Chapter 8 - XYZ Custom Plus
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Real Numbers andAlgebraic Expressions8Introduction© 2008 Tele AtlasImage © 2008 TerraMetricsImage © 2008 DigitalGlobe<strong>Chapter</strong> Outline8.1 Positive and NegativeNumbers8.2 Addition with NegativeNumbers8.3 Subtraction with NegativeNumbers8.4 Multiplication withNegative Numbers8.5 Division with NegativeNumbers8.6 Simplifying AlgebraicExpressionsThe Grand Canyon, located in the state of Arizona, is a large gorge created by theColorado River over millions of years. Much of the Grand Canyon is located in theGrand Canyon National Park, which receives over four million visitors per year.Visitors come to hike trails and view the magnificent rock formations.The Grand Canyon Hiking Trails+North Rim TrailheadYaki PointChange in altitudeBright Angel Trailhead–Colorado RiverMany of the hiking trails have significant changes in altitude. We sometimesrepresent changes in altitude with negative numbers. In this chapter we will workproblems involving both negative numbers and some of the trails found in theGrand Canyon.447
Introduction . . .Positive and Negative NumbersBefore the late nineteenth century, time zones did not exist. Each town would settheir clocks according to the motions of the Sun. It was not until the late 1800sthat a system of worldwide time zones was developed. This system divides theearth into 24 time zones with Greenwich, England designated as the center of thetime zones (GMT). This location is assigned a value of zero. Each of the WorldTime Zones is assigned a number ranging from 212 to 112 depending on its positioneast or west of Greenwich, England.8.1ObjectivesA u se the number line and inequalitysymbols to compare numbers.B Find the absolute value of anumber.C Find the opposite of a number.D Solve applications involving negativenumbers.‒11 ‒10 ‒9 ‒8 ‒7 ‒6 ‒5 ‒4 ‒3 ‒2 ‒1 0 1 2 3 4 5 6 7 8 9 10 11 12Examples now playing atMathTV.com/booksIf New York is 5 time zones to the left of GMT, this would be noted as 25:00 GMT.A Comparing NumbersTo see the relationship between negative and positive numbers, we can extendthe number line as shown in Figure 1. We first draw a straight line and label aconvenient point with 0. This is called the origin, and it is usually in the middleof the line. We then label positive numbers to the right (as we have done previously),and negative numbers to the left.Negative directionPositive direction−5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5Negative numbersPositive numbersOriginNoteA number, otherthan 0, with no sign(1 or 2) in front of itis assumed to be positive. Thatis, 5 5 15.FIGuRE 1The numbers increase going from left to right. If we move to the right, we aremoving in the positive direction. If we move to the left, we are moving in the negativedirection. Any number to the left of another number is considered to be smallerthan the number to its right.−4 < −2−5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5−4 is less than −2 because −4 is to the left of −2 on the number lineFIGuRE 2We see from the line that every negative number is less than every positivenumber.8.1 Positive and Negative Numbers449
450<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsIn algebra we can use inequality symbols when comparing numbers.NotationIf a and b are any two numbers on the number line, thena , b is read “a is less than b”a . b is read “a is greater than b”Practice ProblemsWrite each statement in words.1. 2 , 82. 5 . 10 (Is this a true statement?)As you can see, the inequality symbols always point to the smaller of the twonumbers being compared. Here are some examples that illustrate how we use theinequality symbols.Example 13 , 5 is read “3 is less than 5.” Note that it would also becorrect to write 5 . 3. Both statements, “3 is less than 5” and “5 is greater than 3,”have the same meaning. The inequality symbols always point to the smaller number.Example 20 . 100 is a false statement, because 0 is less than 100,not greater than 100. To write a true inequality statement using the numbers 0and 100, we would have to write either 0 , 100 or 100 . 0.3. 24 , 4Example 323 , 5 is a true statement, because 23 is to the left of 5on the number line, and, therefore, it must be less than 5. Another statement thatmeans the same thing is 5 . 23.4. 27 , 22Example 425 , 22 is a true statement, because 25 is to the left of22 on the number line, meaning that 25 is less than 22. Both statements25 , 22 and 22 . 25 have the same meaning; they both say that 25 is a smallernumber than 22.BAbsolute ValueIt is sometimes convenient to talk about only the numerical part of a numberand disregard the sign (1 or 2) in front of it. The following definition gives us away of doing this.DefinitionGive the absolute value of each ofthe following.5. u6u6. u25uAnswers1. 2 is less than 8.2. 5 is greater than 10. (No.)3. 24 is less than 4.4. 27 is less than 22.5. 6 6. 5The absolute value of a number is its distance from 0 on the number line.We denote the absolute value of a number with vertical lines. For example,the absolute value of 23 is written u23u.The absolute value of a number is never negative because it is a distance, and adistance is always measured in positive units (unless it happens to be 0).Here are some examples of absolute value problems.Example 5u5u 5 5 The number 5 is 5 units from 0.Example 6u 23u 5 3 The number 23 is 3 units from 0.
8.1 Positive and Negative Numbers451Example 7u 27u 5 7 The number 27 is 7 units from 0.Give the absolute value.7. u28uCOppositesDefinitionTwo numbers that are the same distance from 0 but in opposite directionsfrom 0 are called opposites.* The notation for the opposite of a is 2a.Example 8Give the opposite of each of the following numbers:5, 7, 1, 25, 288. Give the opposite of each of thefollowing numbers: 8, 10, 0, 24.Solution The opposite of 5 is 25.The opposite of 7 is 27.The opposite of 1 is 21.The opposite of 25 is 2(25), or 5.The opposite of 28 is 2(28), or 8.We see from this example that the opposite of every positive number is a negativenumber, and likewise, the opposite of every negative number is a positivenumber. The last two parts of Example 8 illustrate the following property:PropertyIf a represents any positive number, then it is always true that2(2a) 5 aIn other words, this property states that the opposite of a negative number is apositive number.It should be evident now that the symbols 1 and 2 can be used to indicateseveral different ideas in mathematics. In the past we have used them to indicateaddition and subtraction. They can also be used to indicate the direction anumber is from 0 on the number line. For instance, the number 13 (read “positive3”) is the number that is 3 units from zero in the positive direction. On the otherhand, the number 23 (read “negative 3”) is the number that is 3 units from 0 inthe negative direction. The symbol 2 can also be used to indicate the opposite ofa number, as in 2(22) 5 2. The interpretation of the symbols 1 and 2 depends onthe situation in which they are used. For example:3 1 5 The 1 sign indicates addition.7 2 2 The 2 sign indicates subtraction.27 The 2 sign is read “negative” 7.2(25) The first 2 sign is read “the opposite of.” The second 2sign is read “negative” 5.This may seem confusing at first, but as you work through the problems in thischapter you will get used to the different interpretations of the symbols 1 and 2.We should mention here that the set of whole numbers along with their oppositesforms the set of integers. That is:Integers 5 {. . . , 23, 22, 21, 0, 1, 2, 3, . . .}*In some books opposites are called additive inverses.Answers7. 8 8. 28, 210, 0, 4
454<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsC Give the opposite of each of the following numbers. [Example 8]51. 357. 052. 2558. 153. 2259. 20.12354. 1560. 23.4555. 7561.56. 23271} 62. } 8 100Simplify each of the following.63. 2(22)64. 2(25)65. 2(28)66. 2(23)67. 2u22u68. 2u25u69. 2u28u70. 2u23u71. What number is its own opposite?72. Is uau 5 a always a true statement?73. If n is a negative number, is 2n positive or negative?74. If n is a positive number, is 2n positive or negative?EstimatingWork Problems 75–80 mentally, without pencil and paper or a calculator.75. Is 260 closer to 0 or 2100?76. Is 220 closer to 0 or 230?77. Is 210 closer to 220 or 20?78. Is 220 closer to 240 or 10?79. Is 2362 closer to 2360 or 2370?80. Is 2368 closer to 2360 or 2370?
8.1 Problem Set455D Applying the Concepts [Example 9]81. The London Eye has aheight of 450 feet. Describethe location ofsomeone standing onthe ground in relation tosomeone at the top of theLondon Eye.© 2008 Infoterra ltd & blueskyImage © 2008 bluesky82. The Eiffel Tower has severallevels visitors canwalk around on. The firstis 57 meters above theground, the second is 115meters high, and the thirdlevel is 276 meters high.What is the location ofsomeone standing on thefirst level in relation tosomeone standing on thethird level?Image © Aerodata InternationalSurveys© Cnes/Spot ImageImage © 2008 DigitalGlobe83. The Bright Angel trail at Grand Canyon National Parkends at Indian Garden, 3,060 feet below the trailhead.Write this as a negative number with respect to thetrailhead.84. The South Kaibab Trail at Grand Canyon National Parkends at Cedar Ridge, 1,140 feet below the trailhead.Write this as a negative number with respect to thetrailhead.85. Car Depreciation Depreciation refers to the decline in acar’s market value during the time you own the car.According to sources such as Kelley Blue Book andEdmunds.com, not all cars depreciate at the samerate. Suppose you pay $25,000 for a new car whichhas a high rate of depreciation. Your car loses about$5,000 in value per year. Represent this loss in valueas a negative number. A car with a low rate of depreciationloses about $2,750 in value each year.Represent this loss as a negative number.86. Census figures In June, 2007 the U.S. Census Bureau releasedpopulation estimates for the twenty-five citieswith the largest population loss between July 1, 2005and July 1, 2006. New Orleans had the largest populationloss. The city’s population fell by 228,782 people.Detroit, Michigan experienced a population loss of12,344 people during the same time period. Representthe loss of population for New Orleans and for Detroitas a negative number.87. temperature and Altitude Yamina is flying from Phoenixto San Francisco on a Boeing 737 jet. When the planereaches an altitude of 33,000 feet, the temperatureoutside the plane is 61 degrees below zero Fahrenheit.Represent this temperature with a negative number. Ifthe temperature outside the plane gets warmer by 10degrees, what will the new temperature be?88. temperature Change At 11:00 in the morning in Superior,Wisconsin, Jim notices the temperature is 15degrees below zero Fahrenheit. Write this temperatureas a negative number. At noon it has warmed up by 8degrees. What is the temperature at noon?
456<strong>Chapter</strong> 8 Real Numbers and Algebraic Expressions89. Temperature Change At 10:00 in the morning in WhiteBear Lake, Minnesota, Zach notices the temperature is5 degrees below zero Fahrenheit. Write this temperatureas a negative number. By noon the temperaturehas dropped another 10 degrees. What is the temperatureat noon?90. Snorkeling Steve is snorkeling in the ocean near hishome in Maui. At one point he is 6 feet below the surface.Represent this situation with a negative number.If he descends another 6 feet, what negative numberwill represent his new position?91. Time Zones New Orleans, Louisiana, is 1 time zone westof New York City. Represent this time zone as a negativenumber, as discussed in the introduction to thischapter.92. Time Zones Seattle, Washington, is 2 time zones west ofNew Orleans, Louisiana. Represent this time zone asa negative number, as discussed in the introduction tothis chapter.-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12New Orleans, LANew York, NYSeattle, WANew Orleans, LATable 2 lists various wind chill temperatures. The top row gives air temperature, while the first column gives wind speedin miles per hour. The numbers within the table indicate how cold the weather will feel. For example, if the thermometerreads 308F and the wind is blowing at 15 miles per hour, the wind chill temperature is 98F.Table 2Wind chill temperaturesAir temperatures (°F)Wind Speed 30° 25° 20° 15° 10° 5° 0° 25°10 mph 16° 10° 3° 23° 29° 215° 222° 227°15 mph 9° 2° 25° 211° 218° 225° 231° 238°20 mph 4° 23° 210° 217° 224° 231° 239° 246°25 mph 1° 27° 215° 222° 229° 236° 244° 251°30 mph 22° 210° 218° 225° 233° 241° 249° 256°93. Wind Chill Find the wind chill temperature if the thermometerreads 258F and the wind is blowing at 25miles per hour.94. Wind Chill Find the wind chill temperature if the thermometerreads 108F and the wind is blowing at 25miles per hour.95. Wind Chill Which will feel colder: a day with an air temperatureof 108F and a 25-mph wind, or a day with anair temperature of 258F and a 10-mph wind?96. Wind Chill Which will feel colder: a day with an air temperatureof 158F and a 20-mph wind, or a day with anair temperature of 58F and a 10-mph wind?
8.1 Problem Set457Table 3 lists the record low temperatures for each month of the year for Lake Placid, New York. Table 4 lists the recordhigh temperatures for the same city.Table 3Record low temperatures forlake placid, New yorkTable 4Record high temperatures forlake placid, new yorkMonthtemperatureMonthtemperatureJanuary 236°FFebruary 230°FMarch 214°FApril 22°FMay 19°FJune 22°FJuly 35°FAugust 30°FSeptember 19°FOctober 15°FNovember 211°FDecember 226°FJanuary 54°FFebruary 59°FMarch 69°FApril 82°FMay 90°FJune 93°FJuly 97°FAugust 93°FSeptember 90°FOctober 87°FNovember 67°FDecember 60°F97. Temperature Figure 5 is a bar chart of the information in Table 3. Use the template in Figure 6 to construct a scatter diagramof the same information. Then connect the dots in the scatter diagram to obtain a line graph of that same information.(Notice that we have used the numbers 1 through 12 to represent the months January through December.)40°40°Temperature (Fahrenheit)30°20°10°0°-10°-20°-30°Temperature (Fahrenheit)30°20°10°0°-10°-20°-30°-40°-40°-50°1 2 3 4 5 6 7 8 9 10 11 12MonthsFigure 5 A bar chart of Table 31 2 3 4 5 6 7 8 9 10 11 12MonthsFigure 6 A scatter diagram, then line graph of Table 398. Temperature Figure 7 is a bar chart of the information in Table 4. Use the template in Figure 8 to construct a scatterdiagram of the same information. Then connect the dots in the scatter diagram to obtain a line graph of that same information.(Again, we have used the numbers 1 through 12 to represent the months January through December.)100°100°Temperature (Fahrenheit)80°60°40°20°Temperature (Fahrenheit)80°60°40°20°0°1 2 3 4 5 6 7 8 9 10 11 12Months0°1 2 3 4 5 6 7 8 9 10 11 12MonthsFigure 7 A bar chart of Table 4Figure 8 A scatter diagram, then line graph of Table 4
458<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsGetting Ready for the Next SectionAdd or subtract.99. 10 1 15100. 12 1 15101. 15 2 10102. 15 2 12103. 10 2 5 2 3 1 4104. 12 2 3 2 7 1 5105. [3 1 10] 1 [8 2 2]106. [2 1 12] 1 [7 2 5]107. 276 1 32 1 4,005108. 17 1 3 1 152 1 1,200109. 635 2 579110. 2,987 2 1,130Maintaining Your SkillsComplete each statement using the commutative property of addition.111. 3 1 5 5 112. 9 1 x 5Complete each statement using the associative property of addition.113. 7 1 (2 1 6) 5 114. (x 1 3) 1 5 5Write each of the following in symbols.115. The sum of x and 4116. The sum of x and 4 is 9.117. 5 more than y118. x increased by 8Extending the Concepts119. There are two numbers that are 5 units from 2 on thenumber line. One of them is 7. What is the other one?120. There are two numbers that are 5 units from 22 on thenumber line. One of them is 3. What is the other one?121. In your own words and in complete sentences, explainwhat the opposite of a number is.122. In your own words and in complete sentences, explainwhat the absolute value of a number is.123. The expression 2(23) is read “the opposite of negative3,” and it simplifies to just 3. Give a similar written descriptionof the expression 2u23u, and then simplify it.124. Give written descriptions of the expressions 2(24) and2u24u and then simplify each of them.
Introduction . . .Addition with Negative NumbersSuppose you are in Las Vegas playing blackjack and youlose $3 on the first hand and then you lose $5 on the nexthand. If you represent winning with positive numbers andlosing with negative numbers, how will you represent theresults from your first two hands? Since you lost $3 and $5for a total of $8, one way to represent the situation is withaddition of negative numbers:(2$3) 1 (2$5) 5 2$8From this example we see that the sum of two negativenumbers is a negative number. To generalize addition of positive and negativenumbers, we can use the number line.AAdding with a Number lineWe can think of each number on the number line as having two characteristics:(1) a distance from 0 (absolute value) and (2) a direction from 0 (positiveor negative). The distance from 0 is represented by the numerical part of thenumber (like the 5 in the number 25), and its direction is represented by the 1or 2 sign in front of the number.We can visualize addition of numbers on the number line by thinking in termsof distance and direction from 0. Let’s begin with a simple problem we know theanswer to. We interpret the sum 3 1 5 on the number line as follows:1. The first number is 3, which tells us “start at the origin, and move 3 units in thepositive direction.”2. The 1 sign is read “and then move.”3. The 5 means “5 units in the positive direction.”Start3 units 5 unitsJ♣♣JEnd8.2ObjectivesA u se the number line to add positiveand negative numbers.B Add positive and negative numbersusing a rule.C Solve applications involvingaddition with positive and negativenumbers.NoteExamples now playing atMathTV.com/booksThis method of addingnumbers may seema little complicated atfirst, but it will allow us to addnumbers we couldn’t otherwiseadd.−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8FIGuRE 1Figure 1 shows these steps. To summarize, 3 1 5 means to start at the origin(0), move 3 units in the positive direction, and then move 5 units in the positivedirection. We end up at 8, which is the sum we are looking for: 3 1 5 5 8.ExAmplE 1Add 3 1 (25) using the number line.sOlutiON We start at the origin, move 3 units in the positive direction, andthen move 5 units in the negative direction, as shown in Figure 2. The last arrowends at 22, which must be the sum of 3 and 25. That is:3 1 (25) 5 22prACtiCE prOBlEms1. Add: 2 1 (25)EndStart3 units5 units−8 −7 −6 −5 −4 −3 −2 −1 1 0 1 23 4 5 6 7 8FIGuRE 28.2 Addition with Negative NumbersAnswer1. 23459
462<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsThe following examples show how the rule is used. You will find that the rulefor addition is consistent with all the results obtained using the number line.8. Add all combinations of positiveand negative 12 and 15.Example 8Add all combinations of positive and negative 10 and 15.Solution 10 1 15 5 2510 1 (215) 5 25210 1 15 5 5210 1 (215) 5 225Notice that when we add two numbers with the same sign, the answer also hasthat sign. When the signs are not the same, the answer has the sign of the numberwith the larger absolute value.Once you have become familiar with the rule for adding positive and negativenumbers, you can apply it to more complicated sums.9. Simplify: 12 1 (23) 1 (27) 1 5Example 9Simplify: 10 1 (25) 1 (23) 1 4SolutionAdding left to right, we have:10 1 (25) 1 (23) 1 4 5 5 1 (23) 1 4 10 1 (25) 5 55 2 1 4 5 1 (23) 5 25 610. Simplify:[22 1 (212)] 1 [7 1 (25)]Example 10Simplify: [23 1 (210)] 1 [8 1 (22)]SolutionWe begin by adding the numbers inside the brackets.[23 1 (210)] 1 [8 1 (22)] 5 [213] 1 [6]5 2711. Add: 25.76 1 (23.24)Example 11Add: 24.75 1 (22.25)Solution Because both signs are negative, we add absolute values. The answerwill be negative.24.75 1 (22.25) 5 27.0012. Add: 6.88 1 (28.55)Example 12Add: 3.42 1 (26.89)Solution The signs are different, so we subtract the smaller absolute valuefrom the larger absolute value. The answer will be negative, because 6.89 islarger than 3.42 and the sign in front of 6.89 is 2.3.42 1 (26.89) 5 23.4713. Add: 5 } 611 2 2 } 6 2Example 13Add: 3 } 81 12 1 } 8 2Solution} 3 8 } is positive.We subtract absolute values. The answer will be positive, becauseAnswers8. See solutions section.9. 7 10. 212 11. 29.00112. 21.67 13. } 23} 81 12 1 } 8 2 5 2 } 85 1 } 4Reduce to lowest terms
8.2 Addition with Negative Numbers463Example 14Add: } 110SolutionLCD of 20.UsingCalculator NoteTechnology} 1 12 4 } 5 2 1 1 2 3 } 20 2To begin, change each fraction to an equivalent fraction with an1}} 1 1 0 12 } 4 5 2 1 1 2 } 3 20 2 5 1 ? 2} 1 10 ? 2 12} 4 ? 45 ? 4 2 1 1 2 } 3 20 2There are a number of different ways in which calculators display negativenumbers. Some calculators use a key labeled 1/2, whereas others use akey labeled (2) . You will need to consult with the manual that came withyour calculator to see how your calculator does the job.Here are a couple of ways to find the sum 210 1 (215) on a calculator:Scientific Calculator: 10 1/2 1 15 1/2 5Graphing Calculator: (2) 10 1 (2) 15 ENT52} 201 12 16 } 20 2 1 1 2 3 } 20 25 2 14 } 201 12 3 } 20 25 2 17} 2014. Add: 1 } 211 2 }3 4 } 2 1 1 2 }5 8 } 2Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Explain how you would use the number line to add 3 and 5.2. If two numbers are negative, such as 23 and 25, what sign will their sumhave?3. If you add two numbers with different signs, how do you determine thesign of the answer?4. With respect to addition with positive and negative numbers, does thephrase “two negatives make a positive” make any sense?Answer14. 2} 7 8 }
8.2 Problem Set465Problem Set 8.2A Draw a number line from 210 to 110 and use it to add the following numbers. [Examples 1–7]1. 2 1 32. 2 1 (23)3. 22 1 34. 22 1 (23)5. 5 1 (27)6. 25 1 77. 24 1 (22)8. 28 1 (22)9. 10 1 (26)10. 29 1 311. 7 1 (23)12. 27 1 313. 24 1 (25) 14. 22 1 (27)B Combine the following by using the rule for addition of positive and negative numbers. (Your goal is to be fast and accurateat addition, with the latter being more important.) [Example 8]15. 7 1 816. 9 1 1217. 5 1 (28)18. 4 1 (211)19. 26 1 (25)20. 27 1 (22)21. 210 1 322. 214 1 723. 21 1 (22)24. 25 1 (24)25. 211 1 (25)26. 216 1 (210)27. 4 1 (212)28. 9 1 (21)29. 285 1 (242)30. 296 1 (231)31. 2121 1 17032. 2130 1 15833. 2375 1 40934. 2765 1 213Complete the following tables.35.First Second theirNumber Number suma b a1b36.First Second theirNumber Number suma b a1b5 235 245 255 265 2725 325 425 525 625 737.First Second theirNumber Number sumx y x1y38.First Second theirNumber Number sumx y x1y25 2325 2425 2525 2625 2730 220230 20230 22030 20230 0
466<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsB Add the following numbers left to right. [Example 9]39. 24 1 (26) 1 (28)40. 35 1 (25) 1 (230)41. 2201 1 (2143) 1 (2101)42. 227 1 (256) 1 (289)43. 2321 1 752 1 (2324)44. 2571 1 437 1 (2502)45. 22 1 (25) 1 (26) 1 (27)46. 28 1 (23) 1 (24) 1 (27)47. 15 1 (230) 1 18 1 (220)48. 20 1 (215) 1 30 1 (218)49. 278 1 (242) 1 57 1 1350. 289 1 (251) 1 65 1 17B Use the rule for order of operations to simplify each of the following. [Example 10]51. (28 1 5) 1 (26 1 2)52. (23 1 1) 1 (29 1 4)53. (210 1 4) 1 (23 1 12)54. (211 1 5) 1 (23 1 2)55. 20 1 (230 1 50) 1 1056. 30 1 (240 1 20) 1 5057. 108 1 (2456 1 275)58. 106 1 (2512 1 318)59. [5 1 (28)] 1 [3 1 (211)]60. [8 1 (22)] 1 [5 1 (27)]61. [57 1 (235)] 1 [19 1 (224)]62. [63 1 (227)] 1 [18 1 (224)]Use the rule for addition of numbers to add the following fractions and decimals. [Examples 11–14]63. 21.3 1 (22.5)64. 29.1 1 (24.5)65. 24.8 1 (210.4)66. 29.5 1 (221.3)67. 25.35 1 2.35 1 (26.89)68. 29.48 1 5.48 1 (24.28)69. 2 5 } 61 12 1 } 6 270. 2 7 } 91 12 2 } 9 271.3} 1 7 12 } 5 7 272. 11 } 131 12 12 } 13 273. 2 2 } 51 3 } 51 12 4 } 5 274. 2 6 } 71 4 } 71 12 1 } 7 2
8.2 Problem Set46775. 23.8 1 2.54 1 0.476. 29.6 1 5.15 1 0.877. 22.89 1 (21.4) 1 0.0978. 23.99 1 (21.42) 1 0.0679.1} 1 2 12 } 3 4 280.3} 1 5 12} 7 10 263. Find the sum of 28, 210, and 23.64. Find the sum of 24, 17, and 26.65. What number do you add to 8 to get 3?66. What number do you add to 10 to get 4?67. What number do you add to 23 to get 27?68. What number do you add to 25 to get 28?69. What number do you add to 24 to get 3?70. What number do you add to 27 to get 2?71. If the sum of 23 and 5 is increased by 8, what numberresults?72. If the sum of 29 and 22 is increased by 10, what numberresults?CApplying the Concepts81. One of the trails at the Grand Canyon starts at BrightAngel Trailhead and then drops 4,060 feet to the ColoradoRiver and then climbs 4,440 feet to Yaki Point.What is the trail’s ending position in relation to theBright Angel Trailhead? If the trail ends below the startingposition write the answer as a negative number.82. One of the trails in the Grand Canyon starts at theNorth Rim trailhead and drops 5,490 feet to the ColoradoRiver. The trail then climbs 4,060 feet to the BrightAngel Trailhead. What is the Bright Angel Trailhead’sposition in relation to the North Rim Trailhead? If thetrail ends below the starting position write the answeras a negative number.Yaki Point4,440 ftNorth RimTrailheadBright Angel Trailhead4,060 ftYaki PointBright AngelTrailhead4,060 ft5,490 ftNorth RimTrailheadColorado RiverColorado River
468<strong>Chapter</strong> 8 Real Numbers and Algebraic Expressions83. Checkbook Balance Ethan has a balance of 2$40 inhis checkbook. If he deposits $100 and then writes acheck for $50, what is the new balance in his checkbook?84. Checkbook Balance Kendra has a balance of 2$20 inher checkbook. If she deposits $45 and then writes acheck for $15, what is the new balance in her checkbook?RECORD ALL CHARGES OR CREDITS THAT AFFECT YOUR ACCOUNTNUMBER DATE DESCRIPTION OF TRANSACTIONPAYMENT/DEBITDEPOSIT/CREDIT(-)(+)p $1001502 Vons Market $50 009/20 Deposit$1009/2100BALANCE-$4000RECORD ALL CHARGES OR CREDITS THAT AFFECT YOUR ACCOUNTNUMBER DATE DESCRIPTION OF TRANSACTIONPAYMENT/DEBITDEPOSIT/CREDIT(–) (+)9/25 Deposit$45009/28p $451504 SLO Soccer $15 002 00BALANCE-$2000Getting Ready for the Next SectionGive the opposite of each number.85. 290.86. 33} 91. 230887. 2492. 21588. 2593. 60.389. 2 _5 94. 70.495. Subtract 3 from 5.96. Subtract 2 from 8.97. Find the difference of 7 and 4.98. Find the difference of 8 and 6.Maintaining Your SkillsThe problems below review subtraction with whole numbers.Subtract.99. 763 2 159 100. 1,007 2 136 101. 465 2 462 2 3 102. 481 2 479 2 2Write each of the following statements in symbols.103. The difference of 10 and x.104. The difference of x and 10.105. 17 subtracted from y.106. y subtracted from 17.
Introduction . . .Subtraction with Negative NumbersHow would we represent the final balance in a checkbook if the original balancewas $20 and we wrote a check for $30? The final balance would be 2$10. We cansummarize the whole situation with subtraction:$20 2 $30 5 2$108.3ObjectivesA Subtract numbers by thinking ofsubtraction as addition of theopposite.B Solve applications involvingsubtraction with positive andnegative numbers.RECORD ALL CHARGES OR CREDITS THAT AFFECT YOUR ACCOUNTPAYMENT/DEBITDEPOSIT/CREDITNUMBER DATE DESCRIPTION OF TRANSACTION(-)(+)1501 9/15Campus Bookstore $30 00BALANCE$20 00-$10 00Examples now playing atMathTV.com/booksFrom this we see that subtracting 30 from 20 gives us 210. Another example thatgives the same answer but involves addition is this:20 1 (230) 5 210A SubtractionFrom the two examples above, we find that subtracting 30 gives the same resultas adding 230. We use this kind of reasoning to give a definition for subtractionthat will allow us to use the rules we developed for addition to do our subtractionproblems. Here is that definition:DefinitionSubtraction If a and b represent any two numbers, then it is always true that8888na 2 b 5 a 1 (2b)m8888To subtract b Add its opposite, 2bIn words: Subtracting a number is equivalent to adding its opposite.Let’s see if this definition conflicts with what we already know to be true aboutsubtraction.ExAmplE 1Subtract: 5 2 2sOlutiON From previous experience we know that5 2 2 5 3NoteThis definition of subtractionmay seem alittle strange at first.In Example 1 you will notice thatusing the definition gives us thesame results we are used to gettingwith subtraction. As we progressfurther into the section, wewill use the definition to subtractnumbers we haven’t been able tosubtract before.prACtiCE prOBlEms1. Subtract: 7 2 3We can get the same answer by using the definition we just gave for subtraction.Instead of subtracting 2, we can add its opposite, 22. Here is how it looks:5 2 2 5 5 1 (22) Change subtraction toaddition of the opposite5 3 Apply the rule for addition ofpositive and negative numbersThe result is the same whether we use our previous knowledge of subtractionor the new definition. The new definition is essential when the problems begin toget more complicated.8.3 Subtraction with Negative NumbersAnswer1. 4469
470<strong>Chapter</strong> 8 Real Numbers and Algebraic Expressions2. Subtract: 27 2 3NoteA real-life analogy toExample 2 would be:“If the temperaturewere 78 below 0 and then itdropped another 28, what wouldthe temperature be then?”3. Subtract: 28 2 6ExAmplE 2Subtract: 27 2 2sOlutiONWe have never subtracted a positive number from a negative numberbefore. We must apply our definition of subtraction:27 2 2 5 27 1 (22) Instead of subtracting 2,we add its opposite, 225 29 Apply the rule for additionExAmplE 3Subtract: 210 2 5sOlutiONWe apply the definition of subtraction (if you don’t know the definitionof subtraction yet, go back and read it) and add as usual.210 2 5 5 210 1 (25) Definition of subtraction5 215 Addition4. Subtract: 10 2 (26)ExAmplE 4Subtract: 12 2 (26)sOlutiONThe first 2 sign is read “subtract,” and the second one is read “negative.”The problem in words is “12 subtract negative 6.” We can use the definitionof subtraction to change this to the addition of positive 6:12 2 (26) 5 12 1 6 Subtracting 26 is equivalentto adding 165 18 Addition5. Subtract: 210 2 (215)NoteExamples 4 and 5 maygive results you are notused to getting. Butyou must realize that the resultsare correct. That is, 12 2 (26) is18, and 220 2 (230) is 10. If youthink these results should be different,then you are not thinking ofsubtraction correctly.ExAmplE 5Subtract: 220 2 (230)sOlutiONInstead of subtracting 230, we can use the definition of subtractionto write the problem again as the addition of 30:220 2 (230) 5 220 1 30 Definition of subtraction5 10 AdditionExamples 1–5 illustrate all the possible combinations of subtraction with positiveand negative numbers. There are no new rules for subtraction. We apply thedefinition to change each subtraction problem into an equivalent addition problem.The rule for addition can then be used to obtain the correct answer.6. Subtract each of the following.a. 8 2 5b. 28 2 5c. 8 2 (25)d. 28 2 (25)e. 12 2 10f. 212 2 10g. 12 2 (210)h. 212 2 (210)ExAmplE 6The following table shows the relationship between subtractionand addition:subtraction Addition of the Opposite Answer7 2 9 7 1 (29) 2227 2 9 27 1 (29) 2167 2 (29) 7 1 9 1627 2 (29) 27 1 9 215 2 10 15 1 (210) 5Answers2. 210 3. 214 4. 16 5. 56. a. 3 b. 213 c. 13 d. 23e. 2 f. 222 g. 22 h. 22215 2 10 215 1 (210) 22515 2 (210) 15 1 10 25215 2 (210) 215 1 10 25
8.3 Subtraction with Negative Numbers471Example 7Combine: 23 1 6 2 27. Combine: 24 1 6 2 7SolutionThe first step is to change subtraction to addition of the opposite.After that has been done, we add left to right.23 1 6 2 2 5 23 1 6 1 (22) Subtracting 2 is equivalentto adding 225 3 1 (22) Add left to right5 1Example 8Combine: 10 2 (24) 2 88. Combine: 15 2 (25) 2 8SolutionChanging subtraction to addition of the opposite, we have10 2 (24) 2 8 5 10 1 4 1 (28)5 14 1 (28)5 6Example 9Subtract 3 from 25.9. Subtract 2 from 28.Solution Subtracting 3 is equivalent to adding 23.25 2 3 5 25 1 (23) 5 28Subtracting 3 from 25 gives us 28.Example 10Subtract 24 from 9.10. Subtract 25 from 7.Solution Subtracting 24 is the same as adding 14:9 2 (24) 5 9 1 4 5 13Subtracting 24 from 9 gives us 13.Example 11Find the difference of 27 and 24.11. Find the difference of 28 and 26.SolutionSubtracting 24 from 27 looks like this:27 2 (24) 5 27 1 4 5 23The difference of 27 and 24 is 23.Example 12Subtract 60.3 from 249.8.Subtract.12. 257.8 2 70.4Solution 249.8 2 60.3 5 249.8 1 (260.3)5 2110.1Example 13Find the difference of 2} 3 5 } and }2 5 }.13. 2 5 } 82 3 } 8SOLUTION 2 3 } 52 2 } 55 2 3 } 51 12 2 } 5 25 2 5 } 55 21Answers7. 25 8. 12 9. 210 10. 1211. 22 12. 2128.2 13. 21
472<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsBApplication14. Suppose the temperature is428F at takeoff and then dropsto 2428F when the planereaches its cruising altitude.Find the difference in temperatureat takeoff and at cruisingaltitude.Example 14Many of theplanes used by the United States duringWorld War II were not pressurizedor sealed from outside air. As a result,the temperature inside these planeswas the same as the surrounding airtemperature outside. Suppose the temperatureinside a B-17 Flying Fortress is508F at takeoff and then drops to 2308Fwhen the plane reaches its cruising altitude of 28,000 feet. Find the difference intemperature inside this plane at takeoff and at 28,000 feet.Courtesy of the U.S. Air Force MuseumSolution The temperature at takeoff is 508F, whereas the temperature at28,000 feet is 2308F. To find the difference we subtract, with the numbers in thesame order as they are given in the problem:50 2 (230) 5 50 1 30 5 80The difference in temperature is 808F.Subtraction and Taking AwaySome people may believe that the answer to 25 2 9 should be 24 or 4, not 214.If this is happening to you, you are probably thinking of subtraction in terms oftaking one number away from another. Thinking of subtraction in this way workswell with positive numbers if you always subtract the smaller number from thelarger. In algebra, however, we encounter many situations other than this. Thedefinition of subtraction, that a 2 b 5 a 1 (2b) clearly indicates the correct way touse subtraction. That is, when working subtraction problems, you should think“addition of the opposite,” not “taking one number away from another.”UsingTechnologyCalculator NoteHere is how we work the subtraction problem shown in Example 11 on acalculator.Scientific Calculator: 7 1/2 2 4 1/2 5Graphing Calculator: (2) 7 2 (2) 4 ENTGetting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.Answer14. 848F1. Write the subtraction problem 5 2 3 as an equivalent addition problem.2. Explain the process you would use to subtract 2 from 27.3. Write an addition problem that is equivalent to the subtraction problem220 2 (230).4. To find the difference of 27 and 24 we subtract what number from 27?
8.3 Problem Set473Problem Set 8.3A Subtract. [Examples 1–5]1. 7 2 52. 5 2 73. 8 2 64. 6 2 85. 23 2 56. 25 2 37. 24 2 18. 21 2 49. 5 2 (22)10. 2 2 (25)11. 3 2 (29)12. 9 2 (23)13. 24 2 (27)14. 27 2 (24)15. 210 2 (23)16. 23 2 (210)17. 15 2 1818. 20 2 3219. 100 2 11320. 121 2 2121. 230 2 2022. 250 2 6023. 279 2 2124. 286 2 3125. 156 2 (2243)26. 292 2 (2841)27. 235 2 (214)28. 229 2 (24)29. 29.01 2 2.433. 2} 1 6 } 2 }5 6 }30. 28.23 2 5.434. 2} 4 7 } 2 }3 7 }31. 20.89 2 1.0135. } 152736. }} 2 } 4 1 5 5 }} 2 } 5 6 }32. 20.42 2 2.0437. 2} 1 3} 2 } 2 }3704238. 2} 1 7} 2 } 1 }76090
474<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsASimplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right.[Examples 7, 8]39. 4 2 5 2 640. 7 2 3 2 241. 28 1 3 2 442. 210 2 1 1 1643. 28 2 4 2 244. 27 2 3 2 645. 12 2 30 2 4746. 229 2 53 2 3747. 33 2 (222) 2 6648. 44 2 (211) 1 5549. 101 2 (295) 1 650. 2211 2 (2207) 1 351. 2900 1 400 2 (2100)55. } 1 2 } 2 }1 3 } 2 }1 4 } 52. 2300 1 600 2 (2200)56. } 1 5 } 2 }1 6 } 2 }1 7 } 53. 23.4 2 5.6 2 8.5 54. 22.1 2 3.1 2 4.1A Translate each of the following and simplify the result. [Examples 9–11]57. Subtract 26 from 5.58. Subtract 8 from 22.59. Find the difference of 25 and 21.60. Find the difference of 27 and 23.61. Subtract 24 from the sum of 28 and 12.62. Subtract 27 from the sum of 7 and 212.63. What number do you subtract from 23 to get 29?64. What number do you subtract from 5 to get 8?EstimatingWork Problems 65–70 mentally, without pencil and paper or a calculator.65. The answer to the problem 52 2 49 is closest to whichof the following numbers?a. 100 b. 0 c. 210066. The answer to the problem 252 2 49 is closest to whichof the following numbers?a. 100 b. 0 c. 210067. The answer to the problem 52 2 (249) is closest towhich of the following numbers?a. 100 b. 0 c. 210068. The answer to the problem 252 2 (249) is closest towhich of the following numbers?a. 100 b. 0 c. 210069. Is 2161 2 (262) closer to 2200 or 2100?70. Is 2553 2 50 closer to 2600 or 2500?
8.3 Problem Set475B Applying the Concepts [Example 12]71. The graph shows the record low temperatures for theGrand Canyon. What is the temperature difference betweenJanuary and July?72. The graph shows the lowest and highest points in theGrand Canyon and Death Valley. What is the differencebetween the lowest point in the Grand Canyon and thelowest point in Death Valley?Record Low TemperaturesTemperature (Celsius)8˚4˚0˚–4˚–8˚JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DECSource: National Park ServiceLowest and Highest PointsGrand CanyonDeath ValleyLake Mead 1,200Badwater Basin –282Point Imperial 8,803Telescope Peak 11,0490 5,000 10,000 15,000Elevation (feet)Source: National Park Service73. The highest point in Grand Canyon National Park isat Point Imperial with an elevation of 8,803 feet. Thelowest point in the park is at Lake Mead at 1,200 feet.What is the difference between the highest and thelowest points?74. Temperature On Monday the temperature reached ahigh of 288 above 0. That night it dropped to 168 below0. What is the difference between the high and the lowtemperatures for Monday?75. Tracking Inventory By definition, inventory is the totalamount of goods contained in a store or warehouse atany given time. It is helpful for store owners to knowthe number of items they have available for sale inorder to accommodate customer demand. This tableshows the beginning inventory on May 1st and tracksthe number of items bought and sold for one month.Determine the number of items in inventory at the endof the month.Date Transaction Number of Number ofunits Available Units Sold76. Profit and Loss You own a small business which providescomputer support to homeowners who wish tocreate their own in-house computer network. In additionto setting up the network you also maintain andtroubleshoot home PCs. Business gets off to a slowstart. You record a profit of $2,298 during the firstquarter of the year, a loss of $2,854 during the secondquarter, a profit of $3,057 during the third quarter, anda profit of $1,250 for the last quarter of the year. Do youend the year with a net profit or a net loss? Representthat profit or loss as a positive or negative value.May 1 Beginning Inventory 400May 3 Purchase 100May 8 Sale 700May 15 Purchase 600May 19 Purchase 200May 25 Sale 400May 27 Sale 300May 31 Ending Inventory
476<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsTuition Cost The chart shows the cost of college tuition and fees at public four-year universities. Because of tax breaks,along with federal and state grants, the actual cost per student is much less than the total cost of tuition and fees. Use theinformation in this chart to answers Questions 77 through 80.77. Find the difference in student grants in 1998 and studentgrants and tax deductions in 2008.Tuition and Fees at 4-year Public Universities78. Find the difference in actual costs in 1998 and actual costs in2008.79. Find the difference in total costs in 1998 and total costs in2008.19982008Actual cost Student grants$1,636 $1,940 $3,576Actual cost Tax deductions/ grants$2,885 $3,700 $6,585Sources: College Board80. What has increased more from 1998 to 2008, student grantsand tax deductions or actual student costs?Repeated below is the table of wind chill temperatures that we used previously. Use it for Problems 81–84.Air Temperature (°F)Wind speed 30° 25° 20° 15° 10° 5° 0° 25°10 mph 16° 10° 3° 23° 29° 215° 222° 227°15 mph 9° 2° 25° 211° 218° 225° 231° 238°20 mph 4° 23° 210° 217° 224° 231° 239° 246°25 mph 1° 27° 215° 222° 229° 236° 244° 251°30 mph 22° 210° 218° 225° 233° 241° 249° 256°81. Wind Chill If the temperature outside is 158F, what is thedifference in wind chill temperature between a 15-mileper-hourwind and a 25-mile-per-hour wind?82. Wind Chill If the temperature outside is 08F, what is thedifference in wind chill temperature between a 15-mileper-hourwind and a 25-mile-per-hour wind?83. Wind Chill Find the difference in temperature between aday in which the air temperature is 208F and the wind isblowing at 10 miles per hour and a day in which the airtemperature is 108F and the wind is blowing at 20 milesper hour.84. Wind Chill Find the difference in temperature betweena day in which the air temperature is 08F and the windis blowing at 10 miles per hour and a day in which theair temperature is 258F and the wind is blowing at 20miles per hour.
8.3 Problem Set477Use the tables below to work Problems 85–88.Record low temperatures forLake Placid, New YorkRecord high temperatures forLake Placid, New YorkMonthtemperatureMonthtemperatureJanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember2368F2308F2148F228F198F228F358F308F198F158F2118F2268FJanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember548F598F698F828F908F938F978F938F908F878F678F608F85. Temperature Difference Find the difference between therecord high temperature and the record low temperaturefor the month of December.86. Temperature Difference Find the difference between therecord high temperature and the record low temperaturefor the month of March.87. Temperature Difference Find the difference between therecord low temperatures of March and December.88. Temperature Difference Find the difference between therecord high temperatures of March and December.Getting Ready for the Next SectionPerform the indicated operations.89. 3(2)(5)90. 5(2)(4)91. 6 292. 8 293. 4 394. 3 395. 6(3 1 5)96. 2(5 1 8)97. 3(9 2 2) 1 4(7 2 2)98. 2(5 2 3) 2 7(4 2 2)99. (3 1 7)(6 2 2)100. (6 1 1)(9 2 4)Simplify each of the following.101. 2 1 3(4 1 1)105. 5 2 102. 6 1 5(2 1 3)106. 2 3 103. (6 1 2)(6 2 2)104. (7 1 1)(7 2 1)107. 2 3 ? 3 2 108. 2 3 1 3 2
478<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsMaintaining Your SkillsWrite each of the following in symbols.109. The product of 3 and 5.110. The product of 5 and 3.111. The product of 7 and x.112. The product of 2 and y.Rewrite the following using the commutative property of multiplication.113. 3(5) 5 114. 7(x) 5Rewrite the following using the associative property of multiplication.115. 5(7 ? 8) 5 116. 4(6 ? y)Apply the distributive property to each expression and then simplify the result.117. 2(3 1 4) 118. 5(6 1 7)Extending the Concepts119. Give an example that shows that subtraction is not acommutative operation.120. Why is the expression “two negatives make a positive”not correct?121. Give an example of an everyday situation that is modeledby the subtraction problem$10 2 $12 5 2$2.122. Give an example of an everyday situation that is modeledby the subtraction problem2$10 2 $12 5 2$22.In <strong>Chapter</strong> 1 we defined an arithmetic sequence as a sequence of numbers in which each number, after the first number, isobtained from the previous number by adding the same amount each time.Find the next two numbers in each arithmetic sequence below.123. 10, 5, 0, . . .124. 8, 3, 22, . . .125. 210, 26, 22, . . .126. 24, 21, 2, . . .
Introduction . . .Multiplication with Negative NumbersSuppose you buy three shares of a certain stockon Monday, and by Friday the price per share hasdropped $5. How much money have you lost? Theanswer is $15. Because it is a loss, we can expressit as 2$15. The multiplication problem below canbe used to describe the relationship among thenumbers.3 shares each loses $5 for a total of 2$1588888n3(25) 5 215From this we conclude that it is reasonable to say that the product of a positivenumber and a negative number is a negative number.A MultiplicationIn order to generalize multiplication with negative numbers, recall that we firstdefined multiplication by whole numbers to be repeated addition. That is:3 ? 5 5 5 1 5 1 5h h hMultiplication Repeated additionThis concept is very helpful when it comes to developing the rule for multiplicationproblems that involve negative numbers. For the first example we look atwhat happens when we multiply a negative number by a positive number.Example 1Multiply: 3(25)88nm88888.4ObjectivesA Multiply positive and negativenumbers.B Apply the rule for order ofoperations to expressionscontaining positive and negativenumbers.C Solve applications involvingmultiplication with positive andnegative numbers.Examples now playing atMathTV.com/booksPractice Problems1. Multiply: 2(26)Solution Writing this product as repeated addition, we have3(25) 5 (25) 1 (25) 1 (25)5 210 1 (25)5 215The result, 215, is obtained by adding the three negative 5’s.Example 2Multiply: 23(5)2. Multiply: 22(6)SolutionIn order to write this multiplication problem in terms of repeated addition,we will have to reverse the order of the two numbers. This is easily done,because multiplication is a commutative operation.23(5) 5 5(23) Commutative property5 (23) 1 (23) 1 (23) 1 (23) 1 (23) Repeated addition5 215 AdditionThe product of 23 and 5 is 215.Example 3Multiply: 23(25)Solution It is impossible to write this product in terms of repeated addition.We will find the answer to 23(25) by solving a different problem. Look at the followingproblem:23[5 1 (25)] 5 23[0] 5 03. Multiply: 22(26)Answers1. 212 2. 2128.4 Multiplication with Negative Numbers479
480<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsNoteThe discussion hereexplains why23(25) 5 15. Wewant to be able to justify everythingwe do in mathematics. Thediscussion tells why23(215) 5 15.The result is 0, because multiplying by 0 always produces 0. Now we can workthe same problem another way, and in the process find the answer to 23(25).Applying the distributive property to the same expression, we have23[5 1 (25)] 5 23(5) 1 (23)(25) Distributive property5 215 1 (?) 23(5) 5 215The question mark must be 115, because we already know that the answer tothe problem is 0, and 115 is the only number we can add to 215 to get 0. So, ourproblem is solved:23(25) 5 115Table 1 gives a summary of what we have done so far in this section.TAblE 1Original Numbers have for Example the Answer isSame signs 3(5) 5 15 PositiveDifferent signs 23(5) 5 215 NegativeDifferent signs 3(25) 5 215 NegativeSame signs 23(25) 5 15 PositiveFrom the examples we have done so far in this section and their summaries inTable 1, we write the following rule for multiplication of positive and negativenumbers:ruleTo multiply any two numbers, we multiply their absolute values.1. The answer is positive if both the original numbers have the same sign.That is, the product of two numbers with the same sign is positive.2. The answer is negative if the original two numbers have different signs.The product of two numbers with different signs is negative.This rule should be memorized. By the time you have finished reading this sectionand working the problems at the end of the section, you should be fast andaccurate at multiplication with positive and negative numbers.Multiply.4. 3(2)5. 23(22)6. 3(22)7. 23(2)8. 8(29)9. 26(24)10. 25(2)(24)ExAmplE 4ExAmplE 5ExAmplE 6ExAmplE 7ExAmplE 8ExAmplE 92(4) 5 8 like signs; positive answer22(24) 5 8 like signs; positive answer2(24) 5 28 unlike signs; negative answer22(4) 5 28 unlike signs; negative answer7(26) 5 242 unlike signs; negative answer25(28) 5 40 like signs; positive answerAnswers3. 12 4. 6 5. 6 6. 26 7. 268. 272 9. 24 10. 40ExAmplE 1023(2)(25) 5 26(25) Multiply 23 and 2 to get 265 30
8.4 Multiplication with Negative Numbers481Example 11Use the definition of exponents to expand each expression.Then simplify by multiplying.a. (26) 2 5 (26)(26) Definition of exponents5 36 Multiplyb. 26 2 5 26 ? 6 Definition of exponents5 236 Multiplyc. (24) 3 5 (24)(24)(24) Definition of exponents5 264 Multiplyd. 24 3 5 24 ? 4 ? 4 Definition of exponents5 264 Multiply11. Use the definition of exponentsto expand each expression.Then simplify by multiplying.a. (28) 2b. 28 2c. (23) 3d. 23 3In Example 11, the base is a negative number in Parts a and c, but not in Parts band d. We know this is true because of the use of parentheses.BOrder of OperationsExample 12Simplify: 26[3 1 (25)]12. Simplify: 22[5 1 (28)]SolutionWe begin inside the brackets and work our way out:26[3 1 (25)] 5 26[22]5 12Example 13Simplify: 24 1 5(26 1 2)13. Simplify: 23 1 4(27 1 3)SolutionSimplifying inside the parentheses first, we have24 1 5(26 1 2) 5 24 1 5(24) Simplify inside parentheses5 24 1 (220) Multiply5 224 AddExample 14Simplify: 22(7) 1 3(26)14. Simplify: 23(5) 1 4(24)SolutionMultiplying left to right before we add gives us22(7) 1 3(26) 5 214 1 (218)5 232Example 15Simplify: 23(2 2 9) 1 4(27 2 2)15. Simplify: 22(3 2 5) 2 7(22 2 4)SolutionWe begin by subtracting inside the parentheses:23(2 2 9) 1 4(27 2 2) 5 23(27) 1 4(29)5 21 1 (236)5 215Example 16Simplify: (23 2 7)(2 2 6)16. Simplify: (26 2 1)(4 2 9)SolutionAgain, we begin by simplifying inside the parentheses:(23 2 7)(2 2 6) 5 (210)(24)5 40Answers11. a. 64 b. 264 c. 227 d. 22712. 6 13. 219 14. 231 15. 4616. 35
482<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsUsing TechnologyCalculator NoteHere is how we work the problem shown in Example 16 on a calculator.(The 3 key on the first line may, or may not, be necessary. Try your calculatorwithout it and see.)Scientific Calculator: ( 3 1/2 2 7 ) 3 ( 2 2 6 ) 5Graphing Calculator: ( (2) 3 2 7 ) ( 2 2 6 ) ENTHere are a few more multiplication problems involving fractions and decimals.17. } 3 4 } 1 2}4 7 } 2Example 171 2 } 3 21 2 3 } 5 2 5 2 6 } 155 2 2 } 5The rule for multiplication alsoholds for fractions18.1 2}5 6 } 21 2} 219. (23)(6.7)9 }02Example 181 2 7 } 8 21 2 5 } 14 2 5 35} 1125Example 19(25)(3.4) 5 217.0 The rule for multiplication also holds fordecimals5} 1620. (20.6)(20.5)Example 20(20.4)(20.8) 5 0.32Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Write the multiplication problem 3(25) as an addition problem.2. Write the multiplication problem 2(4) as an addition problem.3. If two numbers have the same sign, then their product will have whatsign?4. If two numbers have different signs, then their product will have whatsign?Answers17. 2} 3 7 } 18. }3 } 19. 220.1820. 0.30
8.4 Problem Set483Problem Set 8.4A Find each of the following products. (Multiply.) [Examples 1–10]1. 7(28)2. 23(5)3. 26(10)4. 4(28)5. 27(28)6. 24(27)7. 29(29)8. 26(23)9. 22.1(4.3)10. 26.8(5.7)11. 2 4 } 5 1 2 15 } 28 212. 2 8 } 9 1 2 27 } 32 213. 212 12 }3214. 218 15 }6215. 3(22)(4)16. 5(21)(3)17. 24(3)(22)18. 24(5)(26)19. 21(22)(23)20. 22(23)(24)Use the definition of exponents to expand each of the following expressions. Then multiply according to the rule for multiplication.[Example 11]21. a. (24) 2b. 24 2 22. a. (25) 2b. 25 2 23. a. (25) 3b. 25 3 24. a. (24) 3b. 24 3 25. a. (22) 426. a. (21) 4b. 22 4 b. 21 4Complete the following tables. Remember, if x 5 25, then x 2 5 (25) 2 5 25. [Example 11]27.Number Square28.x x 22322210123Number Cubex x 3232221012329.First Second Their30.Number Number Productx y xy6 26 16 06 216 22First Second TheirNumber Number Producta b ab25 325 225 125 025 2125 2225 23
484<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsB Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify eachof the following expressions. [Examples 12–16]31. 4(23 1 2)32. 7(26 1 3)33. 210(22 2 3)34. 25(26 2 2)35. 23 1 2(5 2 3)36. 27 1 3(6 2 2)37. 27 1 2[25 2 9]38. 28 1 3[24 2 1]39. 2(25) 1 3(24)40. 6(21) 1 2(27)41. 3(22)4 1 3(22)42. 2(21)(23) 1 4(26)43. (8 2 3)(2 2 7)44. (9 2 3)(2 2 6)45. (2 2 5)(3 2 6)46. (3 2 7)(2 2 8)47. 3(5 2 8) 1 4(6 2 7)48. 2(3 2 7) 1 3(5 2 6)49. 22(8 2 10) 1 3(4 2 9)50. 23(6 2 9) 1 2(3 2 8)51. 23(4 2 7) 2 2(23 2 2)52. 25(22 2 8) 2 4(6 2 10)53. 3(22)(6 2 7)54. 4(23)(2 2 5)55. Find the product of 23, 22, and 21.56. Find the product of 27, 21, and 0.57. What number do you multiply by 23 to get 12?58. What number do you multiply by 27 to get 221?59. Subtract 23 from the product of 25 and 4.60. Subtract 5 from the product of 28 and 1.Work Problems 61–68 mentally, without pencil and paper or a calculator.61. The product 232(2522) is closest to which of the followingnumbers?a. 15,000 b. 2500 c. 21,500 d. 215,00062. The product 32(2522) is closest to which of the followingnumbers?a. 15,000 b. 2500 c. 21,500 d. 215,00063. The product 247(470) is closest to which of the followingnumbers?a. 25,000 b. 420 c. 22,500 d. 225,00064. The product 247(2470) is closest to which of the followingnumbers?a. 25,000 b. 420 c. 22,500 d. 225,00065. The product 2222(2987) is closest to which of the followingnumbers?a. 200,000 b. 800 c. 2800 d. 21,20066. The sum 2222 1 (2987) is closest to which of the followingnumbers?a. 200,000 b. 800 c. 2800 d. 21,20067. The difference 2222 2 (2987) is closest to which of thefollowing numbers?a. 200,000 b. 800 c. 2800 d. 21,20068. The difference 2222 2 987 is closest to which of thefollowing numbers?a. 200,000 b. 800 c. 2800 d. 21,200
8.4 Problem Set485CApplying the Concepts69. The chart shows the record low temperatures for GrandCanyon National Park, by month. Write the record lowtemperature for March.70. The chart shows the cities with the highest annual insurancerates.Record Low TemperaturesTemperature (Celsius)8˚4˚0˚–4˚–8˚JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DECPriciest Cities for Auto InsuranceDetroit$5,894Philadelphia$4,440Newark, N.J.$3,977Los Angeles$3,430New York City$3,3030 $1000 $2000 $3000 $4000 $5000 $6000Source: Runzheimer InternationalSource: National Park Servicea. What is the monthly payment for a driver in Philadelphia?b. Use negative numbers to write an expression for thecost of three months of auto insurance for a driverliving in Philadelphia.71. Temperature Change A hot-air balloon is rising to itscruising altitude. Suppose the air temperature aroundthe balloon drops 4 degrees each time the balloon rises1,000 feet. What is the net change in air temperaturearound the balloon as it rises from 2,000 feet to 6,000feet?72. Temperature Change A small airplane is rising to itscruising altitude. Suppose the air temperature aroundthe plane drops 4 degrees each time the plane increasesits altitude by 1,000 feet. What is the netchange in air temperature around the plane as it risesfrom 5,000 feet to 12,000 feet?12,000 ft6,000 ft2,000 ft5,000 ft73. Expense Account A business woman has a travel expenseaccount of $1,000. If she spends $75 a week for 8weeks what will the balance of her expense account beat the end of this time.74. Gas Prices Two local gas stations offer different pricesfor a gallon of regular gasoline. The Exxon Mobil stationis currently selling their gas at $3.99 per gallon.The Getty station is currently selling their gas for $3.85per gallon. Represent the net savings to you on a purchaseof 15 gallons of regular gas if you buy gas fromthe Getty gas station.
486<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsGetting Ready for the Next SectionPerform the indicated operations.75. 35 4 576. 32 4 477. 20 _4 78. 30 } 579. 12 2 1780. 7 2 1181. (6 ? 3) 4 282. (8 ? 5) 4 483. 80 4 10 4 284. 80 4 2 4 1085. 15 1 5(4) 4 1086. [20 1 6(2)] 4 (11 2 7)87. 4(10 2 ) 1 20 4 4 88. 3(4 2 ) 1 10 4 5Maintaining Your SkillsWrite each of the following statements in symbols.89. The quotient of 12 and 6 90. The quotient of x and 5Rewrite each of the following multiplication problems as an equivalent division problem.91. 2(3) 5 6 92. 5 ? 4 5 20Rewrite each of the following division problems as an equivalent multiplication problem.93. 10 4 5 5 2 94. } 6 3} 5 79Divide.95. 4,984 4 56 96. 4,994 4 56Extending the ConceptsIn <strong>Chapter</strong> 1 we defined a geometric sequence to be a sequence of numbers in which each number, after the first number,is obtained from the previous number by multiplying by the same amount each time.Find the next two terms in each of the following geometric sequences.97. 2, 26, 18, . . . 98. 1, 24, 16, . . . 99. 22, 6, 218, . . . 100. 21, 4, 216, . . .Simplify each of the following according to the rule for order of operations.101. 5(22) 2 2 3(22) 3102. 8(21) 3 2 6(23) 2103. 7 2 3(4 2 8)104. 6 2 2(9 2 11)105. 5 2 2[3 2 4(6 2 8)]106. 7 2 4[6 2 3(2 2 9)]
Introduction . . .Division with Negative NumbersSuppose four friends invest equal amounts ofmoney in a moving truck to start a small business.After 2 years the truck has dropped $10,000in value. If we represent this change with thenumber 2$10,000, then the loss to each of thefour partners can be found with division:(2$10,000) 4 4 5 2$2,500MOVERS$10,000 drop in 2 yearsFrom this example it seems reasonable to assume that a negative number dividedby a positive number will give a negative answer.To cover all the possible situations we can encounter with division of negativenumbers, we use the relationship between multiplication and division. If we let nbe the answer to the problem 12 4 (22), then we know that12 4 (22) 5 n and 22(n) 5 12From our work with multiplication, we know that n must be 26 in the multiplicationproblem above, because 26 is the only number we can multiply 22 by to get12. Because of the relationship between the two problems above, it must be truethat 12 divided by 22 is 26.The following pairs of problems show more quotients of positive and negativenumbers. In each case the multiplication problem on the right justifies the answerto the division problem on the left.6 4 3 5 2 because 3(2) 5 66 4 (23) 5 22 because 23(22) 5 626 4 3 5 22 because 3(22) 5 2626 4 (23) 5 2 because 23(2) 5 26The results given above can be used to write the rule for division with negativenumbers.8.5ObjectivesA State the place value for numbers instandard notation.B Write a whole number in expandedform.C Write a number in words.D Write a number from words.Examples now playing atMathTV.com/booksADivisionRuleTo divide two numbers, we divide their absolute values.1. The answer is positive if both the original numbers have the same sign.That is, the quotient of two numbers with the same signs is positive.2. The answer is negative if the original two numbers have different signs.That is, the quotient of two numbers with different signs is negative.Example 1212 4 4 5 23 Unlike signs, negative answerExample 212 4 (24) 5 23 Unlike signs; negative answerExample 3212 4 (24) 5 3 Like signs; positive answerExample 4 12}} 5 23 Unlike signs; negative answer2 4Example 5} 2 20} 5 5 Like signs; positive answer248.5 Division with Negative NumbersPractice ProblemsDivide.1. 28 4 22. 8 4 (22)3. 28 4 (22)204. }}2 55. } 2 30}25Answers1. 24 2. 24 3. 4 4. 24 5. 6487
488<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsFrom the examples we have done so far, we can make the following generalizationabout quotients that contain negative signs:If a and b are numbers and b is not equal to 0, then} 2 a a a 2aa} 5 }} 5 2} } and }2 } 5 } }b 2 b b b bB Order of OperationsThe last examples in this section involve more than one operation. We use therules developed previously in this chapter and the rule for order of operations tosimplify each.6. Simplify: } 8( 25)24Example 6Simplify: } 6( 223)2Solution We begin by multiplying 6 and 23:} 6( 23)5 } 2 18}2222Multiplication; 6(23) 5 2185 9 Like signs; positive answer7. Simplify: } 220 1 6(22)7 2 11Example 7Simplify: } 21 51Solution1 5(24)2 2 17Simplifying above and below the fraction bar, we have} 21 5 1 5(24) 215 1 (220) 2355 } 5 } } 5 7122 1725 258. Simplify: 23(4 2 ) 1 10 4 (25)Example 8Simplify: 24(10 2 ) 1 20 4 (24)SolutionApplying the rule for order of operations, we have24(10 2 ) 1 20 4 (24) 5 24(100) 1 20 4 (24) Exponents first5 2400 1 (25) Multiply and divide5 2405 Add9. Simplify: 280 4 2 4 10Example 9Simplify: 280 4 10 4 2Solution In a situation like this, the rule for order of operations states that weare to divide left to right.280 4 10 4 2 5 28 4 2 Divide 280 by 105 24Answers6. 10 7. 8 8. 250 9. 24Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Write a multiplication problem that is equivalent to the division problem212 4 4 5 23.2. Write a multiplication problem that is equivalent to the division problem212 4 (24) 5 3.3. If two numbers have the same sign, then their quotient will have whatsign?4. Dividing a negative number by 0 always results in what kind of expression?
8.5 Problem Set489Problem Set 8.5A Find each of the following quotients. (Divide.) [Examples 1–5]1. 215 4 52. 15 4 (23)3. 20 4 (24)4. 220 4 45. 230 4 (210)6. 250 4 (225)7. } 2 14}278. } 2 18}26129. }}2 31210. }}2 411. 222 4 1112. 235 4 7013. }}2 3014. }}2 515. 125 4 (225)16. 2144 4 (29)Complete the following tables.17.First Second The QuotientNumber Number of a and b18.First Second The QuotientNumber Number of a and ba b } ba }a b } ba }100 25100 210100 225100 25024 2424 2324 2224 2119.First Second The QuotientNumber Number of a and b20.First Second The QuotientNumber Number of a and ba b } ba }a b } ba }2100 252100 5100 25100 5224 22224 24224 26224 2821. Find the quotient of 225 and 5.22. Find the quotient of 238 and 219.23. What number do you divide by 25 to get 27?24. What number do you divide by 6 to get 27?25. Subtract 23 from the quotient of 27 and 9.26. Subtract 27 from the quotient of 272 and 29.
490<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsB Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the followingexpressions as much as possible. [Examples 6–9]27. } 4 (27)22831. } 2 (23)6 2 335. } 2(23 ) 1 102428. } 6 (23)21832. } 2 (23)3 2 636. } 7(2 2)2 621029. } 23 (210)30. } 24 (212)252633. } 4 2 834. } 9 2 58 2 45 2 937. }}2 1 3(26)38. } 3 1 9(21)4 2 12 5 2 739.6(27) 1 3(22)}}20 2 440.9(28) 1 5(21)}}12 2 141. 3(27)(24) }} 6(22)42. 22(4)(28) }} (22)(22)43. (25) 2 1 20 4 444. 6 2 1 36 4 945. 100 4 (25) 246. 400 4 (24) 247. 2100 4 10 4 248. 2500 4 50 4 1049. 2100 4 (10 4 2)50. 2500 4 (50 4 10)51. (2100 4 10) 4 2 52. (2500 4 50) 4 10EstimatingWork Problems 53–60 mentally, without pencil and paper or a calculator.53. Is 397 4 (2401) closer to 1 or 21?54. Is 2751 4 (2749) closer to 1 or 21?55. The quotient 2121 4 27 is closest to which of the followingnumbers?a. 2150 b. 2100 c. 24 d. 656. The quotient 1,000 4 (2337) is closest to which of thefollowing numbers?a. 663 b. 23 c. 230 d. 266357. Which number is closest to the sum 2151 1 (249)?a. 2200 b. 2100 c. 3 d. 7,50058. Which number is closest to 2151 2 (249)?a. 2200 b. 2100 c. 3 d. 7,50059. Which number is closest to the product 2151(249)?a. 2200 b. 2100 c. 3 d. 7,50060. Which number is closest to the quotient 2151 4 (249)?a. 2200 b. 2100 c. 3 d. 7,500
8.5 Problem Set491CApplying the Concepts61. The chart shows the most expensive cities to live in.Expenses can also be written as negative numbers.Find the monthly cost to live in Los Angeles. Use negativenumbers.62. The chart shows the cities with the most expensiveauto insurance. Because insurance is an expense, it canbe written as a negative number. What is the monthlycost of insurance in New York City? Use negative numbersand round to the nearest cent.Priciest Cities to Inhabit in the U.S.Priciest Cities for Auto InsuranceManhattanSan FranciscoLos AngelesSan JoseWashington, D.C.S146,060$133,887$117,726$108,506$102,589Annual Cost (dollars)Detroit$5,894Philadelphia$4,440Newark, N.J.$3,977Los Angeles$3,430New York City$3,3030 $1000 $2000 $3000 $4000 $5000 $6000Source: RunzheimerSource: Runzheimer International63. Temperature Line Graph The table below gives the low temperature for each day of one week in White Bear Lake, Minnesota.Use the diagram in the figure to draw a line graph of the information in the table.Low temperatures in White BearLake, MinnesotaDayMondayTuesdayWednesdayThursdayFridaySaturdaySundaytemperature10 8F8 8F25 8F23 8F28 8F5 8F7 8FTemperature (Fahrenheit)10°8°6°4°2°0°-2°-4°-6°-8°-10°Mon Tue Wed Thu Fri Sat Sun64. Temperature Line Graph The table below gives the low temperature for each day of one week in Fairbanks, Alaska. Usethe diagram in the figure to draw a line graph of the information in the table.Low temperatures in Fairbanks,AlaskaDayMondayTuesdayWednesdayThursdayFridaySaturdaySundaytemperature226 8F25 8F9 8F12 8F3 8F215 8F220 8FTemperature (Fahrenheit)30°25°20°15°10°5°0°-5°-10°-15°-20°-25°-30°Mon Tue Wed Thu Fri Sat Sun
492<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsGetting Ready for the Next SectionThe problems below review some of the properties of addition and multiplication we covered in <strong>Chapter</strong> 1.Rewrite each expression using the commutative property of addition or multiplication.65. 3 1 x 66. 4yRewrite each expression using the associative property of addition or multiplication.67. 5 1 (7 1 a) 68. (x 1 4) 1 6 69. 3(4y) 70. (3y)8Apply the distributive property to each expression.71. 5(3 1 7) 72. 8(4 1 2)Simplify.73. 6 274. 12 275. 4 376. 5 277. 2(100) 1 2(75)78. 2(100) 1 2(53)79. 100(75)80. 100(53)Maintaining Your SkillsThe problems below review addition, subtraction, multiplication, and division of positive and negative numbers, as coveredin this chapter.Perform the indicated operations.81. 8 1 (24)82. 28 1 483. 28 1 (24)84. 28 2 485. 8 2 (24)86. 28 2 (24)87. 8(24)88. 28(4)89. 28(24)90. 8 4 (24)91. 28 4 492. 28 4 (24)Extending the ConceptsFind the next term in each sequence below.93. 32, 216, 8, . . . 94. 243, 281, 27, . . . 95. 232, 16, 28, . . . 96. 2243, 81, 227, . . .Simplify each of the following expressions.97. } 6 2 3(2 2 11)6 2 3(2 1 11)98. } 8 1 4(3 2 5)8 2 4(3 1 5)99.6 2 (3 2 4) 2 3}}1 2 2 2 3100.7 2 (3 2 6) 2 4}}21 2 2 2 3
Introduction . . .Simplifying Algebraic ExpressionsThe woodcut shown here depicts QueenDido of Carthage around 900 b.c., havingan ox hide cut into small strips that willbe tied together to make a long rope. Therope will be used to enclose her territory.The question, which has become knownas the Queen Dido problem, is: what shapewill enclose the largest territory?To translate the problem into somethingwe are more familiar with, suppose we have 24 yards of fencing that we are touse to build a rectangular dog run. If we want the dog run to have the largestarea possible then we want the rectangle, with perimeter 24 yards, that enclosesthe largest area. The diagram below shows six dog runs, each of which has a perimeterof 24 yards. Notice how the length decreases as the width increases.8.6ObjectivesA Simplify expressions by using theassociative property.B Apply the distributive property toexpressions containing numbersand variables.C Use the distributive property tocombine similar terms.D Use the formulas for areaand perimeter of squares andrectangles.Examples now playing atMathTV.com/booksDog Runs with Perimeter 24 yards111098761 23456Since area is length times width, we can build a table and a line graph thatshow how the area changes as we change the width of the dog run.Area enclosed by rectangle ofperimeter 24 yards40Area Enclosed by Fixed PerimeterWidth(Yards)Area(Square Yards)1 112 203 274 32Area (square yards)363228242016125 356 36840 1 2 3 4 5 6 7Width (yards)8.6 Simplifying Algebraic Expressions493
494<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsNoteAn algebraic expressiondoes not containan equal signIn this section we want to simplify expressions containing variables—that is,algebraic expressions. An algebraic expression is a combination of constants andvariables joined by arithmetic operations such as addition, subtraction, multiplicationand division.Ausing the Associative PropertyTo begin let’s review how we use the associative properties for addition and multiplicationto simplify expressions.Consider the expression 4(5x). We can apply the associative property of multiplicationto this expression to change the grouping so that the 4 and the 5 aregrouped together, instead of the 5 and the x. Here’s how it looks:4(5x) 5 (4 ?5)x Associative property5 20xMultiplication: 4 ∙ 5 = 20prACtiCE prOBlEmsMultiply.1. 5(7a)We have simplified the expression to 20x, which in most cases in algebra will beeasier to work with than the original expression.Here are some more examples.ExAmplE 17(3a) 5 (7 ? 3)a Associative property5 21a 7 times 3 is 212. 23(9x)ExAmplE 222(5x2(5x) 5 (22 ? 5)x Associative property5 210xxThe product of −2 and 5 is −103. 5(28y)ExAmplE 33(24y) 5 [3(24)] y Associative property5 212y3 times −4 is −12We can use the associative property of addition to simplify expressions also.Simplify.4. 6 1 (9 1 x)ExAmplE 43 1 (8 1 x) 5 (3 1 8) 1 x Associative property5 11 1 x The sum of 3 and 8 is 115. (3x1 7) 1 4ExAmplE 5(2x 1 5) 1 10 5 2x 1 (5 1 10) Associative property5 2x 1 15 AdditionB using the Distributive PropertyIn <strong>Chapter</strong> 1 we introduced the distributive property. In symbols it looks like this:a(b 1 c) 5 ab 1 acBecause subtraction is defined as addition of the opposite, the distributive propertyholds for subtraction as well as addition. That is,a(b 2 c) 5 ab 2 acApply the distributive property.6. 6(x1 4)Answers1. 35a 2. 227x27xx 3. 240yy4. 15 1 x 5. 3x1 11 6. 6x1 24We say that multiplication distributes over addition and subtraction. Here aresome examples that review how the distributive property is applied to expressionsthat contain variables.ExAmplE 6 4(x4( 1 5) 5 4(x) 1 4(5) Distributive property5 4x1 20 Multiplication
8.6 Simplifying Algebraic Expressions495Example 72(a 2 3) 5 2(a) 2 2(3) Distributive property5 2a 2 6 Multiplication7. 7(a 2 5)In Examples 1–3 we simplified expressions such as 4(5x) by using the associativeproperty. Here are some examples that use a combination of the associativeproperty and the distributive property.Example 84(5x 1 3) 5 4(5x) 1 4(3) Distributive property5 (4 ? 5)x 1 4(3) Associative property5 20x 1 12 Multiplication8. 6(4x 1 5)Example 97(3a 2 6) 5 7(3a) 2 7(6) Distributive property5 21a 2 42 Associative property andmultiplication9. 3(8a 2 4)Example 105(2x 1 3y) 5 5(2x) 1 5(3y) Distributive property5 10x 1 15y Associative property andmultiplication10. 8(3x 1 4y)We can also use the distributive property to simplify expressions like 4x 1 3x.Because multiplication is a commutative operation, we can also rewrite the distributiveproperty like this:b ? a 1 c ? a 5 (b 1 c)aApplying the distributive property in this form to the expression 4x 1 3x, wehave4x 1 3x 5 (4 1 3)x Distributive property5 7x AdditionCSimilar TermsExpressions like 4x and 3x are called similar terms because the variable parts arethe same. Some other examples of similar terms are 5y and 26y and the terms 7a,213a, and } 3 }a. To simplify an algebraic expression (an expression that involves4both numbers and variables), we combine similar terms by applying the distributiveproperty. Table 1 shows several pairs of similar terms and how they can becombined using the distributive property.Table 1Original Apply Distributive simplifiedExpression property Expression4x 1 3x 5 (4 1 3)x 5 7x7a 1 a 5 (7 1 1)a 5 8a25x 1 7x 5 (25 1 7)x 5 2x8y 2 y 5 (8 2 1)y 5 7y24a 2 2a 5 (24 2 2)a 5 26a3x 2 7x 5 (3 2 7)x 5 24xAs you can see from the table, the distributive property can be applied to anycombination of positive and negative terms so long as they are similar terms.Answers7. 7a 2 35 8. 24x 1 309. 24a 2 12 10. 24x 1 32y
496<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsDAlgebraic Expressions Representing Area and PerimeterBelow are a square with a side of length s and a rectangle with a length of l and awidth of w. The table that follows the figures gives the formulas for the area andperimeter of each.SquareRectanglewslSquareRectangleArea A s 2 lwPerimeter P 4s 2l 1 2w11. Find the area and perimeterof a square if its side is 12 feetlong.Example 11Find the area and perimeter of a square with a side 6inches long.Solution Substituting 6 for s in the formulas for area and perimeter of asquare, we haveArea 5 A 5 s 2 5 6 2 5 36 square inchesPerimeter 5 P 5 4s 5 4(6) 5 24 inches12. A football field is 100 yardslong and approximately 53yards wide. Find the area andperimeter.Example 12A soccer field is 100 yards long and 75 yards wide. Find thearea and perimeter.100 yd75 ydSolution Substituting 100 for l and 75 for w in the formulas for area and perimeterof a rectangle, we haveArea 5 A 5 lw 5 100(75) 5 7,500 square yardsPerimeter 5 P 5 2l 1 2w 5 2(100) 1 2(75) 5 200 1 150 5 350 yardsGetting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.Answers11. A 5 144 sq ft, P 5 48 ft12. A 5 5,300 sq yd, P 5 306 yd1. Without actually multiplying, how do you apply the associative propertyto the expression 4(5x)?2. What are similar terms?3. Explain why 2a 2 a is a, rather than 1.4. Can two rectangles with the same perimeter have different areas? Explainyour answer.
8.6 Problem Set497Problem Set 8.6A Apply the associative property to each expression, and then simplify the result. [Examples 1–5]1. 5(4a)2. 8(9a)3. 6(8a)4. 3(2a)5. 26(3x)6. 22(7x)7. 23(9x)8. 24(6x)9. 5(22y)10. 3(28y)11. 6(210y)12. 5(25y)13. 2 1 (3 1 x)14. 9 1 (6 1 x)15. 5 1 (8 1 x)16. 3 1 (9 1 x)17. 4 1 (6 1 y)18. 2 1 (8 1 y)19. 7 1 (1 1 y)20. 4 1 (1 1 y)21. (5x 1 2) 1 422. (8x 1 3) 1 1023. (6y 1 4) 1 324. (3y 1 7) 1 825. (12a 1 2) 1 1926. (6a 1 3) 1 1427. (7x 1 8) 1 2028. (14x 1 3) 1 15B Apply the distributive property to each expression, and then simplify. [Examples 6–10]29. 7(x 1 5)30. 8(x 1 3)31. 6(a 2 7)32. 4(a 2 9)33. 2(x 2 y)34. 5(x 2 a)35. 4(5 1 x)36. 8(3 1 x)37. 3(2x 1 5)38. 8(5x 1 4)39. 6(3a 1 1)40. 4(8a 1 3)41. 2(6x 2 3y)42. 7(5x 2 y)43. 5(7 2 4y)44. 8(6 2 3y)C Use the distributive property to combine similar terms. (See Table 1.)45. 3x 1 5x46. 7x 1 8x47. 3a 1 a48. 8a 1 a49. 22x 1 6x50. 23x 1 9x51. 6y 2 y52. 3y 2 y53. 28a 2 2a54. 27a 2 5a55. 4x 2 9x56. 5x 2 11x
498<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsApplying the Concepts57. A farmer is replacing several turbines on his windmills. He plansto replace x turbines, and he is going to get $300 off each turbinehe buys. Also, he’ll get a $250 rebate on his entire purchase.Write an expression that describes this situation and then simplify.58. A homeowner is replacing 4 solar modules. She is going to receivea discount of some amount x off each module and a $350mail-in rebate. Write an expression that describes this situationand then simplify.Solar Versus Wind Energy CostsEquipment Cost:Modules $6200Fixed Rack $1570Charge Controller $971Cable $440TOTAL $9181Equipment Cost:Turbine $3300Tower $3000Cable $715TOTAL $7015Source: a Limited 2006DArea and Perimeter Find the area and perimeter of each square if the length of each side is as given below.[Example 11]59. s 5 6 feet 60. s 5 14 yards 61. s 5 9 inches 62. s 5 15 metersD Area and Perimeter Find the area and perimeter for a rectangle if the length and width are as given below. [Example 12]63. l 5 20 inches, w 5 10 inches64. l 5 40 yards, w 5 20 yards65. l 5 25 feet, w 5 12 feet66. l 5 210 meters, w 5 120 metersTemperature Scales In the metric system, the scale we use to measure temperature is the Celsius scale. On this scale waterboils at 100 degrees and freezes at 0 degrees. When we write 100 degrees measured on the Celsius scale, we use the notation100°C, which is read “100 degrees Celsius.” If we know the temperature in degrees Fahrenheit, we can convert todegrees Celsius by using the formulaC 55(F 2 32)}} 9where F is the temperature in degrees Fahrenheit. Use this formula to find the temperature in degrees Celsius for each ofthe following Fahrenheit temperatures.67. 68°F 68. 59°F 69. 41°F 70. 23°F 71. 14°F 72. 32°F
<strong>Chapter</strong> 8 SummaryAbsolute Value [8.1]The absolute value of a number is its distance from 0 on the number line. It is thenumerical part of a number. The absolute value of a number is never negative.Examples1. u3u 5 3 and u23u 5 3Opposites [8.1]Two numbers are called opposites if they are the same distance from 0 on thenumber line but in opposite directions from 0. The opposite of a positive number isa negative number, and the opposite of a negative number is a positive number.2. 2(5) 5 25 and 2(25) 5 5Addition of Positive and Negative Numbers [8.2]1. To add two numbers with the same sign: Simply add absolute values and usethe common sign. If both numbers are positive, the answer is positive. If bothnumbers are negative, the answer is negative.2. To add two numbers with different signs: Subtract the smaller absolute valuefrom the larger absolute value. The answer has the same sign as the numberwith the larger absolute value.3. 3 1 5 5 823 1 (25) 5 285 1 (23) 5 225 1 3 5 22Subtraction [8.3]Subtracting a number is equivalent to adding its opposite. If a and b representnumbers, then subtraction is defined in terms of addition as follows:a 2 b 5 a 1 (2b)h hSubtraction Addition of the opposite4. 3 2 5 5 3 1 (25) 5 2223 2 5 5 23 1 (25) 5 283 2 (25) 5 3 1 5 5 823 2 (25) 5 23 1 5 5 2Multiplication with Positive and Negative Numbers [8.4]To multiply two numbers, multiply their absolute values.1. The answer is positive if both numbers have the same sign.2. The answer is negative if the numbers have different signs.5. 3(5) 5 153(25) 5 21523(5) 5 21523(25) 5 15Division [8.5]The rule for assigning the correct sign to the answer in a division problem is thesame as the rule for multiplication. That is, like signs give a positive answer, andunlike signs give a negative answer.6. 12 _4 5 3−125 23_4_ 125 23−4_ −12−4 5 3<strong>Chapter</strong> 8Summary499
500<strong>Chapter</strong> 8 Real Numbers and Algebraic ExpressionsSimplifying Expressions [8.6]7. Simplify.a. 22(5x) 5 (22 ? 5)x 5 210xb. 4(2a 2 8) 5 4(2a) 2 4(8)5 8a 2 32We simplify algebraic expressions by applying the commutative, associative, anddistributive properties.Combining Similar Terms [8.6]8. Combine similar terms.a. 5x 1 7x 5 (5 1 7)x 5 12xb. 2y 2 8y 5 (2 2 8)y 5 26yWe combine similar terms by applying the distributive property.
<strong>Chapter</strong> 8 ReviewGive the opposite of each number. [8.1]1. 17 2. 232 3. 24.6 4. } 3 5 }For each pair of numbers, name the smaller number. [8.1]5. 6; 26 6. 28; 23 7. u23u; 2 8. u24u; u6uSimplify each expression. [8.1]9. 2(24) 10. 2u24u 11. u26u 12. u19uPerform the indicated operations. [8.2, 8.3, 8.4, 8.5]13. 5 1 (27)17. 7 2 9 2 4 2 621. 5(24)4825. }}2 1614. 23 1 818. 27 2 5 2 2 2 322. 24(23)26. } 220515. 2345 1 (2626)19. 4 2 (23)23. (56)(231)27. } 2142716. 223 1 5820. 30 2 4224. (20)(24)28. } 2255Simplify the following expressions as much as possible. [8.2, 8.3, 8.4, 8.5]29. (26) 230.1 2}3 4 } 2 231. (22) 332. (20.2) 433. 7 1 4(6 2 9)34. (23)(24) 1 2(25) 35. (7 2 3)(7 2 9)36. 3(26) 1 8(2 2 5)8 2 424 1 2(25)8(22) 1 5(24)37. } 38. }}39. }}40. } 22(5 ) 1 4(23)28 1 4 6 2 4 12 2 3 102 8<strong>Chapter</strong> 8Review501
502<strong>Chapter</strong> 8 Real Numbers and Algebraic Expressions41. Give the sum of 219 and 223. [8.2]42. Give the sum of 278 and 251. [8.2]43. Find the difference of 26 and 5. [8.3]44. Subtract 28 from 210. [8.3]45. What is the product of 29 and 3? [8.4]46. What is 23 times the sum of 29 and 24? [8.2, 8.4]47. Divide the product of 8 and 24 by 216. [8.4, 8.5]48. Give the quotient of 238 and 2. [8.5]Indicate whether each statement is True or False. [8.2, 8.3, 8.4, 8.5]49. } 2 10} 5 22 50. 10 2 (25) 5 15 51. 2(23) 5 23 1 (23) 52. 26 2 (22) 5 28 53. 3 2 5 5 5 2 32554. Reaction Distance The table below shows how many feet your car will travel from the time you decide you want tostop to the time it takes you to hit the brake pedal. Use the template to construct a line graph of the information in thetable. [8.1]100Speed (mi/hr)REACTION DISTANCESDistance (ft)0 010 1120 2230 3340 4450 5560 6670 7780 88Distance (ft)908070605040302010010 20 30 40 50 60 70 80 90Speed (mph)55. Gambling A gambler wins $58 Saturday night and thenloses $86 on Sunday. Use positive and negative numbersto describe this situation. Then give the gambler’snet loss or gain as a positive or negative number. [8.2]56. Name two numbers that are 7 units from 28 on thenumber line. [8.1]57. Temperature On Wednesday, the temperature reachesa high of 17° above 0 and a low of 7° below 0. What isthe difference between the high and low temperaturesfor Wednesday? [8.3]58. If the difference between two numbers is 23, and oneof the numbers is 5, what is the other number? [8.3]Use the associative properties to simplify each expression. [8.6]59. (3x 1 4) 1 8 60. 8(3x) 61. 23(7a) 62. 6(25y)Apply the distributive property and then simplify if possible. [8.6]63. 4(x 1 3) 64. 2(x 2 5) 65. 7(3y 2 8) 66. 3(2a 1 5b)Combine similar terms. [8.6]67. 7x 2 4x 68. 28a 1 10a 69. 5y 2 y 70. 12x 1 4x
<strong>Chapter</strong> 8 Cumulative ReviewSimplify.1.3} 51 2 } 72.7} 2 } 5 3. 613 2 297 4. 6 6 1 3 } 1 1 } 1 2 3 21 4 } 1 2 } 2 6 3 25.1 2 } 3 2 4 6. 53(807) 7.5} 84 (210)8. Round 37.6451 to the nearest hundredth.9. Change 4 7 } 8to an improper fraction.10. Write the number 38,609 in words.11. Identify the property or properties used in the following:5(x 1 9) 5 5(x) 1 5(9)Simplify:12. 6(3) 3 2 9(2) 2 13. !36} 4914.3} 82 1 } 41 5 } 615. (0.2) 3 1 (0.3) 216.9 2 5} 29 1 517. 2(26)Write each ratio as a fraction in lowest terms.18. 24 seconds to 1 minute19.2} 3to 3 } 420. Change 49 } 6to a mixed number.21. Write 4 7 } 8as a decimal.22. Change } 3 } to a percent.823. Change 76% to a fraction.24. What is 2.5% of 40?25. 17 is what percent of 42.5?Make the following conversions.26. 350 m to kilometers 27. 14 gal to liters 28. Write } 1 4} as a decimal.2529. 10 is 50% of what number? 30. Reduce } 9 9}.36<strong>Chapter</strong> 8Cumulative Review503
504<strong>Chapter</strong> 8 Real Numbers and Algebraic Expressions31. Temperature On Thursday, Arturo notices that the temperaturereaches a high of 9° above 0 and a low of 8°below 0. What is the difference between the high andlow temperatures for Thursday?32. Sale Price A dress that normally sells for $129 is on salefor 20% off the normal price. What is the sale price ofthe dress?33. Ratio If the ratio of men to women in a self-defenseclass is 3 to 4, and there are 15 men in the class, howmany women are in the class?34. Surfboard Length A surfing company decides that asurfboard would be more efficient if its length were reducedby 3 }5 8 } inches. If the original length was 7 feet3}} 1inches, what will be the new length of the board (in6inches)?35. Average Distance A bicyclist on a cross-country trip travels72 miles the first day, 113 miles the second day, 108miles the third day, and 95 miles the fourth day. Whatis her average distance traveled during the four days?36. Area and Perimeter Find the area and perimeter of thetriangle below.59 6 ft6 ft17 3 ft11 ft37. Cost of Chocolate If white chocolate sells for $4.32 perpound, how much will 2.5 pounds cost?38. Number Line The distance between two numbers on thenumber line is 9. If one of the numbers is 24, what arethe two possibilities for the other number?39. Basketball Shots Erica makes a total of 8 two-point basketsin her first 18 games of the season. If she continuesat the same rate, how many two-point baskets willshe make in 45 games?40. Stopping Distances The bar chart below shows how many feet it takes to stop a car traveling at different rates of speed,once the brakes are applied. Use this information in the bar chart to fill in the table.400350352Speed(mi/hr)Distance(ft)Stopping distance (ft)3002502001501005022498813719826920 22304050198020 3040 50 60 70 80Speed (mi/hr)80269
<strong>Chapter</strong> 8 TestGive the opposite of each number.1. 14 2. 2 } 2 3 }Place an inequality symbol (, or .) between each pair of numbers so that the resulting statement is true.3. 21 24 4. u24u u2uSimplify each expression.5. 2(27) 6. 2u22uPerform the indicated operations.7. 8 1 (217)11. (26)(27)8. 24.2 2 1.79. 2} 2 3 } 1 1 2 }4 5 } 212. 2} 1 3 }(218) 13. } 2 801610. 265 2 (229)14. } 2 3.520.7Simplify the following expressions as much as possible.15. (23) 2 16. (22) 3 17. (27)(3) 1 (22)(25) 18. (8 2 5)(6 2 11)19. } 25 1 3( 23)5 2 720. } 23(2 ) 1 5(22)7 2 321. Give the sum of 215 and 246.22. Subtract 25 from 212.23. What is the product of 28 and 23?24. Give the quotient of 45 and 29.<strong>Chapter</strong> 8Test505
506<strong>Chapter</strong> 8 Real Numbers and Algebraic Expressions25. Garbage Production The table and bar chart below give the annual production of garbage in the United States for somespecific years.Yeargarbage(Millions of Tons)1960 881970 1211980 1521990 205Garbage (millions of tons)25020015010050The Growing Garbage Problem215205152121882000 217019601970 1980 1990 2000Use the information from the table and bar chart to construct a line graph using the template below.250Garbage (millions of tons)20015010050019601970 1980 1990 200026. Gambling A gambler loses $100 Saturday night andwins $65 on Sunday. Give the gambler’s net loss orgain as a positive or negative number.27. Temperature On Friday, the temperature reaches a highof 21° above 0 and a low of 4° below 0. What is thedifference between the high and low temperatures forFriday?Apply the distributive property and simplify.28. 7(x 2 5) 29. 4(5x 2 1) 30. 2(9x 2 8y)Combine similar terms.31. 12x 1 20x 32. 9a 2 a
<strong>Chapter</strong> 8 ProjectsReal Numbers and Algebraic Expressionsgroup PROJECTRandom MotionNumber of PeopleTime NeededEquipmentBackground315 minutesCoins, dice, pencil, and paperMicroscopic atoms and molecules move randomly.We use random movement models tohelp us understand their motion. Random motionalso helps us understand things like thestock market and computer science.In a random walk, an ant starts at a lamppostand takes steps of equal length along the street.We can think of the lamppost as the origin.The ant either takes a step in the negative orpositive direction. Mathematicians have studiedquestions such as where the ant is likely to endup after taking a certain number of steps.positionStage Coin Die of Ant0 — — 012345678910ProcedureYou will use a coin and die to simulate random motion. The ant will start at 0 on the number line.Roll the die and flip the coin. The ant will move the number of steps shown on the die. If the coincomes up heads, the ant moves in the positive direction. If the coin comes up tails, the ant movesin the negative direction. Repeat this process 10 times. Start each stage from the ending position ofthe previous stage. For example, if the ant ends up at 23 after Stage 1, then in Stage 2 the ant startsat 23. Record your results in the table.<strong>Chapter</strong> 8Projects507
RESEARCH PROJECTDavid Harold BlackwellAt age 22, David Blackwell earned his doctorate,becoming the seventh African Americanto earn a Ph.D. in mathematics. In high school,Blackwell did not care for algebra and trigonometry.When he took a course in analysis,he really became interested in math. AlthoughBlackwell faced a good deal of racism duringhis career, he became a successful teacher, author,and mathematician. Research the life andwork of Dr. Blackwell, and then present yourresults in an essay.Courtesy of David Harold Blackwell508<strong>Chapter</strong> 8 Real Numbers and Algebraic Expressions
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