Math 8. Chapter 8. Name

Transcription

1 Math 8 Chapter 8 Name

2 WHJH Math

3 8 NAME DATE PERIOD Student-Built Glossary This is an alphabetical list of new vocabulary terms you will learn in Chapter 8. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Add this page to your math study notebook to review vocabulary at the end of the chapter. Vocabulary Term coefficient Found on Page Definition/Description/Example Chapter Resources constant equivalent expressions like terms simplest form simplifying the expression terms of the expression two-step equation Chapter 8 1 Course 3

4 8 NAME DATE PERIOD Family Letter Dear Parent or Guardian: Often real-world situations can be modeled using algebraic expressions, equations, and inequalities. Knowing how to write an equation to represent a problem situation from life such as comparing varying costs of several long-distance phone companies can make the problem easier to solve. In Chapter 8, Algebra: More Equations and Inequalities, your child will learn how to simplify algebraic expressions, solve and write two-step equations that represent real-world situations, to solve equations with variables on each side, about inequalities, and to use the guess and check problem-solving method. In the study of this chapter, your child will complete a variety of daily classroom assignments and activities and possibly produce a chapter project. By signing this letter and returning it with your child, you agree to encourage your child by getting involved. Enclosed is an activity you can do with your child that practices how the math we will be learning in Chapter 8 might be tested. You may also wish to log on to glencoe.com for self-check quizzes and other study help. If you have any questions or comments, feel free to contact me at school. Sincerely, Chapter Resources Signature of Parent or Guardian Date Chapter 8 3 Course 3

5 8 NAME DATE PERIOD Anticipation Guide Algebra: More Equations and Inequalities Step 1 Read each statement. Before you begin Chapter 8 Decide whether you Agree (A) or Disagree (D) with the statement. Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). Chapter Resources STEP 1 STEP 2 Statement A, D, or NS A or D 1. The expression 6y 3(x 2) is in simplest form because it has no like terms. 2. The expressions 4(x 3) and 4x 12 are equivalent. 3. When solving equations, undo each operation in the same order as the order of operations. 4. To solve the equation 4 2x 10, first subtract 4 from each side, and then divide each side by Three times a number decreased by 1 is 11 and 11 equals 1 less than three times a number are equivalent statements. 6. A variable can be added to or subtracted from both sides of an equation with the equation remaining true. 7. The phrase a minimum of $25 can be written as m If t 8, then the inequality 3t 6 20 is false. Step 2 After you complete Chapter 8 Reread each statement and complete the last column by entering an A or a D. Did any of your opinions about the statements change from the first column? For those statements that you mark with a D, use a piece of paper to write an example of why you disagree. Chapter 8 7 Course 3

6 8-1 NAME DATE PERIOD Lesson Reading Guide Simplifying Algebraic Expressions Get Ready for the Lesson Do the Mini Lab at the top of page 416 in your textbook. Write your answers below. 1. Choose two positive and one negative value for x. Then evaluate 2(x 3) and 2x 6 for each of these values. What do you notice? 2. Use algebra tiles to rewrite the expression 3(x 2). (Hint: Use one green x-tile and 2 red 1-tiles to represent x 2.) Lesson 8 1 Read the Lesson 3. When is the Distributive Property used to simplify an algebraic expression? 4. Explain how to simplify the expression 5(x 3). 5. Explain what it means for two expressions to be equivalent. 6. Give an example of an expression containing three terms, one of which is a constant. Remember What You Learned 7. One of your classmates was absent from school today and has not studied the lesson. Write a letter to your classmate explaining how to simplify an expression and how to identify terms and constants. Chapter 8 9 Course 3

7 Notes and Important Things

8 Notes and Important Things

9 8-1 NAME DATE PERIOD Study Guide and Intervention Simplifying Algebraic Expressions The Distributive Property can be used to simplify algebraic expressions. Examples Use the Distributive Property to rewrite each expression. 3(a 5) 2(d 3) 3(a 5) 3(a) 3(5) Distributive Property 2(d 3) 2[d ( 3)] Rewrite d 3 as 3a 15 Simplify. d ( 3). 2(d) ( 2)( 3) Distributive Property 2(d) 6 Simplify. When a plus sign separates an algebraic expression into parts, each part is called a term. In terms that contain a variable, the numerical part of the term is called the coefficient of the variable. A term without a variable is called a constant. Like terms contain the same variables, such as 3x and 2x. Example 3 Identify the terms, like terms, coefficients, and constants in the expression 7x 5 x 3x. 7x 5 x 3x 7x ( 5) x ( 3x) 7x ( 5) 1x ( 3x) Definition of subtraction Identity Property; x 1x The terms are 7x, 5, x, and 3x. The like terms are 7x, x, and 3x. The coefficients are 7, 1, and 3. The constant is 5. An algebraic expression is in simplest form if it has no like terms and no parentheses. Example 4 Simplify the expression 2m 5 6m 3. 2m and 6m are like terms. 5 and 3 are also like terms. 2m 5 6m 3 2m 5 6m ( 3) 2m 6m 5 ( 3) ( 2 6)m 5 ( 3) 4m 2 Exercises Definition of subtraction Commutative Property Distributive Property Simplify. Use the Distributive Property to rewrite each expression. 1. 2(c 6) 2. 4(w 6) 3. (b 4)( 3) 4. Identify the terms, like terms, coefficients, and constants in the expression 4m 2 3m 5. Simplify each expression. 5. 3d 6d s z 3 9z 8 Chapter 8 10 Course 3

10 NAME DATE PERIOD 8-1 Skills Practice Simplifying Algebraic Expressions Use the Distributive Property to rewrite each expression. 1. 4(j 4) 2. 5(n 2) 3. (c 9)3 4. 2(w 8) 5. (s 7)7 6. 4(e 6) 7. (b 3)( 7) 8. 8(v 7) 9. (2n 3) (c d) 11. 7(3x 1) 12. (e f) ( 3m 1) 14. (2b 3)( 9) 15. 5(s 7) 16. (t 7) ( 2v 4) 18. (m n)( 3) Identify the terms, like terms, coefficients, and constants in each expression e 7e 5 Lesson x h 2h 6h y k 7 k z 3 2z z Simplify each expression t 6t 26. 4r r 27. 7f 2f 28. 9a 8a 29. 5c 8c 30. 2g 5g 31. 8k 3 4k 32. 7m 5m x p 1 9p b 3b 8b h 6 8 7h 37. 8b 6 8b t 5 2t w 5w w 40. 6m 7 2m f 7f f y 8 4y y 43. 9a 5 7a 2a 44. 6g 7g x 6 9x 3 Chapter 8 11 Course 3

11 8-1 NAME DATE PERIOD Practice Simplifying Algebraic Expressions Use the Distributive Property to rewrite each expression. 1. 6(z 4) 2. 7(c + 2) 3. (d + 5)9 4. (h + 8)( 3) 5. 5(y 2) 6. 3(6 n) 7. 4(s 4) 8. 9(2 p) 9. 2(3x 1) 10. 5(4n 5) 11. 8(u 2v) 12. 3a(7b 6c) Identify the terms, like terms, coefficients, and constants in each expression b 7b t 3t t 15. 5x 4 x 1 Simplify each expression. 16. h 6h k k 18. 3b 8 2b v v 20. 2f 3 2f s 5 7s x x c 3d 12c d 24. y 9z 16y 25z 2 Write two equivalent expressions for the area of each figure x 5 x + 1 x PAINTING Mr. Torres paid $43 for supplies to paint his office. He paid one person $8 per hour to prepare the office to be painted and another person $10 per hour to paint the office. If both people worked h hours, write two expressions that you could use to represent the total cost of painting the office. Chapter 8 12 Course 3

12 8-1 NAME DATE PERIOD Word Problem Practice Simplifying Algebraic Expressions 1. GAMES At the Beltway Outlet store, you buy x computer games for $13 each and a magazine for $4. Write an expression in simplest form that represents the total amount of money you spend. 2. TENNIS Two weeks ago, James bought 3 cans of tennis balls. Last week, he bought 4 cans of tennis balls. This week, he bought 2 cans of tennis balls. The tennis balls cost d dollars per can. Write an expression in simplest form that represents the total amount that James spent. Lesson AMUSEMENT PARKS Sari and her friends are going to play miniature golf. There are p people in the group. Each person pays $5 for a round of golf and together they spend $9 on snacks. Write an expression in simplest form that represents the total amount that Sari and her friends spent. 5. GEOMETRY Write an expression in simplest form for the perimeter of the triangle below. 2x 4x 2 2x 3 4. BICYCLING The bicycle path at the park is a loop that covers a distance of m miles. Jorge biked 2 loops each on Monday and Wednesday and 3 loops on Friday. On Sunday, Jorge biked 10 miles. Write an expression in simplest form that represents the total distance that Jorge biked this week. 6. SIBLINGS Mala is y years old. Her sister is 4 years older than Mala. Write an expression in simplest form that represents the sum of the ages of the sisters. Chapter 8 13 Course 3

13 8-1 NAME DATE PERIOD Enrichment Olga Taussky-Todd Olga Taussky-Todd ( ) had a rich and varied career as a research mathematician, mathematics professor, and author and editor of mathematical texts. Born in eastern Europe, she lived and worked in Austria, Germany, England, and the United States. She served as the consultant in mathematics for the National Bureau of Standards in Washington, D.C., for ten years. In 1957, she became the first woman appointed to the mathematics department of the California Institute of Technology. Dr. Taussky-Todd made contributions in many areas of mathematics and physics. The exercises below will help you learn some more about her life. Find each product. Circle the correct solution. The phrase following the solution will complete the statement correctly. 1. (x 3)(x 7) Her paper on sums of squares won the Ford Prize of the Mathematical Association of America in? x 2 10x 21: 1971 x 2 21x: (x 4)(2x 1) In 1978, she was elected? of the Austrian Academy of Sciences. 3x 5: Correspondent 2x 2 9x 4: Corresponding Member 3. (2x 1)(x 7) In 1978, the government of Austria awarded her the?. 2x 2 15x 7: Cross of Honor in Science and Arts, First Class 3x 2 8x 7: Purple Cross 4. (x 6)(x 6) In 1988, the University of Southern California awarded her an?. 2x 12: honorary science degree x 2 12x 36: honorary Doctor of Science degree Chapter 8 14 Course 3

14 8-2 NAME DATE PERIOD Lesson Reading Guide Solving Two-Step Equations Get Ready for the Lesson Read the introduction at the top of page 422 in your textbook. Write your answers below. 1. Explain how you could use the work backward strategy to find the cost of each bag of dog treats. Find the cost. 2. Find the cost of each bag. Read the Lesson 3. Define two-step equation. Lesson 8 2 Determine whether each equation is a two-step equation. Explain. c 4. n x What is the first step in solving each equation? 7. 3y x p 11 7 Remember What You Learned 10. Draw a diagram that shows how the equation 2x 3 8 can be modeled using algebra tiles. Chapter 8 15 Course 3

15 Notes and Important Things

16 Notes and Important Things

17 8-2 NAME DATE PERIOD Study Guide and Intervention Solving Two-Step Equations A two-step equation contains two operations. To solve a two-step equation, undo each operation in reverse order. Example 1 Solve 2a Check your solution. Method 1 Vertical Method Method 2 Horizontal Method 2a 6 14 Write the equation. 2a Subtract 6 from each side. 2a a 8 Simplify. 2a 8 2a 8 Divide each side by 2. 2a a 4 Simplify. a 4 Check 2a 6 14 Write the equation. 2( 4) 6 14 Replace a with 4 to see if the sentence is true The sentence is true. The solution is 4. Sometimes it is necessary to combine like terms before solving an equation. Example 2 Solve 5 8x 2x 7. Check your solution. 5 8x 2x 7 Write the equation. 5 6x 7 Combine like terms x 7 7 Add 7 to each side. 12 6x Simplify x 6 6 Divide each side by 6. 2 x Simplify. The solution is 2. Check this solution. Exercises Solve each equation. Check your solution. 1. 2d z s r p x x c 2 3c n 2n r 5 7r b m 12. 3t a q v 0 5 Chapter 8 16 Course 3

18 NAME DATE PERIOD 8-2 Skills Practice Solving Two-Step Equations Solve each equation. Check your solution. 1. 3n s c t f v d k m z a s x x Lesson x x x x x x x 22. 4x 2x y 5y a 15 5a 25. 3z 17 2z e 2e r 1 7r 28. 8k 8 k c 7 6c f 4f Chapter 8 17 Course 3

19 8-2 NAME DATE PERIOD Practice Solving Two-Step Equations Solve each equation. Check your solution. 1. 3g = 4a = 5m = 2t k 5 = = 4x 11 z n = = y = b = y 6 = r = = 5d 8d 14. w + 3w = m + 9m = = 8x 9 5x = s s 18. 6a + 7 a = (y 5) (p 3) (v 2) 22. k z t SHOPPING Mrs. Williams shops at a store that has an annual membership fee of $30. Today she paid her annual membership and bought several fruit baskets costing $15 each as gifts for her coworkers. Her total was $105. Solve the equation 15b to find the number of fruit baskets Mrs. Williams purchased. 26. GAMES A card game has 50 cards. After dealing 7 cards to each player, Tupi has 15 cards left over. Solve the equation 50 7p 15 to find the number of players. 27. GEOMETRY Write an equation to represent the length of P Q. Then find the value of y y 3y P Q Chapter 8 18 Course 3

20 8-2 NAME DATE PERIOD Word Problem Practice Solving Two-Step Equations 1. SHOPPING Jenna bought 5 reams of paper at the store for a total of $21. The tax on her purchase was $1. Solve 5x 1 21 to find the price for each ream of paper. 2. CARS It took Lisa 85 minutes to wash three cars. She spent x minutes on each car and 10 minutes putting everything away. Solve 3x to find how long it took to wash each car. 3. EXERCISE Rick jogged the same distance on Tuesday and Friday, and 8 miles on Sunday, for a total of 20 miles for the week. Solve 2x 8 20 to find the distance Rick jogged on Tuesday and Friday. 4. MOVING Heather has a collection of 26 mugs. When packing to move, she put the same number of mugs in each of the first 4 boxes and 2 mugs in the last box. Solve 4x 2 26 to find the number of mugs in each of the first four boxes. Lesson TELEVISION Burt s parents allow him to watch a total of 10 hours of television per week. This week, Burt is planning to watch several two hour movies and four hours of sports. Solve 2x 4 10 to find the number of movies Burt is planning to watch this week. 7. MONEY McKenna had $32 when she got to the carnival. After riding 6 rides, she had $20 left. Solve 32 6x 20 to find the price for each ride. 6. TRAVEL Lawrence drives the same distance Monday through Friday commuting to work. Last week, Lawrence drove 25 miles on the weekend, for a total of 60 miles for the week. Solve 5x to find the distance Lawrence drives each day commuting to work. 8. GARDENING Jack has 15 rosebushes. He has the same number of yellow, red, and pink bushes, and 3 multicolored bushes. Solve 3x 3 15 to find the number of yellow rosebushes Jack has. Chapter 8 19 Course 3

21 8-2 NAME DATE PERIOD Enrichment Rational Numbers Systems of Equations Sometimes it takes more than one equation to solve a problem. A group of such equations is called a system of simultaneous equations. Here is one system of equations. y x 2 3x 5 16 The solution of this system must be a pair of numbers x and y such that the numbers make both equations true. To solve a system of this type, solve the equation with one variable. Then substitute that answer into the second equation and solve for the other variable. For the system above, x 7 and y 9. Example Solve the system of equations. c d 2 3d 1 17 Solve 3d 1 17 for d. 3d d d 18 Write the equation. Add 1 to both sides. Simplify. 3 d 18 Divide both sides by d 6 Substitute 6 for d in c d 2. c 6 2 c 4 Simplify. The solution is c 4 and d 6. Solve each system of equations t a 2b s 3t s a 8 d 3s m x y c 5 p 4 7m n x p 3 Chapter 8 20 Course 3

22 8-3 NAME DATE PERIOD Lesson Reading Guide Writing Two-Step Equations Get Ready for the Lesson Read the introduction at the top of page 427 in your textbook. Write your answers below. 1. Let n represent the number of payments. Write an expression that represents the amount of the camp fee paid after n payments. 2. Write and solve an equation to find the number of payments you will have to make in order to pay off the balance of the camp. 3. What type of equation did you write for Exercise 2? Explain your reasoning. Read the Lesson Jennifer bought 3 CDs, each having the same price. Her total for the purchase was $51.84, which includes $3.84 in sales tax. Find the price of each CD. 4. Explain how to define the variable in the problem. Then define the variable. 5. The next step is to write an equation for the problem. Assuming that the total, 51.84, will be on the right side of the equals sign by itself, determine which two operations will be represented on the left side of the equals sign. Which is performed first? Explain. 6. Complete the equation. Then solve it. How much does each CD cost? $51.84 Remember What You Learned 7. Work with a partner. Have one partner write a word problem that involves a two-step equation and solve it. Have the other partner check the solution. Then have partners switch tasks. Chapter 8 22 Course 3

23 Notes and Important Things

24 Notes and Important Things

25 8-3 NAME DATE PERIOD Study Guide and Intervention Writing Two-Step Equations Some verbal sentences translate to two-step equations. Example 1 Translate each sentence into an equation. Sentence Equation Four more than three times a number is 19. 3n 4 19 Five is seven less than twice a number. 5 2n 7 Seven more than the quotient of a number and 3 is n 10 3 After a sentence has been translated into a two-step equation, you can solve the equation. Example 2 Translate the sentence into an equation. Then find the number. Thirteen more than five times a number is 28. Words Thirteen more than five times a number is 28. Variable Let n the number. Equation 5n Write the equation. 5n Subtract 13 from each side. 5n 15 Simplify. 5 n Divide each side by 5. n 3 Simplify. Therefore, the number is 3. Exercises Translate each sentence into an equation. Then find each number. 1. Five more than twice a number is Fourteen more than three times a number is Seven less than twice a number is 5. Lesson Two more than four times a number is Eight less than three times a number is Three more than the quotient of a number and 2 is 7. Chapter 8 23 Course 3

26 8-3 NAME DATE PERIOD Skills Practice Writing Two-Step Equations Translate each sentence into an equation. Then find each number. 1. Four more than twice a number is Three more than four times a number is Five less than twice a number is One less than four times a number is Seven more than the quotient of a number and 2 is Six less than six times a number is Five less than the quotient of a number and 3 is Seven more than twice a number is The difference between 5 times a number and 3 is Nine more than three times a number is Nine more than the quotient of a number and 4 is Four less than the quotient of a number and 3 is Nine less than six times a number is Three less than the quotient of a number and 6 is Eight more than the quotient of a number and 5 is The difference between twice a number and 11 is 23. Chapter 8 24 Course 3

27 NAME DATE PERIOD 8-3 Practice Writing Two-Step Equations Translate each sentence into an equation. 1. Three more than eight times a number is equal to Twelve less than seven times a number is Four more than twice a number is Nine less than five times a number is equal to ART Ishi bought a canvas and 8 tubes of paint for $ If the canvas cost $6.95, how much did each tube of paint cost? 6. ENGINEERING The world s two highest dams are both in Tajikistan. The Rogun dam is 35 meters taller than the Nurek dam. Together they are 635 meters tall. Find the height of the Nurek dam. U.S. PRESIDENTS For Exercises 7 and 8, use the information at the right. 7. If you double President Reagan s age at the time of his first inauguration and subtract his age at the time he died, the result is 45 years. How old was President Reagan when he died? 8. If you divide the age of the first President Bush when he was inaugurated by 2 and add 14 years, you get the age of President Clinton when he was first inaugurated. How old was President G. H. W. Bush when he was inaugurated? 9. GEOMETRY Find the value of x in the triangle at the right. x x President Age at First Inauguration J. Carter 52 R. Reagan 69 G. H. W. Bush W. Clinton 46 G. W. Bush Lesson ALGEBRA Three consecutive integers can be represented by n, n 1, and n 2. If the sum of three consecutive integers is 57, what are the integers? Chapter 8 25 Course 3

28 NAME DATE PERIOD 8-3 Word Problem Practice Writing Two-Step Equations Solve each problem by writing and solving an equation. 1. CONSTRUCTION Carlos is building a screen door. The height of the door is 1 foot more than twice its width. What is the width of the door if it is 7 feet high? 2. GEOMETRY A rectangle has a width of 6 inches and a perimeter of 26 inches. What is the length of the rectangle? 3. EXERCISE Ella swims four times a week at her club s pool. She swims the same number of laps on Monday, Wednesday, and Friday, and 15 laps on Saturday. She swims a total of 51 laps each week. How many laps does she swim on Monday? 4. SHOPPING While at the music store, Drew bought 5 CDs, all at the same price. The tax on his purchase was $6, and the total was $61. What was the price of each CD? 5. STUDYING Over the weekend, Koko spent 2 hours on an assignment, and she spent equal amounts of time studying for 4 exams for a total of 16 hours. How much time did she spend studying for each exam? 7. HOME IMPROVEMENT Laura is making a patio in her backyard using paving stones. She buys 44 paving stones and a flowerpot worth $7 for a total of $73. How much did each paving stone cost? 6. FOOD At the market, Meyer buys a bunch of bananas for $0.35 per pound and a frozen pizza for $4.99. The total for his purchase was $6.04, without tax. How many pounds of bananas did Meyer buy? 8. TAXI A taxi service charges you $1.50 plus $0.60 per minute for a trip to the airport. The distance to the airport is 10 miles, and the total charge is $ How many minutes did the ride to the airport take? Chapter 8 26 Course 3

29 8-3 NAME DATE PERIOD Enrichment An Wang An Wang ( ) was an Asian American who became one of the pioneers of the computer industry in the United States. In 1948, he invented a magnetic-pulse controlling device that vastly increased the storage capacity of computers. He later founded his own company, Wang Laboratories, and became a leader in the development of desktop calculators and word-processing systems. In 1988, Wang received a special honor for his contributions to the advancement of the computer industry. To find out what the honor was, solve each equation. If the solution appears at the bottom of this page, write the variable on the line directly above the solution each time it appears. If you have solved the equations correctly, the variables will spell out the honor. R 1. 3V M S H D F O W T I A E N L Lesson Chapter 8 27 Course 3

30 8-4 NAME DATE PERIOD Lesson Reading Guide Solving Equations with Variables on Each Side Get Ready for the Lesson Read the introduction at the top of page 434 in your textbook. Write your answers below. 1. Copy the table. Continue filling in rows to find how many days until Tanner and Jordan sell the same number of packages. 2. Write an expression for Jordan s gift wrap sales after d days. 3. Write an expression for Tanner s gift wrap sales after d days. Time (days) 4. On which day will Tanner s sales pass Jordan s sales? 5. Write an equation that could be used to find how many days it will take until Tanner and Jordan sell the same number of packages.... Jordan s Sales Tanner s Sales 0 8 4(0) 8 5(0) (1) 12 5(1) (2) (3) (2) 10 5(3) Read the Lesson 6. What is the first step in solving an equation with variables on each side? 7. What does it mean to isolate the variable when solving an equation? Determine whether the variable is isolated in each equation. Then determine whether the equation is solved for the variable. 8. c x d 15 Remember What You Learned 11. Create a general set of guidelines to solve any type of equation. Chapter 8 28 Course 3

31 Notes and Important Things

32 Notes and Important Things

33 8-4 NAME DATE PERIOD Study Guide and Intervention Solving Equations with Variables on Each Side Some equations, such as 3x 9 6x, have variables on each side of the equals sign. Use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side of the equals sign. Then solve the equation. Example 1 Solve 3x 9 6x. Check your solution. 3x 9 6x Write the equation. 3x 3x 9 6x 3x Subtract 3x from each side. 9 3x Simplify. 3 x Mentally divide each side by 3. To check your solution, replace x with 3 in the original equation. Check 3x 9 6x Write the equation. 3( 3) 9 6( 3) Replace x with The sentence is true. The solution is 3. Example 2 Solve 4a 7 5 2a. 4a 7 5 2a Write the equation. 4a 2a 7 5 2a 2a Add 2a to each side. 6a 7 5 Simplify. 6a Add 7 to each side. 6a 12 Simplify. a 2 Mentally divide each side by 6. The solution is 2. Check this solution. Exercises Solve each equation. Check your solution. 1. 6s 10 s 2. 8r 4r u 2u 4. 14t 8 6t 5. k 20 9k m 13 m 23 Lesson b 5 3b y 1 27 y h 72 4h z 8z x 8 5x d 3d 2 Chapter 8 29 Course 3

34 NAME DATE PERIOD 8-4 Skills Practice Solving Equations with Variables on Each Side Solve each equation. Check your solution. 1. 3w 6 4w 2. a 18 7a 3. 8c 5c d 10 6d 5. 2e 4e v 2v n 6 10n 8. 2y 27 5y 9. 8h 6h g 4g 11. 4x 9 6x c 15 2c t 10 7t z 6 7z e 12 7e k 6 8k d 10 6d a 9 6a k 3k t 4 10t c c n 5n y y b 2 7b m 2 6m g 5 7g s 1 8 2s 28. 9w 3 4w z 7 2z a 4a 12 Chapter 8 30 Course 3

35 8-4 NAME DATE PERIOD Practice Solving Equations with Variables on Each Side Solve each equation. Check your solution. 1. 9m 14 2m 2. 13x 32 5x 3. 8d 25 3d 4. t 27 4t 5. 7p 5 6p z 5 9z h h f f y 17 3y x 32 7x a 16 4a v 6v Find each number. 13. Fourteen less than five times a number is three times the number. Define a variable, write an equation, and solve to find the number. 14. Twelve more than seven times a number equals the number less six. Define a variable, write an equation, and solve to find the number. Write an equation to find the value of x so that each pair of polygons has the same perimeter. Then solve x + 6 x + 3 x + 9 x + 4 x GOLF For an annual membership fee of $500, Mr. Bailey can join a country club that would allow him to play a round of golf for $35. Without the membership, the country club charges $55 for each round of golf. Write and solve an equation to determine how many rounds of golf Mr. Bailey would have to play for the cost to be the same with and without a membership. 5x 5x 5x 5x 5x 5x x + 14 x x + 9 Lesson MUSIC Marc has 45 CDs in his collection, and Andrea has 61. If Marc buys 4 new CDs each month and Andrea buys 2 new CDs each month, after how many months will Marc and Andrea have the same number of CDs? Chapter 8 31 Course 3

36 8-4 NAME DATE PERIOD Word Problem Practice Solving Equations with Variables on Each Side Solve each problem by writing and solving an equation. 1. PLUMBING A1 Plumbing Service charges $35 per hour plus a $25 travel charge for a service call. Good Guys Plumbing Repair charges $40 per hour for a service call with no travel charge. How long must a service call be for the two companies to charge the same amount? 2. EXERCISE Mike s Fitness Center charges $30 per month for a membership. All-Day Fitness Club charges $22 per month plus an $80 initiation fee for a membership. After how many months will the total amount paid to the two fitness clubs be the same? 3. SHIPPING The Lone Star Shipping Company charges $14 plus $2 a pound to ship an overnight package. Discount Shipping Company charges $20 plus $1.50 a pound to ship an overnight package. For what weight is the charge the same for the two companies? 5. MONEY The Wayside Hotel charges its guests $1 plus $0.80 per minute for long distance calls. Across the street, the Blue Sky Hotel charges its guests $2 plus $0.75 per minute for long distance calls. Find the length of a call for which the two hotels charge the same amount. 4. MONEY Julia and Lise are playing games at the arcade. Julia started with $15, and the machine she is playing costs $0.75 per game. Lise started with $13, and her machine costs $0.50 per game. After how many games will the two girls have the same amount of money remaining? 6. COLLEGE Jeff is a part-time student at Horizon Community College. He currently has 22 credits, and he plans to take 6 credits per semester until he is finished. Jeff s friend Kila is also a student at the college. She has 4 credits and plans to take 12 credits per semester. After how many semesters will Jeff and Kila have the same number of credits? Chapter 8 32 Course 3

37 8-4 NAME DATE PERIOD Enrichment Famous Scientific Equations Many important laws or principles in physical science are described by equations. You may have already studied some of the equations on this page, or you may learn about them in future science classes. Match each statement with its equation. Then write the variables for the quantities listed. SCIENTIFIC PRINCIPLE EQUATION AND VARIABLES 1. Law of the Lever A lever will balance if the mass of object 1 times its distance from the fulcrum equals the mass of object 2 times its distance from the fulcrum. P 1 V 1 P 2 V 2 pressure at first time volume at first time pressure at second time 2. Newton s Second Law of Motion The acceleration on an object equals the applied force divided by the object s mass. volume at second time I R V voltage resistance 3. Ohm s Law The amount of current in an electrical circuit equals the voltage divided by the resistance. 4. Boyle s Law For a gas at a constant temperature, the product of the pressure and the volume remains constant. 5. Law of Universal Gravitation To compute the force of gravity between two objects, multiply their masses by the gravitational constant and then divide by the square of the distance between the objects. A m F m F G 1 m 2 d 2 current applied force mass of object acceleration mass of first object mass of second object distance between objects gravitational constant force of gravity m 1 d 1 m 2 d 2 mass of first object distance of first object from fulcrum Lesson 8 4 mass of second object distance of second object from fulcrum Chapter 8 33 Course 3

38 8-5 NAME DATE PERIOD Study Guide and Intervention Problem-Solving Investigation: Guess and Check You may need to use the guess and check strategy to solve some problems. Understand Plan Solve Check Example Determine what information is given in the problem and what you need to find. Select a strategy including a possible estimate. Solve the problem by carrying out your plan. Examine your answer to see if it seems reasonable. The school booster club spent $776 on ski passes for the school ski trip. Adult tickets cost $25 each and student tickets cost $18 each. They bought four times as many student tickets as adult tickets. Find the number of adult and student tickets purchased. Understand Plan Adult tickets cost $25 each and student tickets cost $18 each. They bought four times more student tickets than adult tickets. The total amount paid for the tickets was $776. Make a guess and check to see if it is correct. Remember, the number you guess for the student tickets must be four times more than the number you guess for adult tickets. Solve Check You need to find the combination that gives a total of $776. Make a list and use a to represent the number of adult tickets and s to represent the number of student tickets. Guess $25a $18s $776 Check If a 10, then s 4(10) 40 $25(10) $18(40) $970 too high If a 5, then s 4(5) 20 $25(5) $18(20) $485 too low If a 7, then s 4(7) 28 $25(7) $18(28) $679 still too low If a 8, then s 4(8) 32 $25(8) $18(32) $776 correct Exercises The booster club bought 8 adult tickets and 32 student tickets. Thirty-two student tickets is 4 times more than the 8 adult tickets. Since the cost of 8 adult tickets, $200, plus the cost of 32 student tickets, $576, equals $776, the guess is correct. Use the guess and check strategy to solve each problem. 1. JEWELRY Jana is making necklaces and bracelets. She puts 8 crystals on each necklace and 3 crystals on each bracelet. She needs to make 20 more necklaces than bracelets. She has 270 crystals. If she uses all the crystals, how many necklaces and bracelets can she make? 2. GIFT BAGS The ninth-grade class is filling gift bags for participants in a school fundraiser. They put 2 raffle tickets in each child s bag and 4 raffle tickets in each adult s bag. They made twice as many adult bags as child bags. If they had 500 raffle tickets, how many child bags and adult bags did they make? Lesson 8 5 Chapter 8 35 Course 3

39 Notes and Important Things

40 Notes and Important Things

41 NAME DATE PERIOD 8-5 Skills Practice Problem-Solving Investigation: Guess and Check Use the guess and check strategy to solve each problem. 1. NUMBER THEORY A number cubed is 1,728. What is the number? 2. MONEY Jackson has exactly $43 in $1, $5, and $10 bills. If he has 8 bills, how many of each bill does he have? 3. NUMBERS Jona is thinking of two numbers. One number is 18 more than twice the other number. The sum of the numbers is 48. What two numbers is Jona thinking of? 4. PACKAGES The packages in a mail driver s truck weigh a total of 950 pounds. The large packages weigh 20 pounds each and the small packages weigh 10 pounds each. If he has 10 more large packages than small packages, how many large and small packages are on the truck? 5. NUMBER THEORY One number is twice the other. The sum of the numbers is 246. What are the two numbers? 6. MOVIE RENTALS A movie rental store rented 3 times as many DVDs as videos. DVDs rent for $5 a day and videos rent for $3 a day. If the total rental income for a weekend was $2,160, how many DVDs and videos did the store rent? Chapter 8 36 Course 3

42 8-5 NAME Practice DATE PERIOD Problem-Solving Investigation: Guess and Check Mixed Problem Solving For Exercises 1 and 2, solve using the guess and check strategy. 1. NUMBER THEORY A number is squared and the result is 676. Find the number. 5. STATES Of the 50 United States, 14 have coastlines on the Atlantic Ocean, 5 have coastlines on the Gulf of Mexico, and one state has coastlines on both. How many states do not have coastlines on either the Atlantic Ocean or the Gulf of Mexico? 2. CRAFTS Sabrina has 12 spools of ribbon. Each spool has either 3 yards of ribbon, 5 yards of ribbon, or 8 yards of ribbon. If Sabrina has a total of 68 yards of ribbon, how many spools of each length of ribbon does she have? 6. TIME Melissa spent 7 1 minutes of the 2 last hour downloading songs from the Internet. What percent of the last hour did she spend downloading songs? Use any strategy to solve Exercises 3 7. Some strategies are shown below. PROBLEM-SOLVING STRATEGIES Draw a diagram. Make a table. Guess and check. 3. NUMBERS Among all pairs of whole numbers with product 66, find the pair with the smallest sum. 4. SHOPPING You are buying a jacket that costs $ If the sales tax rate is 7.75%, would it be more reasonable to expect the sales tax to be about $4.90 or $5.60? 7. VOLUNTEERING Greg helps his mother deliver care baskets to hospital patients each Saturday. Last Saturday at noon they had three times as many baskets left to deliver as they had already delivered. If they were delivering a total of 64 baskets that day, how many had they delivered by noon? Lesson 8 5 Chapter 8 37 Course 3

43 8-5 NAME DATE PERIOD Word Problem Practice Problem-Solving Investigation: Guess and Check Use the guess and check strategy to solve each problem. SKATES For Exercises 1 and 2, use the information below. It shows the income a sporting goods store received in one week for skate sharpening. Skate Sharpening Income for Week 6 Cost to Sharpen Cost to Sharpen Total Pairs of Total Income Hockey Skates Figure Skates Skates Sharpened from Skate Sharpening $6 a pair $4 a pair 214 $1, How many pairs of hockey skates and figure skates were sharpened during the week? 2. How much more did the sporting goods store earn sharpening hockey skates than figure skates? 3. FIELD TRIP At the science museum, the laser light show costs $2 and the aquarium costs $1.50. On a class field trip, each of the 30 students went to either the laser light show or the aquarium. If the teacher spent exactly $51 on tickets for both attractions, how many students went to each attraction? 5. READING MARATHON Mrs. Johnson s class broke the school reading record by reading a total of 9,795 pages in one month. Each student read a book that was either 245 pages or 360 pages. If 32 students participated in the reading marathon, how many students read each book? 4. NUMBERS Mr. Wahl is thinking of two numbers. The sum of the numbers is 27. The product of the numbers is 180. What two numbers is Mr. Wahl thinking of? 6. REWARDS The soccer coaches bought gifts for all their soccer players. Gifts for the girls cost $4 each and gifts for the boys cost $3 each. There were 32 more boy soccer players than girl soccer players. If the coaches spent a total of $411 on gifts for their players, how many boys and girls played soccer? Chapter 8 38 Course 3

44 8-6 NAME DATE PERIOD Lesson Reading Guide Inequalities Get Ready for the Lesson Read the introduction at the top of page 441 in your textbook. Write your answers below. 1. List three envelope sizes that Iko can use. Lesson How much will it cost to mail an invitation that weighs 2.5 ounces? Read the Lesson 3. Complete the table by providing the symbol used to represent each phrase. Phrase Symbol Phrase Symbol is greater than is at most is at least exceeds is fewer than is less than or equal to is more than is no less than 4. Explain the difference between an open and a closed circle on the graph of an inequality. 5. What does the arrow to the right or to the left indicate on the graph of an inequality? 6. Describe how to graph x Describe how to graph x 6. Remember What You Learned 8. Use a newspaper to find real-world situations in which relationships between quantities are described by phrases like no more than, at least, greater than, and at most. Chapter 8 39 Course 3

45 Notes and Important Things

46 Notes and Important Things

47 8-6 NAME DATE PERIOD Study Guide and Intervention Inequalities A mathematical sentence that contains or is called an inequality. When used to compare a variable and a number, inequalities can describe a range of values. Some inequalities use the symbols or. The symbol is read is less than or equal to. The symbol is read is greater than or equal to. Examples Write an inequality for each sentence. SHOPPING Shipping is free on orders of more than $100. Let c the cost of the order. c 100 RESTAURANTS The restaurant seats a maximum of 150 guests. Let g the number of guests. g 150 Inequalities can be graphed on a number line. An open or closed circle is used to show where the solutions start, and an arrow pointing either left or right indicates the rest of the solutions. An open circle is used with inequalities having or. A closed circle is used with inequalities having or. Examples Graph each inequality on a number line. d 2 Place a closed circle at 2. Then draw a line and an arrow to the left d 2 Place an open circle at 2. Then draw a line and an arrow to the right Exercises Write an inequality for each sentence. 1. FOOD Our delivery time is guaranteed to be less than 30 minutes. 2. DRIVING Your speed must be at least 45 miles per hour on the highway. Graph each inequality on a number line. 3. r 7 4. x 1 Chapter 8 40 Course 3

48 8-6 NAME DATE PERIOD Skills Practice Inequalities Write an inequality for each sentence. 1. SPORTS You need to score at least 30 points to take the lead. 2. SEASONS There are less than 12 hours of daylight each day in winter. 3. TRAVEL The bus seats at most 60 people. Lesson MONEY The coupon is good for any item that costs less than $ TESTS A score of at least 92 on the test is considered an A. 6. HEALTH The baby weighed more than 7 pounds at birth. 7. DRIVING Victor drives less than 12,000 miles per year. 8. TRAVEL Your waiting time will be 18 minutes or less. 9. SCHOOL TRIPS At least 15 students must sign up for the school trip. For the given value, state whether each inequality is true or false. 10. y 2 8, y u 1, u p 5 6, p a 3, a s 15, s d, d g 7, g k 4, k , z , m 4 z m Graph each inequality on a number line. 20. v b n w r h 7 Chapter 8 41 Course 3

49 8-6 NAME DATE PERIOD Practice Inequalities Write an inequality for each sentence. 1. JOBS Applicants with less than 5 years of experience must take a test. 2. FOOTBALL The home team needs more than 6 points to win. 3. VOTING The minimum voting age is GAMES You must answer at least 10 questions correctly to stay in the game. 5. DINING A tip of no less than 10% is considered acceptable. 6. MONEY The cost including tax is no more than $75. For the given value, state whether the inequality is true or false b 16, b f 8, f t 24, t 5 z m, m , z , d 4 5 d Graph each inequality on a number line. 13. y h c t x r 9 For Exercises 19 and 20, use the table that shows the literacy rate in several countries. 19. In which country or countries is the literacy rate less than 90%? 20. In which country or countries is the literacy rate at least 88%? Country Literacy Rate Albania 87% Jamaica 88% Panama 93% Senegal 40% Chapter 8 42 Course 3

50 8-6 NAME DATE PERIOD Word Problem Practice Inequalities 1. SPORTS Colin s time in the 400-meter run was 62 seconds. Alvin was at least 4 seconds ahead of Colin. Write an inequality for Alvin s time in the 400-meter run. 2. RESTAURANTS Before Valerie and her two friends left Mel s Diner, there were more than 25 people seated. Write an inequality for the number of people seated at the diner after Valerie and her two friends left. Lesson FARM LIFE Reggie has 4 dogs on his farm. One of his dogs, Lark, is about to have puppies. Write an inequality for the number of dogs Reggie will have if Lark has fewer than 4 puppies. 4. MONEY Alicia had $25 when she arrived at the fair. She spent t dollars on ride tickets and she spent $6.50 on games. Write an inequality for the amount of money Alicia had when she left the fair. 5. HEALTH Marcus was in the waiting room for 26 minutes before being called. He waited at least another 5 minutes before the doctor entered the examination room. Write an inequality for the amount of time Marcus waited before seeing the doctor. 7. HOMEWORK Nova spent one hour on Thursday, one hour on Saturday, and more than 2 hours on Sunday working on her writing assignment. Write an inequality for the amount of time she worked on the assignment. 6. POPULATION The population of Ellisville was already less than 250 before Bob and Ann Tyler and their three children moved away. Write an inequality for the population of Ellisville after the Tyler family left. 8. YARD WORK Harold was able to mow more than 3 4 of his lawn on Saturday night. Write an inequality for the fraction of the lawn that Harold will mow on Sunday. Chapter 8 43 Course 3

51 8-6 NAME DATE PERIOD Enrichment A Triangle Inequality A well-known inequality in geometry relates the measures of the three sides of any triangle. Here are two different statements of this inequality. A The Triangle Inequality The sum of the measures of any two sides of a triangle is greater than the measure of the third side. In any ABC, AB BC AC. C B Solve. 1. Consider three line segments with the measures 3, 4, and 8. Write a statement using the symbol to show that these three segments do not satisfy the triangle inequality. 2. Try to draw a triangle using the three segments in Problem 1. Describe what happens. Can the three measures be used to make a triangle? Write yes or no m, 2 m, 7 m 4. 5 cm, 8 cm, 11 cm 5. 5 in., 14 in., 7 in cm, 5 cm, 4 cm yd, 10 yd, 10 yd 8. 4 ft, 10 ft, 5 ft For each triangle, describe the possible measures of side AB. 9. B 10. C 11. A A A B 7 C C B Chapter 8 44 Course 3

52 8-7 NAME DATE PERIOD Lesson Reading Guide Solving Inequalities by Adding or Subtracting Get Ready for the Lesson Read the introduction at the top of page 445 in your textbook. Write your answers below. 1. Add 5 to each side of the inequality Write the resulting inequality and decide whether it is true or false. 2. Would it be colder in Erie or Philadelphia if the temperature in both cities dropped 10º? Explain. Reading the Lesson 3. How are solving an inequality and solving an equation similar? Lesson Explain what solving an inequality means. 5. Are x 7 and 7 x equivalent inequalities? Explain. 6. Are x 2 and 2 x equivalent inequalities? Explain. 7. Write an inequality equivalent to n 5, but use the symbol. Helping You Remember 8. Look up inequality in a dictionary. How does its meaning relate to what you have learned in this lesson? Chapter 8 45 Course 3

53 Notes and Important Things

54 Notes and Important Things

55 8-7 NAME DATE PERIOD Study Guide and Intervention Solving Inequalities by Adding or Subtracting Solving an inequality means finding values for the variable that make the inequality true. You can use the Addition and Subtraction Properties of Inequality to help solve an inequality. When you add or subtract the same number from each side of an inequality, the inequality remains true. Examples Solve each inequality. Check your solution. Then graph the solution on a number line. 9 r 5 Write the inequality. 9 5 r 5 5 Subtract 5 from each side. 4 r or r 4 Simplify. Check Solutions to the inequality should be greater than 4. Check this result by replacing r in the original inequality with two different numbers greater than 4. Both replacements should give true statements. To graph the solution, place an open circle at 4 and draw a line and arrow to the right x 7 4 x x 3 Write the inequality. Add 7 to each side. Simplify. Check Replace x in the original inequality with 3 and then with a number greater than 3. The solution is x 3. To graph the solution, place a closed circle at 3 and draw a line and arrow to the right. Exercises Solve each inequality. Check your solution. 1. t b r p a m 6 Solve each inequality and check your solution. Then graph the solution on a number line. 7. s d Chapter 8 46 Course 3

56 8-7 NAME DATE PERIOD Skills Practice Solving Inequalities by Adding or Subtracting Solve each inequality. Check your solution. 1. r e k 5 4. y n g m 8 8. t u x p z q k s 7 13 Write an inequality and solve each problem. 16. Two more than a number is less than eleven. Lesson Five less than a number is at least The difference between a number and 6 is no more than The sum of a number and 7 is more than The difference between a number and ten is greater than Four less than a number is less than 11. Solve each inequality and check your solution. Then graph the solution on a number line p w z s b v v m 7 11 Chapter 8 47 Course 3

57 8-7 NAME DATE PERIOD Practice Solving Inequalities by Adding or Subtracting Solve each inequality. Check your solution. 1. p t b k y n ( 12) 7. r a j h w ( 16) d ( 5) y b f Write an inequality and solve each problem. 16. Five less than a number is more than twenty. 17. Four more than a number is no more than twelve. 18. The sum of a number and 3.5 is at least The difference of a number and 5 is less than The sum of 12 and a number is at least Eleven less than a number is more than fifteen. Solve each inequality and check your solution. Then graph the solution on a number line. 22. n t p ( 5) x a ( 5) m s w 1 4 Chapter 8 48 Course 3