evaluate each side of the equation. 5t + 8t ( ) + 8( ) ( ) ( 4) , false The number is not a solution. 11. a. Begin with the median weekly earnings of male college graduates in 005. Because 005 is 5 years after 000, we substitute 5 for in the formula for earnings of males. M M ( ) 5 ( ) 5 M M + + M + M M 118 The formula indicates that the median weekly earnings of male college graduates in 005 were $118. The bar graph shows earnings of $1167. Thus, the model overestimates weekly earnings by $118 $1167, or by $16. b. Begin with the median weekly earnings of female college graduates in 005. Because 005 is 5 years after 000, we substitute 5 for in the formula for earnings of females. F F ( ) 5 ( ) 5 F F + + F + F F 885 The formula indicates that the median weekly earnings of female college graduates in 005 were $885. The bar graph shows earnings of $88. Thus, the model overestimates weekly earnings by $885 $88, or by $. c. Make a Table: Male Earnings Female Earnings in 005 in 005 Difference (Gap) Mathematical Models $118 $885 $118 $885 $98 Data from Figure $1167 $88 $1167 $88 $84 Copyright 017 Pearson Education, Inc. 79

20 Chapter Integers and Introduction to Solving Equations.5 Concept and Vocabulary Check 1. divide. multiply. subtract 4. add 5. subtract.5 Eercise Set ( ) 6+ ( 15) ( ) 8+ ( 1) ( 0) 4 7+ ( 5) ( 5) 5 9+ ( 7) 5. 4( ) 4( 8) 6. ( ) 5 ( ) ( 10)( ) 50 ( 10)( 9) ( 10)( 4) 0 ( 10)( 16) ( 7) 4 ( 5) 4 ( 15) ( 4 10) 7 ( 6) 7 ( 18) ( 4 )( 7) 1( 5) 5 1. ( 7 )( 4 10) 4( 6) ( 4 ) 5 ( 1) 5 ( 10) ( 4 ) 5 ( 8) 5 ( 5) ( ) + 1 ( 4) 6( 8) + 1 ( 4) 48 + ( ) ( ) + 0 ( 5) 4( 8) + 0 ( 5) + ( 6) ( ) 0 7( 9) ( ) 40 5( 4) Copyright 017 Pearson Education, Inc.

21 Section.5 Order of Operations with Integers; Mathematical Models 19. ( ) 0 4( 16) ( ) 1 ( 5) ( 8+ 18) ( 8 9) ( 10) ( 1) 100 ( 1) 100. ( ) ( 6 7) ( 6) ( 1) 6 ( 1). ( 1) ( 9 1) 7 4. ( 1 4 ) ( 1 16) ( 4) ( 18 ) ( 14 7) ( 6) ( ) ( 6) ( 4 ) ( 15 5) ( 8) ( ) ( 8) ( 5) ( 4) ( 9 ) ( ) 9 [ ] ( 1 ) 5( 4) [ ] ( 10 15) 40 ( 5) [ ] [ ] ( 15 0) 0 ( 5) [ ] [ ] ( ) + 4( 6) 9+ 4( 6) Copyright 017 Pearson Education, Inc. 81

22 Chapter Integers and Introduction to Solving Equations ( 6) + 8( 5) 6+ 8( 5) ( 5) 4 5 4( 5) 45 4( 5) 0 ( 0) ( 6) 5 6 5( 6) 56 5( 6) 0 ( 0) ( 11) ( 11) ( 6) ( 6) Copyright 017 Pearson Education, Inc.

23 Section.5 Order of Operations with Integers; Mathematical Models ( ) 8 + ( 5) ( 6) 8 + ( 5) ( ) 6 + ( 4) ( 1) 6 + ( 4) ( 5) 7 5 ( 15) ( 7) 40 + ( 7) ( 6) 8 6 ( 4) ( 8) 78 + ( 8) y y + ( ) ( 8) Copyright 017 Pearson Education, Inc. 8

24 Chapter Integers and Introduction to Solving Equations 50. y y + ( ) ( 18) m m+ 4( m+ 5) ( 7) ( 7) + 4( 7+ 5) ( 7) ( 7) 4 49 ( 7) 4 49 ( 1) ( 8) 49 1 ( 8) m 5m+ 6( m+ ) ( 8) 5( 8) + 6( 8+ ) ( 8) 5( 8) 6( 6) 64 5( 8) 6( 6) 64 ( 40) ( 6) ( 6) , , , , Copyright 017 Pearson Education, Inc.

25 Section.5 Order of Operations with Integers; Mathematical Models y y m m+ n + 5n ( ) ( ) + ( 10) + 5( 10) ( 8) ( ) ( 10) 4 ( ) ( 50) ( 50) 16 + ( 50) m m+ n + 6n ( ) ( ) + ( 8) + 6( 8) ( 7) ( ) ( 8) 54 ( ) ( 48) ( 48) 11+ ( 48) c+ a 4b ( 6) + ( 8) 4( 0) Copyright 017 Pearson Education, Inc. 85

26 Chapter Integers and Introduction to Solving Equations 6. 5c+ a b 5( 10) + 9 ( 0) b 4ac ( 6) 4( 5) 6 4( 5) 6 ( 60) b 4ac 4( 6)( 4) 9 4( 6)( 4) 9 ( 96) ( ) b 1 c a ( ) b 15 c a ( ) + ( ) ( 7) , true The number is a solution , true The number is a solution ( + 1) ( ) + ( ) ( ) 4 ( 17) , true The number is a solution ( 4+ ) + 1 ( 4+ ) 4 ( ) , true The number is a solution. w w , false The number is not a solution. w w , false The number is not a solution. y+ 6+ 5y 7y 8y ( 10) 14 ( 16) 1 + ( 10) , true The number is a solution. 86 Copyright 017 Pearson Education, Inc.

27 Section.5 Order of Operations with Integers; Mathematical Models 74. 4y 8 y 10y y ( ) ( ) ( ) ( ) 8 8 ( ) 0 ( 6) 8+ ( 8) , true The number is a solution. 75. m ( m ) ( ) , false The number is not a solution. 76. m ( m ) ( ) , false The number is not a solution , true The number is a solution , true The number is a solution y + y ( ) () , false The number is not a solution. y + 5y ( ) , false The number is not a solution. 81. ( 8) , true The number is a solution. 8. ( 8) , true The number is a solution. z 1z , true The number is a solution. z 7z ( ) , true The number is a solution. Copyright 017 Pearson Education, Inc. 87

28 Chapter Integers and Introduction to Solving Equations y y + 8y ( ) + ( ) ( 59) , true The number is a solution. 4+ y y + 1y ( 60) + ( 1) , true The number is a solution. 87. a a. b. b a b ( 5) 6 ( 50) 90. a b ( 5) 9 ( 0) a b a. 9. a b. b e e e e e billion worldwide s in 010 e e e e e billion worldwide s in 009 c. According to the model there were 96 billion 49 billion, or 47 billion more worldwide s in 010. d. According to the graph there were 94 billion 47 billion, or 47 billion more worldwide s in 010. This is the same as the number obtained by the model. 94. a. D ( ) + D D D D 5470 The mean credit card debt in 011 was $ Copyright 017 Pearson Education, Inc.

29 Section.6 Solving Equations: The Addition and Multiplication Properties of Equality b. D ( ) + D D D D 510 The mean credit card debt in 01 was $510. c. According to the model the mean credit card balance in 01 was $5470 $510, or $160 less than in 011. d. According to the model the mean credit card balance in 01 was $5476 $55, or $151 less than in 011. The number obtained by the model is $9 greater Answers will vary. 98. does not make sense; Eplanations will vary. Sample eplanation: is not a solution because when it is substituted for the variable in the equation a false statement occurs. 99. makes sense 100. makes sense 101. makes sense 10. false; Changes to make the statement true will vary. A sample change is: false; Changes to make the statement true will vary true 105. true A sample change is: 106. ( 1) ( 4) ( 4 1) ( 4) ( 8) ( 4) R y , true The number is a solution w 1 8+ ( 0) 1 1 1, true The number is a solution. w w , false The number is not a solution..6 Check Points 1. We can isolate the variable,, by adding 5 to both sides of the equation No additional parentheses are needed Copyright 017 Pearson Education, Inc. 89

30 Chapter Integers and Introduction to Solving Equations. We can isolate the variable, z, by subtracting 0 from both sides of the equation. z + 0 z z 10 z The solution set is { 10 }.. We can isolate the variable, y, by adding 1 to both sides of the equation. 9 y y y The solution set is { 4. } 4. We can isolate the variable, w, by subtracting 1 from both sides of the equation. 1 + w w w w ( 6) We can isolate the variable,, by multiplying both sides of the equation by We can isolate the variable, w, by multiplying both sides of the equation by 6. w 10 6 w w 60 w w a. We can isolate the variable,, by dividing both sides of the equation by b. We can isolate the variable, y, by dividing both sides of the equation by y 44 11y y 4 y 4 11y The solution set is { } Copyright 017 Pearson Education, Inc.

31 Section.6 Solving Equations: The Addition and Multiplication Properties of Equality c. We can isolate the variable, z, by dividing both sides of the equation by z 15 5z 5 5 1z z 15 5z 15 5( ) a. We can isolate the variable,, by multiplying both sides of the equation by ( 11) b. We can isolate the variable, y, by adding 1 to both sides of the equation. y 1 8 y y 6 y The solution set is { 6 }. c. We can isolate the variable, z, by dividing both sides of the equation by z 56 8z z 7 z 56 8z 56 8( 7) d. We can isolate the variable, m, by subtracting 0 from both sides of the equation. 0 m m m 0 m The solution set is { 0 }. 9. In the formula, A represents the child s age, in months. Thus, we substitute 50 for A. Then use the addition property of equality to find V, the number of words in the child s vocabulary. V A V (50) V V V 100 At 50 months, a child will have a vocabulary of 100 words. 10. Substitute 60 for R Then use the multiplication property of equality to find L, the lifespan of mice. RL L L L L At 60 beats per minute, the average lifespan of mice is years. Copyright 017 Pearson Education, Inc. 91

32 Chapter Integers and Introduction to Solving Equations.6 Concept and Vocabulary Check 1. solving. equivalent. b c 4. subtract; solution 5. adding 7 6. subtracting 7 7. bc 8. divide 9. multiplying; dividing; 8.6 Eercise Set The solution set is { } The solution set is { }.. z z z 4 z z z z z The solution set is {. } 5. z z z 0 z The solution set is { } 6. z z z 8 z The solution set is { 8 }. 7. y y y y y y y 1 y The solution set is { 4 }. 9 Copyright 017 Pearson Education, Inc.

33 Section.6 Solving Equations: The Addition and Multiplication Properties of Equality w w w 17 w The solution set is { 1 } w w w 1 w The solution set is { } y 0 6+ y y y 0 6+ ( 14) y 9 8+ y y 1 8+ y 9 8+ ( 1) ( 4) Copyright 017 Pearson Education, Inc. 9

34 Chapter Integers and Introduction to Solving Equations y 11 y 11 ( ) 1y y y The solution set is { } y 8 5 y 58 ( 5) y 40 y y The solution set is { } 19. 5z 5 5z z 7 z 7 5z z 4 6z z 7 z 7 6z y 6 7y y 9 y 9 7y The solution set is { }. 4y 4y 4 4 1y 8 y 8 4y The solution set is { } The solution set is { } 94 Copyright 017 Pearson Education, Inc.

35 Section.6 Solving Equations: The Addition and Multiplication Properties of Equality The solution set is { } z 18 z 6 1z 6 z 18 z z 54 9z z 6 z 54 9z y 17y y y 17 y y 16y y y 16y 16( 0) The solution set is { } The solution set is { 8. } Copyright 017 Pearson Education, Inc. 95

36 Chapter Integers and Introduction to Solving Equations The solution set is { } ( 4) The solution set is { } 7. y 17 4 y y 71 y The solution set is { 71 }. 8. y y y 8 y The solution set is { 4 } The solution set is { 1 }. 96 Copyright 017 Pearson Education, Inc.

37 Section.6 Solving Equations: The Addition and Multiplication Properties of Equality 41. 5w w w 0 5w The solution set is { } 4. 6w 181 6w w 0 6w The solution set is { } z 496 1z z 496 1z z 714 4z z 714 4z z z z z The solution set is { } z z z z The solution set is { } z z 1 1 z 0 1+z The solution set is { } z z 4 4 z 0 4+z The solution set is { } Copyright 017 Pearson Education, Inc. 97

38 Chapter Integers and Introduction to Solving Equations 49. z z z 15,76 z , ,76. The solution set is { } 5. z z 4 ( 4)( 714) 4 z 9, 988 z , , z z z 9,988 z , ,988. The solution set is { } z z 1 ( 1)( 496) 1 z 15, 76 z , , The number is The number is The number is Copyright 017 Pearson Education, Inc.

39 Section.6 Solving Equations: The Addition and Multiplication Properties of Equality The number is The number is ( 15) 45 The number is The number is The number is S 1850, M 150 C+ M S C C C 1700 The cost of the computer is $ C 50, S 650 C+ M S 50 + M 650 M M 10 The markup is $ D ( 65) years after 1980, or in 05 the U.S. diversity inde will be D ( 7) years after 1980, or in 05 the U.S. diversity inde will be 7. n 69. M 5 n 5 n n If you are miles away from the lightning flash, it will take 10 seconds for the sound of thunder to reach you. n 70. M 5 n 5 n n If you are miles away from the lightning flash, it will take 15 seconds for the sound of thunder to reach you. 71. B 7. C Copyright 017 Pearson Education, Inc. 99

40 Chapter Integers and Introduction to Solving Equations 7. a. b. 74. a b. A lw ( 9) 171 l l l The length is 19 yards. P l+ w P P P 56 The perimeter is 56 yards. A lw 161 l ( 7) 161 l ( 7) 7 7 l The length is feet. P l+ w P + ( 7) P P 60 The perimeter is 60 feet. 150 p wa 150 p ( 60)( 5) 150 p p p 10 The dosage is 10 milligrams. 150 p wa 150 p ( 90)( 5) 150 p p p 15 The dosage is 15 milligrams Answers will vary. 81. does not make sense; Eplanations will vary. Sample eplanation: They are both mathematically the same. 8. makes sense 8. makes sense 84. does not make sense; Eplanations will vary. Sample eplanation: To isolate the variable we need to use the multiplication property of equality. 85. false; Changes to make the statement true will vary. 18 A sample change is: If 18, then. 86. true 87. true 88. false; Changes to make the statement true will vary. A sample change is: y Answers will vary. Eample: Answers will vary. Eample: , 56 19,15 7,56 + 7, 56 19,15 + 7,56 18,11 The solution set is { 18,11 } ,59 y + 17,96 67,59 17,96 y + 17,96 17,96 60, 71 y 9. The solution set is { 60,71 }. w w w 1,75,156 The solution set is { 1, 75,156 } , z 860, z z The solution set is { 49 } Copyright 017 Pearson Education, Inc.

41 Chapter Review Eercises ( 1) 4( 1) 1 ( 4) a 7; b a 4; b a ; b 10 Chapter Review Eercises < > > The opposite of 19 is The opposite of is. 10. ( 7) ( 6) ( 17) ( 5) ( 6) Copyright 017 Pearson Education, Inc. 101

42 Chapter Integers and Introduction to Solving Equations ( 5) + ( 1) + 4 ( 7 + 4) + ( 5) + ( 1) 11+ ( 18) ( 15) + ( 7) 1+ ( 41) + ( 15) + ( 7) 1+ ( 18) The person is standing at 800 feet below sea level ( ) ( 4) + ( ) + ( ) + ( 4) 0 + [ 7] The water level is feet ( 1). 9 ( 15) ( 0) ( 1) ( 04) ( 59) ( 75) ( 75) ( 8) ( 5) [ 5 11 ] ( 7) ( 16) ( ) ( 11) ( 0) ( 0) Copyright 017 Pearson Education, Inc.

43 Chapter Review Eercises ( 80) ,500 ( 650) 6, ,150 The difference in elevation is 7,150 feet. 8. 7( 1) ( 5) ( 6) ( 7)( )(10) ( )( )( 4) ( 44) ( 4) ( 50) 5. ( 50) 0 undefined b. Lost money in January, April, and May ( 75) + ( 190) 415 The total loss was $415. c. Made money in February, March, and June The total made was $175. d The mean loss was $40. e. 150 ( 75) 75 You loss $75 more in January. f. ( 150) 450 The total loss would be $ ( ) 5 6+ ( 10) ( ) 0 10( 8) ( 8) ( 4 5) 1 ( 0) 1 ( 17) (16) 6 16 ( 16) a. Lost the most money in May The total loss would be $ ( 4 ) 8 ( 16) 8 ( 14) Copyright 017 Pearson Education, Inc. 10

44 Chapter Integers and Introduction to Solving Equations ( 5) + 5( ) 8 4( ) + 5( ) 81 + ( 15) 8[ ] ( + ) ( 5) ( 5) + 4( 5+ ) 5 ( 5) + 4( ) 5 ( 15) + ( 1) ( 1) 15 + ( 7) 68. b 4ac ( 5) 4( ) 5 4( ) 5 ( 4) ( 6) ( 6) , false The number is not a solution. 70. ( + ) 18 5 ( 4+ ) 18 5( 4) ( 1) , true The number is a solution. 71. a. F ( + 15) F ( 10) ( ) F ( 10) ( ) F ( 10) ( 7) F F F 6470 There was 6470 farmers markets in 010. b. The model overestimates by , or by 8 farmers markets y y y 14 y The solution set is { } z z z Copyright 017 Pearson Education, Inc.

45 Chapter Test w w w w The solution set is { } y 6 9y y 6 9y a. rt 40 0t 40 0t t 14 The time for the trip is 14 hours. b. rt 40 60t 40 60t t 7 The time for the trip is 7 hours. Chapter Test ( 6) ( 11) ( 11) + ( + 6). ( 17) ( 4)( 5)( 1) ( 5 1) 6 ( 7) ( )( 4) ( 7 10) ( )( 4) ( ) 1 ( ) 4 8. ( 6 8) ( 5 7) ( ) ( ) 4( 8) 9. ( ) 8 ( ) > , 00 ( 80) 16, ,00 The difference in elevation is 17,00 feet. 1. a. 7 7 b. 7 7 c. ( 7) 7 c. The difference is 14 hours 7 hours, or 7 hours. Copyright 017 Pearson Education, Inc. 105

46 Chapter Integers and Introduction to Solving Equations ( + ) ( 4) 5( 4) + ( 4+ ) 16 5( 4) + ( 1) 16 ( 0) + ( ) ( ) + ( 18) 14. ( + ) 15 4 ( 9+ ) 15 4( 9) ( 7) , true The number is a solution The solution set is { 7 } y 70 7y y 70 7 y y y y 16 y The solution set is { }. w w w 0 w The solution set is { } a. A A A A 8 The office area per worker in 010 was 08 square feet. b. The model underestimates by 5 square feet 08 square feet, or by 17 square feet. 106 Copyright 017 Pearson Education, Inc.