CHAPTER 4 COMPACTED MATHEMATICS EQUATIONS AND INEQUALITIES TOPICS COVERED: Variables Algebraic Expressions Algebraic Equations Inequalities

Transcription

1 CHAPTER 4 COMPACTED MATHEMATICS EQUATIONS AND INEQUALITIES TOPICS COVERED: Variables Algebraic Epressions Algebraic Equations Inequalities

2 Activity 4-1 Finding the Unknown NAME: Each type of fruit represents a single digit (0-9). Using what you know about the number system, determine the value of each fruit = = What single-digit number do and each represent in the epressions below? + = = What single-digit numbers do,, and represent in the epression below? - = 3 = = 20

3 Activity 4-2 Describing the Ideal School NAME: Vocabulary: A variable is a symbol used to represent one or more numbers. A variable epression is a mathematical phrase that uses numbers, variables, and operation symbols. Variable epressions are made up of one or more terms. A term is a number, a variable, or the product or quotient of a number and a variable. These variables represent information about a particular math class. s = number of students in the class g = number of girls in the class r = number of students in the class with red b = number of students in the class with black hair hair t = number of tetbooks given to each student h = number of hours of classes each day Tell what each of the following epressions represents. 1. s-g 4. s-b 2. 5h 5. st 3. b+r 6. g 100 s Write an epression for each of the following. 7. the number of students who do not have red hair 8. the number of tetbooks handed out to girls 9. the number of minutes of classes each day 10. the number of tetbooks handed out 11. the percentage of students who have black hair Write an equation for each of the following. there are 16 more students with black hair than students 12. with red hair 13. there are twice as many students as girls % of the students have black hair Translate each equation into words. Then make a table showing four pairs of sample values that fit the equation. 15. s = b b = 4r 17. s = 2g Write an equation that says that y is equal to 1.25 times. Then make a table of values that fit the equation. For, pick whole numbers ranging from 1 to 10. Write y as a decimal. Show all work on a separate sheet of paper.

4 Activity 4-3 Operation Code Words NAME: Often times certain code words can help you determine what type of operation is necessary in a word problem. Be careful, though, because the code word does not always mean just one operation. You must still read the problem carefully to make sure your choice of operations makes sense. ADDITION CODE WORDS Word Eample sum What is the sum of 5 and 3? in all There were 5 cats and 3 dogs. How many animals were there in all? altogether There were 5 boys and 3 girls at school. How many students are there altogether? total He scored 5 points and then 3 more points. How many total points did he score? spend/spent She bought a shirt for $25 and shoes for $50. How much money did she spend? combined How much is 5 and 3 combined? joined If 5 pirated joined 3 other pirates, how many pirates are there? plus What is 5 plus 3? both He has 5 cards and she had 3 cards. How many cards do they both have? and How much is 5 and 3? more 5 penguins were playing. 3 more arrived. How many penguins are there? SUBTRACTION CODE WORDS Word Eample difference What is the difference between 8 and 2? left 8 cats were in the room. 2 disappeared. How many cats are left? less What is 8 less 2? minus What is 8 minus 2? change He bought a watch $18 and gave the clerk a $20 bill. How much change did he get? profit She bought a stock for $200 and sold it for $800. How much profit did she make? off The $8 pencil s price was changed to $2. How many dollars off was the pencil? more The house has 8 people and the tent has 2. How many more live in the house? er words One can hold 2 liters and one hold 8 liters. How much bigger is the 8 liter can? MULTIPLY CODE WORDS Word Eample product What is the product of 5 and 7? times What is 5 times 7? twice 5 people ate. Twice as many sang. How many people were singing? Before the question words such as: each, per, every, one, apiece If 5 people each had 7 watermelons, how many watermelons were there? DIVIDE CODE WORDS Word Eample quotient What is the quotient of 12 and 3? separate How many separate groups of 3 can you make with 12 buttons?

5 equal How many equal groups of 3 can you make with 12 buttons? half What is half of 12? In the question words such as: each, per, every, one, apiece If 3 students had 12 strawberries, how many strawberries are there per each student?

6 Activity 4-4 Cow Pastures NAME: Pasture Page & Number: Pasture Page & Number: EQUATIONS: ANSWERS: EQUATIONS: ANSWERS: Pasture Page & Number: Pasture Page & Number: EQUATIONS: ANSWERS: EQUATIONS: ANSWERS: Pasture Page & Number: Pasture Page & Number: EQUATIONS: ANSWERS: EQUATIONS: ANSWERS: Pasture Page & Number: Pasture Page & Number: EQUATIONS: ANSWERS: EQUATIONS: ANSWERS:

7 Activity 4-5 Algebraic Epressions and Variables NAME: Eamples of epressions Phrases Epression Phrases Epression 9 more than a number the sum of 9 and a number a number plus 9 a number increased by 9 the total of and subtracted from a number a number minus 4 4 less than a number a number decreased by 4 the difference of h and 4 h - 4 Phrases Epression Phrases Epression a number divided by 5 t 6g the quotient of t and 5 divide a number by multiplied by g 6 times a number the product of g and 6 Eamples of equations Phrases Sity less than three times a number is $59. Three times the amount less 60 is equal to is equal to 60 subtracted from three times a number. A number times three minus 60 equals 59. Write each phrase as an algebraic epression or equation less than m 2. the quotient of 3 and y 3. the total of 5 and c 4. the difference of 6 and r 5. the product of k and 9 6. a number increased by 7 is the price decreased by $4 is twice as many points as Bob 8. $29 would be 18 points 9. after dividing the money 5 three more than 8 times as 10. ways, each person got $67 many trees is 75 trees 11. seven less than a number is thirteen plus v 13. si less than w divided into y 15. twice Michael s age 16. the quotient of n and fourteen divided by a four times the number of feet 18. number is 21 is 12 feet is seven fewer students the number of cats decreased 20. than last year by 17 is two and one-half times the one hundred increase by a 22. amount of interest is $2,500 number is 537 Write a word phrase for each variable epression a m 25. 8r 26. t + 17 Equation 3n 60 = 59

8 Activity 4-6 Modeling Equivalent Epressions NAME: Solve. 1. Jessica rode 9 miles farther than Roger rode. Let r represent the number of miles Roger rode. Write an epression for the number of miles Jessica rode. 2. Let m represent the number of children playing soccer. Those children are separated into 4 equal teams. Write an epression for the number of children on each team. 3. Glenda bought some apps for her tablet. Each app cost $5. Let n represent the number of apps she bought. Write an epression to show the total amount she spent. Write each phrase as a numerical or algebraic epression multiplied by added to n 6. r divided by 8 7. the product of 7 and m 8. the difference between 48 and the quotient of 18 and subtracted from t 11. the sum of w and 253 Write two word phrases for each epression. 12. t n Solve. 14. Write an epression that has two terms. Your epression should have a variable and a constant.

9 Activity 4-7 Modeling Equivalent Epressions NAME: Solve. 1. Cal bought 2 packs of 100 paper plates and 1 pack of 60 paper plates. Write an epression for the total number of plates that he bought. 2. The temperature dropped 25. Then the temperature went up 17. Let t represent the beginning temperature. Write an epression to show the ending temperature. 3. Jill purchased fruit juice boes for a party. She purchased 1 case of 44 boes and several packs containing 4 boes each. Let p represent the number of 4-bo packs she purchased. Write an epression for the total number of juice boes Jill purchased. Use the figures at the right for Eercises Write an epression for the perimeter of the triangle at the right. 5. Write an epression for the perimeter of the square. 6. Write an epression for the area of the square. Solve. 7. Write an epression that has four terms. Your epression should have three different variables and a constant. 8. Josef said that he could represent the amount of money he made last week with the epression: 24d 8n. Write a problem about the money Josef made last week.

10 Activity 4-8 Modeling Equivalent Epressions NAME: Circle the letter of the correct answer. The first one is done for you. solution: result: answer 1. Which of the following is the solution to an addition problem? A sum B plus C add 2. Which word phrase represents the following epression n 3? A the quotient of n and 3 B 3 less than n C n less than 3 3. Which word phrase represents the following epression 5m? A 5 fewer than m B m groups of 5 C m divided by 5 5. Which word phrase represents the following epression r 6? A the product of r and 6 B the quotient of r and 6 C take away 6 from r 4. Which of the following is the solution to a multiplication problem? A quotient B factor C product 6. Which word phrase represents the following epression 3 p? A 3 increased by p B 3 decreased by p C the difference of 3 and p Match the algebraic epressions A E to Eercises Some letters may be used more than once. Some letters may not be used at all. The first one is done for you. A. 9 B. 9 C. 9 D. 9 E less than C 8. the quotient of and 9 9. the sum of 9 and 10. the product of 9 and 11. more than decreased by 9 Solve. 13. Nicole had 38 beads. She lost some of them. This can be modeled by the epression 38. What does represent? 14. Wilhelm bought some shirts. He paid $12 for each shirt. This can be modeled by the epression 12. What does represent?

11 Activity 4-9 Algebraic Epressions NAME: Translate each phrase below into an algebraic epression. Write the correct letter net to each epression times a number A more than a number B decreased by a number C less than a number D one third of a number E more than 3 times a number F less than 3 times a number G less than 4 times a number H decreased by 4 times a number I less than twice a number J decreased by twice a number K less than half a number L times a number, increased by 4 M times a number, increased by 4 N 9 times the number times a number, increased by 8 O 8( 5) times the sum of a number and 8 P 8(2 5) more than 8 times a number Q times the sum of a number and 5 R 2(5 8) 19. twice the sum of 5 times a number and 8 S more than five-eighths of a number T 5( 8) times the sum of twice a number 5 U 2 and meters higher than altitude V meters per second slower than speed W degrees hotter than temperature X meters shorter than twice length Y years older than twice age Z $9 cheaper than 4 times price AA centimeters less than threefourths of length 4 3 BB 9

12 Activity 4-10 Writing Algebraic Epressions NAME: Connect the answer dots in order. Answers may be used more than once. Start a new line. 1. one more than a number 2. the product of -3 and a number 3. the value in cents of some nickels 4. three less than the product of 7 and a number 5. double a number 6. the sum of a number and its opposite 7. a number increased by 1 8. the product of 4 and three less than a number 9. a number fewer than a number decreased by three more than double a number 12. the result when a number is multiplied by 7 and this value is decreased by 3 Start a new line. 13. the sum of a number and itself 14. two less than a number Start a new line. 15. five decreased by a number 16. the value of some dimes decreased by the value of twice as many nickels Start a new line. 17. half a number 18. the value in cents of some quarters 19. double the sum of a number and si less than the product of five and a number 21. si more than the product of five and a number 22. five less than triple a number 23. five decreased by triple a number 24. a number increased by one quarter of a number 26. the value in dollars of some quarters Start a new line. 27. the product of two and a number increased by the value in cents of some dimes 29. the value in dollars of some dimes 30. two more than a number Start a new line. 31. the value in pennies of 5 pennies 32. the value in cents of some pennies Start a new line. 33. the opposite of two 34. the opposite of the opposite of the opposite of a number Circle: 35. a number 36. the opposite of a number

13 Activity 4-11 Algebraic Epressions and Variables NAME: 1. Which word phrase does not describe the epression 24? a. 24 decreased by b. the difference of 24 and c. 24 minus d. 24 less than Write each phrase as an algebraic epression or equation. 2. nine less than t divided by a number increased by n times the number of muffin pans 6. $15 times the number of hours 7. 8 less than a number 8. m squared 9. the square root of g 10. the quotient of five the sum of b squared and 11. divided by the sum of g h and k two times a number decreased by 5 is equal to 8 two times a number decreased by 8 is 23 a number divided by four is Write two word phrases for each variable epression. 18. q t 100 five less than a number is equal to 5 five times a number divided by 8 is one the difference of si squared and a number is nine Evaluate each epression if y = y y y y 24. 3y y 26. y y y Evaluate each epression if r = 6 and s = r s 31. r + s s 33. s - r 34. s sr 36. 4s 37. s + r + 16 Evaluate each epression if j = 2, k = 7, and m = k + m 39. j m 41. 2m 5j 42. 3m + j k + m 44. j + m k 45. 2j + k m 46. jm - k

14 Activity 4-12 Evaluating Epressions NAME: Evaluate each epression for the given value(s) of the variable(s). 1. a 4 when a b 9 when b 3 3. c 2 when c (9 d) 6 when d 3 5. g 2 23 when g h j when h 8 and j (n 2) m when n 5 and m 9 8. r(s 2 )(t) when r 2, s 3, and t 5 Use the given values to complete each table. 9. p 2(13 p) 10. v w 3v w 11. y 2 y Solve. 12. The sales ta in one town is 8%. So, the total cost of an item can be written as c 0.08c. What is the total cost of an item that sells for $12? 13. To change knots per hour to miles per hour, use the epression 1.15k, where k is the speed in knots per hour. A plane is flying at 300 knots per hour. How fast is that plane flying in miles per hour? 14. Lurinda ordered some boes of greeting cards online. The cost of the cards is $6.50n $3 where n is the number of boes ordered and $3 is the shipping and handling charge. How much will Lurinda pay if she orders 8 boes of cards?

15 Activity 4-13 Evaluating Epressions NAME: Use the given values to complete each table. 1. r 3.14 r 2 2. z a 2z a 3. y 10 2 (y 1) Solve Melinda is hauling water in her pickup truck. An old bridge has a maimum weight limit of 6,000 pounds. To find the weight of her truck, Melinda uses the epression 5, g, where g is the number of gallons of water she is hauling. Can Melinda safely drive her pickup across the bridge if she is hauling 120 gallons of water? Eplain. 5. A certain machine produces parts that are rectangular prisms. The surface area of each part is found by using the epression 2s 2 4sh, where s is the length of a side of the base and h is the height. What is the surface area of that part when s is 0.5 mm and h is 2 mm? mm 2 Three students incorrectly evaluated 4 2 2y for 3 and y 2. Use the table below to complete Eercises 6 9. Grayson 4 2 2y 4(3) 2 2( 2) 144 ( 4) What error did Grayson make? Emily 4 2 2y 4(3) 2 2( 2) 36 2( 2) 38( 2) 76 Pat 4 2 2y 4(3) 2 2(2) What error did Emily make? 8. What error did Pat make? 9. Show the correct way to complete the evaluation of 4 2 2y for 3 and y 2.

16 Activity 4-14 Evaluating Epressions NAME: Evaluate each epression for the given value of the variable. Show each step you used. The first one is done for you. 1. 3n 4 2 when n Substitute 2 for n Evaluate eponents Multiply. 22 Add (a 3) when a 5 2 (5 3) Substitute values. 2 Clear the parentheses. 3. r r 2 4 when r 8 Multiply Substitute values. 8 4 Multiply or divide from left to right, so divide first. 8 Multiply. Add. Use the given values to complete each table. The first one is done for you. 4. w 6(3 w) 5. c 2c 7 6. w w Solve. Show your work. 7. The height of horses is measured in hands. To find the height of a horse in inches, use the epression 4h, where h is the number of hands. Rosa has a horse that is 15 hands tall. How tall is Rosa s horse in inches? Rosa s horse is inches tall.

17 Activity 4-15 Algebraic Epressions and Variables NAME: Evaluate each epression if = 5, y = 4, and z = z z 4. + z 5. y + 3 z y 2 8. yz z + 4 Evaluate each epression if a = 8, b = 4, and c = a + b + c 11. 4b + a 12. cb a a b bc a b c 2a b 17. 3(b + a) - c 18. 2b 3c 4 2b 6(a c) b(b + a) - b c b Evaluate each epression if a = 12, b = 3, and c = 4, m = 9, n = 3. m a mn b n c 3 n (n + n) - m 27. 4c 3b m 2m 3( b c) b(c b) + c n ( b c) Solve the epression 7r for each given value of r. 31. r = r = r = 3,645 Solve the epression s - 11 for each given value of s. 34. s = s = s = 1,400 g Solve the epression for each given value of g g = g = g = Evaluate each algebraic epression for the given values of the variables. 40. (b 2 c) d 4, for b = 6, c = 4, and d = (a + b 2 ) c d, for a = 7, b = 2, c = 6, and d = s 2 3 t v, for s = 5, t = 7, and v = 3

18 Activity 4-16 Writing Equations to Represent Situations NAME: Determine whether the given value is a solution of the equation. Write yes or no ; w 10; w v 1.2; v d; d w 25; w t 32; t s 4; s p 3; p 11 Circle the letter of the equation that each given solution makes true. 9. m a 16 A 10 m 20 C 7m 26 A 2a 18 C 24 a 6 B m 4 15 Write an equation to represent each situation. D 18 m 2 B a D a Seventy-two people signed up for the 12. Mary covered her kitchen floor with soccer league. After the players were 10 tiles. The floor measures 6 feet long evenly divided into teams, there were by 5 feet wide. The tiles are each 3 feet 6 teams in the league and people on long and w feet wide. each team. Solve. 13. The low temperature was 35 F. This was 13 F lower than the daytime high temperature. Write an equation to determine whether the high temperature was 48 F or 42 F. 14. Kayla bought 16 bagels. She paid a total of $20. Write an equation to determine whether each bagel cost $1.50 or $ Write a real-world situation that could be modeled by the equation Then solve the problem. y

19 Activity 4-17 Writing Equations to Represent Situations NAME: Circle the letter of the value that makes each equation true d 8(12 6) m A m 6 C m 9 A d 18 C d 8 B m 3 D m 2 B d 48 D d a 3(10 2) A 6 C 16 A a 15 C a 40 B 8 D 4 B a 60 D a 20 For Eercises 5 7, use the table at the right that shows how many minutes certain mammals can stay under water. 5. A sperm whale can stay under water 7 times as long as minutes more than a platypus can. Write an equation that states the relationship of the minutes these two mammals can stay under water. Seal A sea cow can stay under water y minutes. This is 11 minutes longer than one-third the time a hippopotamus can. Write an equation that states the relationship of the minutes these two Sperm Whale 112 mammals can stay under water. Complete the table with 16 or Write an equation that includes division that relates the number of minutes a seal can stay under water to the number of minutes a sperm whale can stay under water. Animal Min Hippopotamus 15 Platypus 10 Sea Cow Solve. 8. Mr. Sosha teaches 4 math classes, with the same number of students in each class. Of those students, 80 are sith graders and 40 are fifth graders. Write an equation to determine whether there are 22, 25, or 30 students in each class. How many are in each class? 9. Write an equation that involves multiplication, addition, contains a variable, and has a solution of 8.

20 Activity 4-18 Writing Equations to Represent Situations NAME: Is the given value of the variable a solution of the equation? Write yes or no. The first one is done for you ; 4 yes w 10; w v 12; v ; p 7 p 5. 8 w 11; w t 20; t 5 Circle the letter of the equation that each given solution makes true. The first one is done for you g 7 A A 9g 16 B 9 3 B 8 g 1 C 3 18 C 11 g y m 9 A 7 y 1 A m 4 13 B 3y 6 B 7m 36 C y C 18 2 m 11. z a 8 A 5z 20 A 2a B 4 B a z a C z 3 7 C Rhonda has $13. She has one $5 bill, three $1 bills, and one other bill. Is the other bill a $1 bill or a $5 bill? Eplain. (1)$5 (3) Other bill $

21 Activity 4-19 Solving One Step Equations with Whole Numbers (Addition and Subtraction) Name: Solve by showing all the steps to the equations below. Round answers to nearest hundredth if necessary = = = = = = = = = = = = 33

22 Activity 4-20 Solving One Step Equations with Decimals (Addition and Subtraction) Name: Solve by showing all the steps to the equations below. Round answers to nearest hundredth if necessary = = = = = = = = = = = = 24.48

23 Activity 4-21 Solving One Step Equations with Integers (Addition and Subtraction) Name: Solve by showing all the steps to the equations below. Round answers to nearest hundredth if necessary = (-42) = = = = (-8) = = = (-46) = = = = -42

24 Activity 4-22 Solving One Step Algebraic Addition and Subtraction Equations Name: Tell whether the given value of the variable is a solution of the equation. 1. a -12 = 13; a = m = -31; m = = 20; = 5 Solve the Equation. Be sure to check your answer = r a 12 = z 24 = = s m = = t = a 2 7 Write the verbal sentence as an equation. Then solve the equation. 12. The difference of a number, b, and 8 is more than a number,, is A paperback version of a book costs $ This cost is $2.89 less than the cost of the hardcover version of the book. Write and solve an equation to find the hardcover cost of the book. 15. During a recent trip to the gym, you worked out with free weights and rode a stationary bike. You used a stationary bike for 28 minutes of the 75 minutes you spent at the gym. Write and solve an equation to find the number of minutes you spent working out with free weights. 16. The left- and right-hand margins on a sheet of paper are both 1.25 inches wide. The total width of the sheet of paper is 8.5 inches. Write and solve an equation to find the width of the tet area that lies between the margins.

25 Activity 4-23 Addition and Subtraction Equations Name: Solve each equation. Graph the solution on the number line r 2 r w 12 w m 1 8 m 4. t 1 3 t Use the drawing at the right for Eercises Write an equation to represent the measures of the angles. 6. Solve the equation to find the measure of the unknown angle. Use the drawing at the right for Eercises Write an equation to represent the measures of the angles. 8. Solve the equation to find the measure of the unknown angle. Write a problem for the equation 3 8. Then solve the equation and write the answer to your problem. 9.

26 Activity 4-24 Addition and Subtraction Equations Name: Solve each equation. 1. b b 2. s s n 12 n z 1.84 z 5. d ( 3) 7 d g 52 g Use the drawing at the right for Eercises Write an equation to represent the measures of the angles. 8. Solve the equation to find the measure of the unknown angle. Write and solve an equation to answer each question. 9. Kayla is 13 years old. Her uncle says that his age minus 22 is equal to Kayla s age. How old is Kayla s uncle? 10. Gavin wants to buy a jacket that sells for $ An advertisement says that net week that jacket will be on sale for $ How much will Gavin save if he waits until net week to buy the jacket? 11. Sierra sawed inches off the end of a board. The remaining board was inches long. How long was the board that Sierra started with? Write a problem for the equation Then solve the equation and write the answer to your problem. 12.

27 Activity 4-25 Addition and Subtraction Equations Name: Solve each equation. Graph the solution on the number line. The first one is done for you r 1 r 6 5 r r 2. 2 w 3 w 3. 5 m 2 m 4. t 5 0 t Use the drawings at the right for Eercises 5 6. The first one has been done for you. 5. Write an equation to represent the measures of the angles Solve the equation to find the measure of the unknown angle. 7. Mayumi has the boes shown at the right. The total number of objects in two of the boes is the same as the number of objects in the third bo. Write an equation to show the relationship of the number of objects in the boes. 8. How many objects are in the bo marked n? objects Write a problem for the equation 5 2. Then solve the equation and write the answer to your problem. 9.

28 Activity 4-26 Solving One Step Equations with Whole Numbers (Multiplication and Division) Name: Solve by showing all the steps to the equations below. Round answers to nearest hundredth if necessary = = = = = = = = = = = = 6

29 Activity 4-27 Solving One Step Equations with Decimals (Multiplication and Division) Name: Solve by showing all the steps to the equations below. Round answers to nearest hundredth if necessary = = = = = = = = = = = = 6.3

30 Activity 4-28 Solving One Step Algebraic Multiplication and Division Equations Name: Tell whether the given value of the variable is a solution of the equation. n 1. 5r = 125; r = a = 21; a = ; n 14 6 Describe how to solve the equation without actually solving. 8 = 72 14b = 8 m Solve the Equation. Be sure to check your answer. Round the solution to the nearest hundredth if necessary. 7. 4p = y = c d = m z a w r 16. t q = b = 8

31 Activity 4-29 Solving One Step Equations with Integers (Multiplication and Division) Name: Solve by showing all the steps to the equations below. Round answers to nearest hundredth if necessary = = = = = = = = = = = = -5

32 Activity 4-30 Multiplication and Division Equations Name: Solve each equation. Graph the solution on the number line. Check your work e 3 e w w m m 4. 5 k 2 k Use the drawing at the right for Eercises Write an equation you can use to find the length of the rectangle. 6. Solve the equation. Give the length of the rectangle. Solve. 7. Alise separated her pictures into 3 piles. Each pile contained 9 pictures. How many pictures did she have in all? Write and solve an equation to represent the problem. State the answer to the problem.

33 Activity 4-31 Multiplication and Division Equations Name: Solve each equation. 1. 8b 5.6 b s s n n z z d 70 d 6. t 4 t 3 9 Use the drawing at the right for Eercise The perimeter of the square at the right is 48 inches. What is the area of the square at the right? Eplain how you found your answer. Write and solve an equation to answer each question. 8. Jose is making model SUVs. Each SUV takes 5 tires. He used 85 tires for the models. How many model SUVs did Jose make? 9. Renee talked for 6 minutes on the phone. Nathan talked for n minutes. Nathan talked three times as long as Renee. How long did Nathan talk? 10. Sylvia rented a boat for $16.50 per hour. Her total rental fee was $ For how many hours did Sylvia rent the boat? Write a problem for the equation 0.5n Then solve the equation and write the answer to your problem. 11.

34 Activity 4-32 Multiplication and Division Equations Name: Solve each equation. Graph the solution on the number line. Check your work. The first is done for you m m 4 8 2m m a 2 a s s 4. 2 u 5 u Use the situation below to complete Eercises 5 8. The first one is done for you. Jim knows the length of his garden is 12 feet. He knows the area of the garden is 60 ft 2. What is the width of Jim s garden? 5. Fill in the known values in the picture at the right. 6. Write an equation you can use to solve the problem. 7. Solve the equation. w 8. Write the solution to the problem.

35 Activity 4-33 Generating Equivalent Epressions Name: Justify each step used to simplify the epression y y 2 2. (3 2) 2y 2 3. (3 2) 2y y 2 Simplify. 5. 3r n 2 r 5 2n v w 7 8v 2w 7. 4c 2 6c 3c 2 2c 3 8. z 3 5z 3z z 2 Write and simplify an epression for the perimeter of each figure A square has sides of 10. Write and simplify an epression for the perimeter of that square. 12. A rectangle has a length of 2 7 and a width of 3 8y. Write and simplify an epression for the perimeter of that rectangle. 13. In the space at the right, draw a triangle. Use an algebraic epression to label the length of each side. Write an epression for the perimeter of your triangle. Then simplify that epression.

36 Activity 4-34 Generating Equivalent Epressions Name: Simplify. 1. 3a a 2 5(a 2) 2. 8(v w) 7(v 2w) 3. 4c c 2 ) 2c 4. z 3 5(z 3) 4(2 2z 2 ) Write and simplify an epression for the perimeter of each figure A square has sides of 0.4. Write an epression for the perimeter of that square. Simplify the epression. 8. A rectangle has a length of 2( y) and a width of 3( y). Write an epression for the perimeter of that rectangle. Simplify the epression. Solve. 9. Peter collected soup for the food pantry. He packed 6 small boes with n cans of soup in each bo. He packed 4 boes with twice as many cans as in the small boes. Write and simplify an epression for the number of cans that Peter packed. 10. Netta faed n pages from the library. The library charges $1.50 per page. Later the same day, Netta faed n more pages from a local copy shop. The copy shop charges $1.25 per page plus a $2 convenience fee. Write and simplify an epression for the amount Netta spent on faes that day.

37 Activity 4-35 Generating Equivalent Epressions Name: Identify like terms in each list. The first one is done for you. 1. 5a b 43 2a b 2 2b 4 5a and 2a; b and 2b; 43 and 4 2. n 4n 3 2m 6m 5n 2n 3. 2d 5f 2g 7 3g g y Combine like terms to simplify. The first one is done for you. 5. 4r 5n 2 3r 9 2n 2 r 5n 2 7 2n 6. 3v w 8 2v c 2 6c 2c 2 5c 8. z 5e 3z e Perimeter is the distance around a figure. Write an epression for the perimeter of each figure. Be sure to combine like terms. The first one is done for you d 4c Circle the letter of the correct answer. 11. A square has sides of 6. Which epression shows the perimeter of that square? A 6 B 12 C 24 D A rectangle has a length of 4 5 and a width of 8 4. Which epression shows the perimeter of that rectangle? A 4 1 B 12 2 C 12 1 D 24 2

38 Activity 4-36 Distributive Property and Simplifying Epressions Name:

39

40 Activity 4-37 Combine Like Terms and Distributive Property Name: SIMPLIFY EACH EXPRESSION 1) 12r + 5h 7u + 11r + 9h 4r 3) 4(5a 3b) 5) 5(7 + 3) 12 2) 8y 6(3y 7) 4) 2(z + 3) + 3(5 3z) 6)

41 Find the perimeter of each of the shapes pictured below: (Combine Like Terms of the sides) 7) 8) 9) 3 2 4z + 3 4z y 3ab + 4a 2 2a 2 3b 2 6b 2 5ab ) 11) 12) a - b 3a - b 3a - b

42 Activity 4-38 Distributive Property and Simplifying Epressions Name: Simplify by distributing and collecting like terms. Show your work. The 1 st one is done for you. 1. 3(4 + 6) + 7 = 11. 6m + 3(2m + 5) + 7 = = (2 + 3) + 8 = 12. 5(m + 9) m = (4 + 4) = 13. 3m + 2(5 + m) + 5m = ( + 8) = 14. 6m (3m + 7) = 5. 3(7 + 2) + 8 = 15. 4(2m + 6) + 3(3 + 5m) = 6. 3(4 2 y ) + 4( y 3 ) = 16. 2(1 3 y y) + 3(4y y) = Simplify the epression first. Then evaluate the resulting epression for the given value of the variable (2 + 6) = if = (2m + 1) + 2(5m + 3) = if m = = = 13(4) + 30 = (2 + 7) = if = (7 + 5m) + 4(m + 6) = if m = (9 + 4) = if = (4m + 5) + 8(3m + 1) = if m = (4 + 7) + = if = (8 + m) + 2(7m 7) = if m = 3

43 Activity 4-39 Solving Two-Step Equations Name: Directions: Solve problems below mathematically. Steps: 1. Get rid of constants by doing opposite operation 2. Get rid of coefficient by doing opposite operation. 1.) = ) = 74 3.) = 11 4.) = 8 5.) 2-7 = ) = 35 7.) = ) = 23 9.) = ) = 26

44 Activity 4-40 Solving 2 Step Equations Name: Guided Practice: Solve the equations below using inverse operations, distributive property and combining like terms = n = = v = = 12b = e 9 = n = (5-7) = (2 + 4) = 112

45 Activity 4-41 Solving 2 Step Equations Name: Independent Practice: Solve the equations below using inverse operations, distributive property and combining like terms. 1. 4y 3 = n 6 4 = 4 p 2 7 = c 1.7 = d = m 4 = = = = 6 + 2(2 6) = 3 + 6(4 8)

46 Activity 4-42 Writing Equations Name: Equation Solution DIS collected a total of $ for the St. Jude s children s fund. Henry donated $57.50 more than Karen. Write an algebraic epression to determine how much money Karen donated and then solve. While installing water pipes Chaney had an unknown length of piping. If she cut her piping into 5 pieces, write an algebraic equation that can be used to determine the length of each pipe if the total length was 37.5 feet. Solve as well. Evan grew his hair to an unknown length to be donated to charity. Alesei had grown his hair 3 inches less than Evan s. Write an algebraic equation to determine Evan s hair length if Alesei s is 14 inches long. Their total hair length is 31 inches. As a member of a music club, Eric can order CD s for $14.99 each. The music club also charges $4.99 per shipment. Eric s most recent order was for $ Write and solve an equation to determine how many CD s Eric ordered. Samantha works for a Pizza Hut on the weekend delivering pizzas. She earns $7 per hour plus $1.50 for each delivery made. Write and solve an epression to find out how much money she will make after working 15.5 hours and making 8 deliveries? Suppose Helena is able to run twice as fast as Jordan. Write and solve an equation to find out how fast Jordan runs if Helena can run 14.5 mph.

47 Activity 4-43 Solving Multistep Equations Name: