Chapter 1.Variables, Expressions and Integers PREREQUISITE SKILLS QUIZReview Vocabulary sumdifferenceproductquotientfactor Preparing for Success To prepare for success in this chapter, test your knowledgeof these concepts and skills. You may want to look at the pages referred to in bluefor additional review.Vocabulary Copy and complete the statement using a review word.1. In the multiplication equation 12 ×5 =60, 12 and 5 are called and60 iscalled the .2. When you divide one number by another, the result is called the .Find the sum or difference.3. 7.2 +13.7=4. 2.41 +34.6=5. 10.5 -7.3=6. 27.1 -18.6= Find the product or quotient.7. 3.2 ×1.4= 8. 0.5 ×27= 9. 27.88 ÷8.2= 10. 11.9 ÷1.7=NOTETAKING STRATEGIESKEEPING A NOTEBOOK Some useful items to put in your notebook include the following.Note WorthyYou will find a notetaking strategy at the beginning of each chapter. Look for additional notetaking and study strategies throughout the chapter. • assignments • formulas • rules and properties • vocabulary • symbols • worked-out examples When you copy examples into your notebook, you may find it helpful to draw a diagram. Include comments that make the solution process clear. For example, a diagram can help you to order the numbers 3.2, 3.09, 3, 3.15, 3.12, and 3.02 from least to greatest.Draw a number line and graph the numbers:Write the numbers in the order in which they appear from left to right: 3, 3.02, 3.09, 3.12, 3.15, 3.2.In Lesson 1.8, you may want to include a diagram of acoordinate plane in your notebook.Lesson 1.1Expressions and VariablesVocabulary numerical expression variable variable expression evaluateverbal model Before: You evaluated numerical expressions.Now:You’ll evaluate and write variable expressions.Why? So you can find the amount left on a gift card, as in Ex. 39.Blue Whales During its summer feeding season, a blue whale eats about 4 tons of food every day.To find about how many tons of food a blue whale eats in a given number of days, you can multiply the number of days by 4, as shown in the table.A numerical expression consists of numbers and operations. In thet able, the expression 4 p10 is a numerical expression. It can also bewritten as 4 ×10 or 4(10).A variable is a letter used to represent one or more numbers. Av ariable expression consists of numbers, variables, and operations.O ne way you can use a variable expression is to generalize a pattern,a s in the table. The variable expression 4٠d represents the amount offood a blue whale can eat in d days. You can also write 4٠d as 4d.To evaluate a variable expression, substitute a number for each variable and evaluate the resulting numerical expression.Study StrategyExample1. Evaluating a Variable ExpressionWhen you write a variable expression involving multiplication, avoid using the symbol × . It may be confusedw ith the variable x.Evaluate the expression 4 ×d when d = 120 to find about how many tons of food a blue whale eats in a feeding season of 120 days. Solution4٠d = 4٠120 Substitute 120 for d.= 480 Multiply.Answer A blue whale eats about 480 tons of food in 120 days.Example 2. Evaluating Expressions with Two VariablesEvaluate the expression when x =10 and y =4.a. x + y = 10 +4 Substitute 10 for x and 4 for y.= 14 Add.b. xy = 10(4) Substitute 10 for x and 4 for y.=40 Multiply.Checkpoint Evaluate the expression when x =6 and y =12.1. y +82. 9 -x3. y -x4. xyWriting Variable Expressions You can solve a real-world problem by creating a verbal model and using it to write a variableexpression. A verbal model describes a problem using words as labels and using math symbols to relate the words. The tableshows common words and phrases that indicate mathematical operations.Watch OutCommon Words and Phrases that Indicate Operations Addition Subtraction Multiplication DivisionOrder is important in subtraction and division expressions.“The difference ofa number and 7” means n - 7, not7 -n. “The quotient of a number and 5”meansn/5, not 5/n. Plus minus times divided by the sum of the difference of the product of divided into increased by decreased by multiplied by the quotient of total fewer than ofmore than less thanadded to subtracted fromExample 3. Writing a Variable ExpressionB aseball You plan to divide the 120 players in a baseball league intoteams with the same number of players. Use a verbal model to write avariable expression for the number of teams if you know the numberof players on each team.SolutionReading Algebra L et p represent the number of players on each team. The word divide indicates division.W hen you write a variable e xpression involving division, u se a fraction bar instead of t he division symbol ÷. For example, write “the quotient of n and 12 as n/12.in league =120 /pVocabulary Check 1. Identify the variable in the expression 21 +d.2. Compare and contrast the expressions 2 +x and 2 +3.Skill Check E valuate the expression when x =4.3. 10 -x4. x + 75. 2x6. 32/xEvaluate the expression when m 5 and n 6.7. n/2 8. m + n9. n - m10. MnGuided Problem Solving 11.Astronauts In 2002, astronauts Carl Walz and Dan Bursch spent 196 days inorbit. How many sunrises did they see?1An astronaut in orbit circles Earth every 90 minutes and sees16 sunrises each day. Let d be the number ofdays an astronaut is in orbit. Write a variable expression for the number of sunrises seen in d days.2Identify the value of d for Walz’s and Bursch’s 2002 space flight.3Find the number of sunrises Walz and Bursch saw.Practice and Problem SolvingE valuate the expression when x =6.12. x + 3 13. 15 -x14. 2x15. x/316. 20x17. 24/x 18. 30 -x19. 15 +xEvaluate the expression when a =4, b =2, and c =16.20. a + b21. c - a22. ab23. a/b24. bc 25. c/a26. a–b 27. c/b28. b + c29. c - b30. ac31. a + cWrite a variable expression to represent the phrase.32. The product of 72 and 33. The difference of a numbera number and 134. 13 more than a number 35. The sum of a number and 9.436. The quotient of a number and 3 37. A number divided by 4138. Error Analysis Describe and correct thedifference of a number and 31.39. Gift Card You can evaluate the expression 50 -d to find the amount you have left on a $50 gift card after youhave spent d dollars. Find the amount left after you have spent $18.40. Music Competition The doublebar graph shows three students’scores in a music competition. Astudents ’ final score is the sum of the points for technique t and forinterpretation i.a. Write a variable expression for astudent’s final score.b. Find each student’s final score.c. Interpret You earn 35 points fortechnique. At least how many points must you earn for interpretation to have a higherscore than students A, B, and C?Evaluate the expression when a = 2.5, b = 15, and c = 3.5.41. a + b 42. b - c 43. bc 44. a + c45. b /a 46. c – a 47. c /a 48. acWrite a variable expression to represent the phrase.49. The number of inches in x feet 50. The number of pounds in y ounces 51. DVD Rentals You belong to an online DVD rental service. Youryearly rental budget is $200. Each rental costs $4.a. Copy and complete the table.b. Write a variable expression forthe cost of r rentals.c. Write a variable expression forthe amount of your budget leftafter r rentals.d. Writing How many DVDs will you be able to rent before the $200 is spent? Explain how you found your answer. 52. Extended Problem SolvingIn football, each field goal (FG) isworth 3 points. Each kicked pointafter touchdown (PAT ) is worth1 point. The table shows careertotals for three leading kickers.a. Let p be the number of points after touchdown that a kicker scored,and let f be the number of field goals. Write a variable expression for the total number of points.b. Evaluate Find the total number of points for each kicker.c. Compare List the players in order from least total number of pointsto greatest.53. Critical Thinking Are there any values of the variable a for which theexpressions 2 + a and 2a have the same value? Explain.Logical Reasoning Describe the pattern shown in the table. Then write av ariable expression involving n to complete the table. In the table, thet hree dots indicate that the pattern continues.54. Cost of item (dollars) 1.00 2.00 3.00 4.00 ...nCost with tax (dollars) 1.05 2.10 3.15 4.20 ...?55.C ost of item (dollars) 1.00 1.50 2.00 2.50 .. n.Cost with coupon (dollars) 0.50 1.00 1.50 2.00 ...?56.Challenge The plastic tips on the ends of a shoelace are called aglets.Suppose a sneaker factory produces p pairs of shoes each hour and isin operation for h hours each day. Write a variable expression for thenumber of single aglets the factory uses each day. Evaluate theexpression when p = 200 and h = 24. Explain what your answer means. Mixed Review Find the sum or difference.57. 3.2 +4.7 58. 5.1 +6.8 59. 7.3 -2.1 60. 9.9 -5.4Find the product or quotient.61. 8(13.2) 62. 12.5/563. 24.32/3.264. (6.5)(4.3)65. Order the decimals from least to greatest: 8.9, 8.79, 7.98, 9.87, 7.8, 9.78.Standardized Test Practice 66. Multiple Choice Write a variable expression for a length of time inminutes if you know the number s of seconds.A. s + 60B. s/60C. 60sD. 60/s67. Multiple Choice Evaluate the expression x - y when x = 12.8 and y = 4.F.3.2G. 8.8H. 12.4I. 13.2 Lesson 1.2Powers and ExponentsVocabulary power base exponent Before: You multiplied whole numbers and decimals.Now:You’ll use powers to describe repeated multiplication.Why? So you can find the total number of e-mails sent, as in Ex. 29.A power is the result of a repeated multiplication of the same factor.F or example, the number 125 is a power because 125 =5 •5 •5. A powercan be written in a form that has two parts: a number called the base and a number called the exponent. The exponent shows the number of times the base is used as a factor.The table shows how to read and write powers. Numbers raised to thefirst power, such as 121, are usually written without the exponent.Note W orthy E xampl e1.U si n g E xponents Write the product using an exponent.Write additional examples of your own in your notebook. For each product, identify the base and the exponent, then write the product using an exponent. a. 13•13•13•13=134 The base 13 is used as a factor 4 times.b. (0.2)(0.2)(0.2)=(0.2)3 The base 0.2 is used as a factor 3 times.c. n•n•n•n•n•n=n6 The base n is used as a factor 6 times.d. t•t•t•t•t = t5 The base t is used as a factor 5 times.Checkpoint Write the product using an exponent.1. 10•10•102. (4.3)(4.3)3. x•x•x•x4. Critical Thinking Evaluate each power: 02, 03, 04. Use your results towrite a rule for the value of 0 raised to any nonzero whole numberexponent.E xampl e2.E val u ati n g P owers w i t h V ari a bl e sEvaluate the expression x4 when x =0.5.x4 = (0.5)4 Substitute 0.5 for x.=(0.5)(0.5)(0.5)(0.5) Use 0.5 as a factor 4 times.=0.0625 Multiply.Checkpoint Evaluate the expression when m =3.5. m26. m37. m48. m5U sing Formulas A formula describes a relationship between quantities.Some formulas involve powers. For example, you can use a formula tofind the area of a square or the volume of a cube.Area is measured in square units, such as square feet (ft2) or squarec entimeters (cm2). Volume is measured in cubic units, such as cubicinches (in.3) or cubic meters (m3).Exampl e 2. U si n g P owers i n F ormul a s I ce Sculpture An artist uses a cube-shaped block of ice to make an icesculpture for a competition. Find thevolume of the block of ice.SolutionUse the formula for the volume of a cube.V =s 3 Write the formula.=(20)3Substitute 20 for s. In the Real WorldIce Sculpture A cubic inch ofi ce weighs about 0.03 pound. If t he artist carves away about 2000 cubic inches from the b lock in Example 3, about how much does the resultingsculpture weigh?Exercises=8000 Evaluate power. Answer The volume of the block of ice is 8000 cubic inches. Checkpoint Find the area of a square with the given side length. 9. 9 meters 10. 11 inches 11. 1.5 centimeters Guided Practice Vocabulary Check1. Identify the base and the exponent in the expression 135.2. How are the expressions 34 and 43 different? Skill CheckW rite the power in words and as a repeated multiplication. Then evaluate the power. 3. 122 4. (0.3)3 5. (1.2)3 6. 54 Evaluate the expression when k = 6. 7. k 2 8. k 3 9. k 4 10. k 5 11. Gift Box A gift box has the shape of a cube with an edge length of 14 inches. Find the volume of the box.12. Error Analysis Describe and correct the error in writing 23 as a repeated multiplication. Practice and Problem Solving Write the product using an exponent. 13. 32 • 32 14. 11 • 11 •11 15. 6 • 6 •6 • 6• 6 16. 2 • 2 • 2 • 2 17. (5.6)(5.6)(5.6) 18. (1.7)(1.7) 19. z • z • z 20. n •n •n •n Write the power in words and as a repeated multiplication. Then evaluate the power.21. 83 22. 25 23. 106 24. 123 25. 93 26. 44 27. (0.2)2 28. (0.6)4 29. Extended Problem Solving You send an e-mail to 4 friends. Each friend sends the e-mail to 4 more friends. Each of those friends sends it to 4 friends, and so on. a. Copy and complete the table. b. Calculate Find the number of e-mails sent at stage 9. c. Estimate Estimate the stage at which more than 1,000,000 e-mails will be sent. Use a calculator to check your estimate. Evaluate the expression when n 7 and when n 0.4. 30. n 2 31. n 3 32. n 4 33. n 5 34. Writing The square of a number is the second power of the number. The cube of a number is the third power of the number. Explain why these names are reasonable. 35. Critical Thinking Explain why 1 raised to any power is equal to 1. 36. Aquariums An aquarium has a square base with a side length of 15 inches. You fill the aquarium with water to a height of 15 inches. a. Find the volume of the water in the aquarium. b. A cubic inch of water weighs approximately 0.036 pound. Find the approximate weight of the water in the aquarium. 37. Patterns The table shows sums of odd numbers. a. Copy and complete the table. Identify any pattern that you see. b. Write a variable expression for the sum of the first n odd numbers. c. Use your expression from part (b) to find the sum of the first 100 odd numbers. 38. Challenge Find values of x, y, and z so that each of the expressions x 2, y 3, and z 6 has a value of 64.Mixed Review Find the product or quotient.39. (2.5)(7.1) 40. (2.3)(8.4) 41. 1.2 ÷ 2.4 42. 5.2 ÷ 1.2543. Olympics The bar graph shows the number of gold medals won by thefour countries with the most goldmedals in the 2000 OlympicSummer Games. How many goldmedals did the four countries winin all?Evaluate the expression when x = 15.44. x + 4 45. 200 -x46. x - 11 47. 3xStandardized TestPractice 48. Multiple Choice Which expression has a value of 81?A. 43B. 34C. 28D. 27349. Short Response Compare eachnumber in the top row of the table withthe number below it. Describe anypattern you see. Complete the tablewith a variable expression involving n.1 2 3 4 ... n1 8 27 64 ... ?Student ReferenceReview thist opic in preparationf or solving problems i n Lesson 1.3. For a r eview of problems olving strategies.A Problem Solving PlanYou can use the following 4-step plan to solve a problem.1.Read and Understand Read the problem carefully. Identify the questionand any important information.2.Make a Plan Decide on a problem solving strategy.3.Solve the Problem Use the problem solving strategy to answer thequestion.4.Look Back Check that your answer is reasonable.Reading and PlanningExample You plan to ship 5 books to a friend. The table shows the masseso f the books. Is it possible to ship the books in 2 boxes, each with a mass of6 kilograms or less? Explain.Book A B C D EMass (kg) 1.4 2.1 3.8 1.9 2.5Read and UnderstandWhat do you know?The mass of each box must be 6 kilograms or less.The table gives the mass of each book.What do you want to find out?Is it possible to put the books in 2 boxes so that each box has a mass of6 kilograms or less?Make a PlanHow can you relate what you know to what you want to find out?Check that the total mass of the books doesn’t exceed 12 kilograms. If it does, you can’t divide the books as you want.Use the strategy guess, check, and revise to choose books for each box.Solving and Looking BackTo solve the problem from page 14 about shipping books, carry out the plan. Then check the answer.Solve the ProblemThe total mass of the books is 1.4 +2.1 +3.8 +1.9 +2.5 =11.7 kg, so it may be possible to ship the books as you want.N ow use the strategy guess, check, and revise.Put the 3 lightest books in one box and the2 heaviest in the other. The mass of thesecond box is more than 6 kg.Try switching books B and E. The mass ofeach box is less than 6 kg.Answer It is possible to ship the books in 2 boxes, each with a mass of6 kg or less. Put books A, D, and E in one box and books C and B in the other.Look BackIt makes sense that the box with 3 books contains the 2 lightest books andthe box with 2 books contains the heaviest book. So, the answer is reasonable.CheckpointTest your knowledge of the problem solving plan by solving these problems.1. Pool Schedules A community pool offers3 swim sessions each Saturday morning.Each session lasts 35 minutes, with10 minutes between sessions. The final sessionends at 11:05 A.M. At what time does the first session begin?2. Theater Seating The center section of a theater has 10 rows. There are 41 seats inrow 10, 38 seats in row 9, 35 seats in row 8, and so on. How many seats are in row 1?3. Movie Marathon You have invited friends over to watch movies. You have rented 4 movies:one action, one science fiction, one comedy, and one animated. In how many different orders can you watch the movies?Lesson 1.3Order of OperationsVocabulary order of operations,Before：You performed basic operations.Nowadays：You’ll use order of operations to evaluate expressions. Why？So you can find the height of a boojum, as in Ex. 28.Flower Flag There are about2000 plants in each of the50 stars of this flower flag.There are about 64,100 plants in each of the 7 short stripes and 106,700 plants in each of the 6 long stripes. The bluecontains about 198,900plants. You can approximate the total number of plants by evaluating the expression50•2000 + 7•64,100 + 6•106,700 + 198,900.To evaluate expressions involving more than one operation,mathematicians have agreed on a set of rules called the oroperations.Note Worthy Order ofOperations1. Evaluate expressions inside grouping symbols.You should include material that appears on a notebook like the one shown in your own notebook.2. Evaluate powers.3. Multiply and divide from left to right.4. Add and subtract from left to right.Example 1. Using Order of OperationsTo approximate the total number of plants in the flower flag describedabove, use the order of operations to evaluate the expression 50•2000 + 7•64,100 + 6•106,700 + 198,900Write expression.= 100,000 + 448,700 + 640,200 + 198,900 Multiply.=1,387,800 Add.Answer There are approximately 1,387,800 plants in the flower flag.Grouping Symbols Parentheses ( ), brackets [ ], and fraction bars arec ommon grouping symbols. Grouping symbols indicate operations thats hould be performed first. For example, compare the expressions3•2 + 5 and 3(2 + 5). To evaluate 3•2 + 5, you multiply first, then add.To evaluate 3(2 + 5), you add first, then multiply.Example 2. Using Grouping SymbolsEvaluate the expression.a. 8(17 - 2.3) = 8(14.7) Subtract within parentheses.= 117.6 Multiply.Watch Out b. 14+612−7= (14 + 6) ÷ (12 - 7) Rewrite fraction as division.When grouping symbols appear inside other grouping symbols, as in part (c) of Example 2, work from the innermost grouping symbols out. = 20÷5Evaluate within parentheses. = 4 Divide.c. 5 •[36 - (13 + 9)] = 5• [36 - 22] Add within parentheses.= 5 •14Subtract within brackets. = 70 Multiply.Checkpoint Evaluate the expression.1. 28 - 63 ÷ 72. 52 + 12.5•43. 9•6 + 27÷34. 10(1.5 + 0.6)5. 70−9.23+56. 72 ÷ [(11 - 7)•2]Example 3. Evaluating Variable ExpressionsStudy Strategy E valuate the expression when x =2 and y =5.a. 4(x +y) = 4(2 +5) Substitute 2 for x and 5 for y.Y ou can use the first letters of t he words of the sentenceP lease Excuse My Dear Aunt Sally to help you remember t he order of operations.P parenthesesE exponents = 4(7) Add within parentheses. = 28 Multiply.b. 3(x +y)2 = 3(2 +5)2 Substitute 2 for x and 5 for y. = 3(7)2 Add within parentheses. = 3(49) Evaluate power.M/D multiplication and divisionA/S addition and subtraction147 Multiply.Checkpoint Evaluate the expression when x =4 and y =2.7. 1.2(x + 3) 8. 1.2x + 3 9. 3x - 2y10. 0.5[y - (x - 2)] 11. x2 – y 12. 2(x - y)2Example 4. Using a Problem Solving PlanSewing You buy a pattern and enough material to make two pillows.The pattern costs $5. Each pillow requires $3.95 worth of fabric and abutton that costs $.75. Find the total cost.SolutionRead and Understand You buy one pattern plus fabric and buttons for two pillows. Youare asked to find the total cost.Make a Plan Write a verbal model.Solve the Problem Write and evaluate an expression.Total cost =5 +2(3.95 +0.75) Substitute values into verbal model.=5 +2(4.70) Add within parentheses.= 5 +9.40 Multiply.= 14.40 Add.Answer The total cost is $14.40.Look Back Use estimation to check that the answer is reasonable. Thec ost of materials for each pillow is about $4 +$1 =$5. The total cost isabout $5 +2($5) =$15. The answer is reasonable.1.3 E xercisesGuided PracticeVocabulary Check 1. Give three examples of grouping symbols.2. Describe in order the steps you would take to evaluate the expression 12(x - 3)2 when x = 5.Skill Check Evaluate the expression. 3. 15 - 3•4 4. 48 ÷ 6 + 2 5. 3•8 + 5•4 6. 18+127−2 7. 17 - (32 - 2) 8. 4[15 - (2 + 5)]9. Twin Convention The table shows the numbers of sets of twins, triplets, quadruplets, and quintuplets registered at a twin convention. Write and evaluate an expression for the total number of people who registered at the convention. Practice and Problem SolvingE valuate the expression.10. 47.7 - 12•3 11. 11•7 - 9•5 12. 14 ÷ 7 + 36 ÷ 413. 5.8(3) + 3(1.1) 14. 36−122+836 15. 9.8+2.27−59.82 16. 5(21 - 32) 17. 7[2.5 + 3(12 - 7)] 18. 84 ÷ [(18 - 16) •3]Evaluate the expression when x = 3, y = 4, and z = 5.19. 0.25y + x 20. 0.25(y + x) 21. 4(z - x) 22. 6.5y x−1 23. x + 24.4y24.4 24. 7z- x 2 25. x + 2[z - (y - 1)] 26. (x + y)2 - 3.6 27. y + (z - 1)2 28. Plants A boojum is a very slow-growing cactus.One fifty-year-old boojum is 1.5 meters talland has been growing about 0.03 meter each year. Assume the growth pattern continues.a. Write an expression for the height in metersof the boojum y years from now.b. Apply How tall will the boojum be in 50 years?29. Craft Fair Your school is setting up a row of5 tables for a craft fair. Each table is 72 inches long. The space between each pair ofneighboring tables must be 48 inches. Write andevaluate an expression to find the length of thespace needed for the tables from the beginningof the first table to the end of the last table.30. Basketball In basketball, players score points by making free throwsworth 1 point each, field goals worth 2 points each, and field goalsworth 3 points each. A player scores 4 free throws, 7 two-point field goals, and 2 three-point field goals. Write and evaluate an expressionfor the total number of points the player scores.31. Movies You buy 4 videotapes for $14.99 each and 3 DVDs for $19.99each. Find the total cost of the movies.Evaluate the expression when x = 4 and y = 3.26.5y2 32. 5x2 2y33. 7(x2 5y) 34. x935.y2 x1 36. Cell Phone You and your sister share cell phone service. You divide thebill equally, including the monthly fee of $39 plus $.30 for eachadditional minute beyond your free minutes.a. Write an expression for your share of the bill in a month when youare charged for m extra minutes.b. Apply One month, you are charged for 125 additional minutes. Findyour share of the bill.。