Math Answers And Wb 2p6g5t

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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME ______________________________________________ DATE______________ PERIOD _____

1-1

Skills Practice

Practice Variables and Expressions

Write an algebraic expression for each verbal expression.

1. the sum of a number and 10

2. 15 less than k

x 10

1. the difference of 10 and u

k 15

74 3y

5. 15 decreased by twice a number

6. the difference of 17 and 5 times a number

8 3x

4. 74 increased by 3 times y

33j

2m 6

5. 8 increased by three times a number

18 x

3. the product of 33 and j

4. 6 more than twice m

18q

2. the sum of 18 and a number

10 u

7. three fourths the square of b

3 b2 4

8. 9 less than g to the fourth power

2y 2

x2 91

g4 9

8. two fifths the cube of a number

2 x3 5

Evaluate each expression.

11. 53 125

12. 33 27

13. 102 100

14. 24 16

15. 72 49

16. 44 256

17. 73 343

18. 113 1331

Write a verbal expression for each algebraic expression. 19–26. Sample answers 20. 52

19. 9a

the product of 9 and a

5 squared

21. c 2d

22. 4 5h

23. 2b2

24. 7x3 1

the sum of c and twice d

Glencoe Algebra 1

2 times b squared 25. p4 6q

the difference of 4 and 5 times h 1 less than 7 times x cubed

p to the fourth power plus 6 times q

Chapter 1

are given.

26. 3n2 x

3 times n squared minus x

8

Answers

Glencoe Algebra 1


10. 34 81

9. 112 121

10. 83 512

11. 54 625

12. 45 1024

13. 93 729

14. 64 1296

15. 105 100,000

16. 123 1728

17. 1004 100,000,000

Write a verbal expression for each algebraic expression. 18–25. Sample answers are given. 18. 23f 19. 73

the product of 23 and f

seven cubed

2

21. 4d3 10

22. x3 y4 x cubed

23. b2 3c3

k5 24. 6

25.

20.

5m2

2 more than 5 times m squared times y to the fourth power

4 times d cubed minus 10

b squared minus 3 times c cubed 4n2 7

one sixth of the fifth power of k

one seventh of 4 times n squared

26. BOOKS A used bookstore sells paperback fiction books in excellent condition for $2.50 and in fair condition for $0.50. Write an expression for the cost of buying e excellent-condition paperbacks and f fair-condition paperbacks. 2.50e 0.50f 27. GEOMETRY The surface area of the side of a right cylinder can be found by multiplying twice the number by the radius times the height. If a circular cylinder has radius r and height h, write an expression that represents the surface area of its side. 2rh Chapter 1

9

Glencoe Algebra 1

(Lesson 1-1)

A3

9. 82 64



Answers

7. the product of 2 and the second power of y

6. 91 more than the square of a number

15 2x

17 5x

Page A3

3. the product of 18 and q

10:26 AM

Variables and Expressions Write an algebraic expression for each verbal expression.

Lesson 1-1

1-1

5/10/06

Chapter 1

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1-1

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Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

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1-1

Word Problem Practice

Enrichment

Toothpick Triangles

BLOCKS For Exercises 5–7, use the following information. A toy manufacturer produces a set of blocks that can be used by children to build play structures. The product packaging team is analyzing different arrangements for packaging their blocks. One idea they have is to arrange the blocks in the shape of a cube, with b blocks along one edge.

Figure 2

Figure 3

3; 5; 7

b

2. How many toothpicks does it take to make up the perimeter of each image?

3; 4; 5

b

10

Glencoe Algebra 1


Glencoe Algebra 1

7. The team finally decides that their favorite package arrangement is to take 2 layers of blocks off the top of a cube measuring b blocks along one edge. Write an expression representing the number of blocks left behind after the top two layers are removed. b 3 2b 2 or (b 2) b 2

2

Chapter 1

6. The packaging team decides to take one layer of blocks off the top of this package. Write an expression representing the number of blocks in the top layer of the package. b 2


4. TIDES The difference between high and low tides along the Maine coast in November is 19 feet on Monday and x feet on Tuesday. Write an expression to show the average rise and fall of the tide 19 x for Monday and Tuesday.

3. Sketch the next three figures in the pattern.

b3


6136 or about 198 shows per year y

5. Write an expression representing the total number of blocks packaged in a cube measuring b blocks on one edge.

Figure 4

(Lesson 1-1)

A4

3. THEATER Howard Hughes, Professor Emeritus of Texas Wesleyan College, reportedly attended a record 6136 theatrical shows. Write an expression to represent the average number of theater shows attended if he accumulated the record over y years. Use the expression to find the average number of shows Mr. Hughes attended per year if he went to the theater for 31 years.

Figure 5

Figure 6

4. Continue the pattern to complete the table. Image Number

1

2

3

4

5

6

7

8

9

10

Number of toothpicks

3

5

7

9

11

13

15

17

19

21

Number of toothpicks in Perimeter

3

4

5

6

7

8

9

10

11

12

5. Let the variable n represent the figure number. Write an expression that can be used to find the number of toothpicks needed to create figure n. 2n 1 6. Let the variable n represent the figure number. Write an expression that can be used to find the number of toothpicks in the perimeter of figure n. n 2

Chapter 1

11

Glencoe Algebra 1

Page A4

1. How many toothpicks does it take to create each figure? b

10:26 AM

Figure 1

Lesson 1-1

Variable expressions can be used to represent patterns and help solve problems. Consider the problem of creating triangles out of toothpicks shown below.

Answers

2. TECHNOLOGY There are 1024 bytes in a kilobyte. Write an expression that describes the number of bytes in a computer chip with n kilobytes. 1024 n or 1024n

5/10/06

Variables and Expressions 1. SOLAR SYSTEM It takes Earth about 365 days to orbit the sun. It takes Uranus about 85 times as long. Write a numerical expression to describe the number of days it takes Uranus to orbit the sun. 365 85

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NAME ______________________________________________ DATE______________ PERIOD _____

1-2 1-2

Lesson Reading Guide Order of Operations

Study Guide and Intervention

Evaluate Rational Expressions

Numerical expressions often contain more than one operation. To evaluate them, use the rules for order of operations shown below.

Read the introduction to Lesson 1-2 in your textbook.

Order of Operations

represents the number of hours over 100 used by Nicole in a given month.

Step Step Step Step

Example 1

Read the Lesson

1 2 3 4

Evaluate expressions inside grouping symbols. Evaluate all powers. Do all multiplication and/or division from left to right. Do all addition and/or subtraction from left to right.

parentheses, brackets, braces, and fraction bars

Multiply 2 and 4. Add 7 and 8. Subtract 4 from 15.

b. 3(2) 4(2 6) 3(2) 4(2 6) 3(2) 4(8) 6 32

2. What does evaluate powers mean? Use an example to explain.

Sample answer: To evaluate a power means to find the value of the power. To evaluate 43, find the value of 4 4 4.

38

3 23 4 3 38 3 23 42 3 42 3

Add 2 and 6.

19 3 4 62

e. multiplication 51 729 9

f. evaluate powers 2

What You Learned

Glencoe Algebra 1

4. The sentence Please Excuse My Dear Aunt Sally (PEMDAS) is often used to the order of operations. The letter P represents parentheses and other grouping symbols. Write what each of the other letters in PEMDAS means when using the order of operations.

E—exponents (powers), M—multiply, D—divide, A—add, S—subtract

12

Answers

Glencoe Algebra 1


d. 69 57 3 16 4 division


c. 17 3 6 multiplication

Find 4 squared. Add 2 and 16. Multiply 3 and 18.

Multiply left to right. Add 6 and 32.

Evaluate power in numerator.

11 42 3

Add 3 and 8 in the numerator.

11 16 3

Evaluate power in denominator.

11 48

Multiply.

Exercises Evaluate each expression. 1. (8 4) 2 8

2. (12 4) 6 96

3. 10 2 3 16

4. 10 8 1 18

5. 15 12 4 12

6. 3

7. 12(20 17) 3 6 18

8. 24 3 2 32 7

9. 82 (2 8) 2 6

4(52) 4 3 4(4 5 2)

16. 1

Chapter 1

8(2) 4 84

4 32 12 1

12. 6

2 42 82 (5 2) 2

15. 2 35

52 3 1 20(3) 2(3) 3

18. 3

10. 32 3 22 7 20 5 27 11. 1 13. 250 [5(3 7 4)] 2

15 60 30 5

14. 2 17.

13

4 32 3 2

82 22 (2 8) 4

Glencoe Algebra 1

(Lesson 1-2)

A5

b. 26 8 14 subtraction

Divide 12 by 3.

b. 2

3. Read the order of operations on page 11 in your textbook. For each of the following expressions, write addition, subtraction, multiplication, division, or evaluate powers to tell what operation to use first when evaluating the expression. a. 400 5[12 9] addition


a. 3[2 (12 3)2] 3[2 (12 3)2] 3(2 42) 3(2 16) 3(18) 54

Answers

a. 7 2 4 4 7244784 15 4 11

1. The first step in evaluating an expression is to evaluate inside grouping symbols. List four types of grouping symbols found in algebraic expressions.

Example 2


Page A5

4.95 represents the 0.99 represents the cost of each additional hour after 100 hours, and (117 100) In the expression 4.95 0.99(117 100),

regular monthly cost of internet service,

Chapter 1

10:26 AM

Order of Operations

Get Ready for the Lesson

Lesson 1-2

1-2

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

1-2


NAME ______________________________________________ DATE______________ PERIOD _____

1-2 1-2

(continued)

Order of Operations

Skills Practice

5/10/06

Order of Operations Evaluate each expression.

Evaluate Algebraic Expressions

Algebraic expressions may contain more than one operation. Algebraic expressions can be evaluated if the values of the variables are known. First, replace the variables with their values. Then use the order of operations to calculate the value of the resulting numerical expression.

2. (9 2) 3 21

3. 4 6 3 22

4. 28 5 4 8

5. 12 2 2 16

6. (3 5) 5 1 41

7. 9 4(3 1) 25

8. 2 3 5 4 21

Evaluate x3 5( y 3) if x 2 and y 12. 23 5(12 3) 8 5(12 3) 8 5(9) 8 45 53

Replace x with 2 and y with 12. Evaluate 23. Subtract 3 from 12. Multiply 5 and 9. Add 8 and 45.

The solution is 53.

4 5

3 5

4. x3 y z2 27

5. 6a 8b 9

6. 23 (a b) 21

8. 2xyz 5 53

9. x(2y 3z) 36

10. (10x)2 100a 480

z2 y2 7 4 x

7 8

25ab y xz

16. 1

2

Chapter 1

冢 yz 冣

1

2

13 16

5a2b 16

3 5

21 12. a2 2b 1 25 (z y)2 1 x 2

15.

3 5

18. (z x)2 ax 5

17. 25 y

6 xz y 2z 11

20.

14

冢 z y x 冣冢 y z x 冣

1 24

21. 1

Glencoe Algebra 1


Glencoe Algebra 1

冢 xz 冣

3xy 4 11. 7x

14. 6xz 5xy 78

13. 2

19.

3 5


y2 9

7. 2 x 4

3. x y2 11


2. 3x 5 1

11. 14 7 5 32 1

12. 6 3 7 23 22

13. 4[30 (10 2) 3] 24

14. 5 [30 (6 1)2] 10

15. 2[12 (5 2)2] 42

16. [8 2 (3 9)] [8 2 3] 6

Evaluate each expression if x 6, y 8, and z 3. 17. xy z 51

18. yz x 18

19. 2x 3y z 33

20. 2(x z) y 10

21. 5z ( y x) 17

22. 5x ( y 2z) 16

23. x2 y2 10z 70

24. z3 ( y2 4x) 67

y xz 2

25. 13

Chapter 1

3y x2 z

26. 20

15

Glencoe Algebra 1

(Lesson 1-2)

A6

1. x 7 9

10. 10 2 6 4 26

Lesson 1-2

Exercises Evaluate each expression if x 2, y 3, z 4, a , and b .

9. 30 5 4 2 12

Answers

x3 5( y 3)

Page A6

1. (5 4) 7 63

10:26 AM

Example

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944


NAME ______________________________________________ DATE______________ PERIOD _____

1-2 1-2

Practice Order of Operations 2. 9 (3 4) 63

3. 5 7 4 33

4. 12 5 6 2 5

5. 7 9 4(6 7) 11

6. 8 (2 2) 7 14

7. 4(3 5) 5 4 12

8. 22 11 9 32 9

9. 62 3 7 9 48

10. 3[10 (27 9)] 21

11. 2[52 (36 6)] 62

12. 162 [6(7 4)2] 3

52 4 5 4 2 5(4)

5t 100; 1400 students

7 32 1 4 2 2

(2 5)2 4 3 5

14. 26 2

15. 2

17. b2 2a c2 89

18. 2c(a b) 168

19. 4a 2b

20. (a2 4b) c 8

21. c2 (2b a) 96

50

2s 671; 8749 ft

2c3 ab 4

22. 39

23. 5

2(a b)2 9 24. 5c 10

b2 2c2 25. acb

Ann Carlyle is planning a business trip for which she needs to rent a car. The car rental company charges $36 per day plus $0.50 per mile over 100 miles. Suppose Ms. Carlyle rents the car for 5 days and drives 180 miles. 26. Write an expression for how much it will cost Ms. Carlyle to rent the car.

5(36) 0.5(180 100) 27. Evaluate the expression to determine how much Ms. Carlyle must pay the car rental company. $220.00

GEOMETRY For Exercises 28 and 29, use the following information.

Glencoe Algebra 1

The length of a rectangle is 3n 2 and its width is n 1. The perimeter of the rectangle is twice the sum of its length and its width.


CAR RENTAL For Exercises 26 and 27, use the following information.


7

3. TRANSPORTATION The Plaid Taxi Cab Company charges $1.75 per enger plus $3.45 per mile for trips less than 10 miles. Write and evaluate an expression to find the cost for Max to take a Plaid taxi 8 miles to the airport.

6–8, use the following information. During a long weekend, Devon paid a total of x dollars for a rental car so he could visit his family. He rented the car for 4 days at a rate of $36 per day. There was an additional charge of $0.20 per mile after the first 200 miles driven. 6. Write an algebraic expression to represent the amount Devon paid for additional mileage only. x – (36 4)

$1.75 $3.45m; $29.35

7. Write an algebraic expression to represent the number of miles over 200 miles that Devon drove the rented car.

4. GEOMETRY The area of a circle is related to the radius of the circle such that the product of the square of the radius and a number gives the area. Write and evaluate an expression for the area of a circular pizza below. Approximate as 3.14.

x – (36 4) 0.20

8. How many miles did Devon drive overall if he paid a total of $174 for the car rental? 350 mi

r 2; 153.86 in2

7 in.

28. Write an expression that represents the perimeter of the rectangle.

2[(3n 2) (n 1)] 29. Find the perimeter of the rectangle when n 4 inches. 34 in. Chapter 1

16

Answers

Glencoe Algebra 1

Chapter 1

17

Glencoe Algebra 1

(Lesson 1-2)

A7

bc2 a c

c2

CONSUMER SPENDING For Exercises

Answers

2. GEOGRAPHY Guadalupe Peak in Texas has an altitude that is 671 feet more than double the altitude of Mount Sunflower in Kansas. Write and evaluate an expression for the altitude of Guadalupe Peak if Mount Sunflower has an altitude of 4039 feet.

Evaluate each expression if a 12, b 9, and c 4. 16. a2 b c2 137

5. BIOLOGY Lavania is studying the growth of a population of fruit flies in her laboratory. She notices that the number of fruit flies in her experiment is five times as large after any six-day period. She observes 20 fruit flies on October 1. Write and evaluate an expression to predict the population of fruit flies Lavania will observe on October 31. 20 55; 62,500 flies

Page A7

1. (15 5) 2 20

1. SCHOOLS Jefferson High School has 100 less than 5 times as many students as Taft High School. Write and evaluate an expression to find the number of students at Jefferson High School if Taft High School has 300 students.

10:26 AM

Order of Operations


13. 1


Lesson 1-2

1-2

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-2 1-2

Enrichment

Graphing Calculator Activity

The Four Digits Problem

Answers will vary. Sample answers are given.

(4 3) (2 1)

19 3(2 4) 1

3

(4 3) (2 1)

20

4

(4 2) (3 1)

5

(4 2) (3 1)

6

4312

8 9

4 2 (3 1)

10

4321

21 (4 3)

37

31 2 4

21

(4 3) (2 1)

38

42 (3 1)

22

21 (4 3)

39

42 (3 1)

40

41 (3 2)

41

43 (2 1)

42

43 (2 1)

23 31 (4 2) 24

(2 4) (3 1)

25 (2 3) (4 1) 24 (3 1) 26

43 42 13

27

3 (4 1)

44

43 (2 1)

28

21 3 4

45

43 (2 1)

29

2(4 +1) 3

46

43 (2 1)

2

11 (4 3) (2 1) 12 (4 3) (2 1) 13 (4 3) (2 1) 14 (4 3) (2 1)

30 (2 3) (4 1) 34 (2 1) 31

47

31 42

48

4 (3 1) 2

15

2(3 4) 1

32

42 (3 1)

49

41 23

16

(4 2) (3 1)

33

21 (3 4)

50

41 32

17

3(2 4) 1

34

2 (14 3)

Example 2

4y 5x

Evaluate xy if x 4 and y 12.

Evaluate the expression and display the answer as a fraction. ALPHA [:] 12 STO ALPHA [Y] ALPHA [:] Keystrokes: 4 STO ( 4 ALPHA [Y] ) ) ALPHA [Y] — ( 5 MATH 1 ENTER . Exercises Evaluate each expression if a = 4, b = 6, x = 8, and y = 12. For Exercises 4-6, express answers as fractions. 1. bx ay b

40

a2

b 4. 2 2 x b

11 14

2. a[ x (y a)2]

3. a3 (y b)2 x2

2a(x b) 5.

b 3 a b 5b 6.

68

92 3

xy 9b

2 3

22 7

[ (

)

2

]

y a(x 1)

Answers will vary. Using a calculator is a good way to check your solutions.

Chapter 1

18

Glencoe Algebra 1


Glencoe Algebra 1

Does a calculator help in solving these types of puzzles? Give reasons for your opinion.

You can also use a colon, which is the ALPHA function above the decimal key, to chain commands together. This process is called concatenation. Using the colon in Example 1, the keystrokes become 8 STO ALPHA [A] ALPHA [:] ALPHA [A] x 2 — 4 ALPHA [A] + 6 ENTER .


3(4 1) 2 4321

34 (2 1)

Chapter 1

19

Glencoe Algebra 1

(Lesson 1-2)

A8

7

36

Answers

2

35 2(4 +1) 3


(2 3) (4 1)

Page A8

Example 1 Evaluate a2 4a 6 if a 8. Store 8 as the value for a. Keystrokes: 8 STO ALPHA [A] ENTER Enter the expression and press ENTER to evaluate. Keystrokes: ALPHA [A] x 2 — 4 ALPHA [A] + 6 ENTER

Express each number as a combination of the digits 1, 2, 3, and 4. 18

Key

10:26 AM

One well-known mathematic problem is to write expressions for consecutive numbers beginning with 1. On this page, you will use the digits 1, 2, 3, and 4. Each digit is used only once. You may use addition, subtraction, multiplication (not division), exponents, and parentheses in any way you wish. Also, you can use two digits to make one number, such as 12 or 34.

1 (3 1) (4 2)

STO

5/10/06

Using The

When evaluating algebraic expressions, it is sometimes helpful to use the store key STO on the calculator, especially to check solutions.

Lesson 1-2

1-2

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Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

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NAME ______________________________________________ DATE______________ PERIOD _____

1-3 1-3

Lesson Reading Guide


Open Sentences

Solve Equations

How is the open sentence different from the expression 15.50 5n?

The open sentence has two expressions ed by the symbol.

Example 1 Find the solution set of 3a 12 39 if the replacement set is {6, 7, 8, 9, 10}.

Read the Lesson 1. How can you tell whether a mathematical sentence is or is not an open sentence?

An open sentence must contain one or more variables.

Replace a in 3a 12 39 with each value in the replacement set.

2. How would you read each inequality symbol in words? Words

is less than

is greater than is less than or equal to

is greater than or equal to

a. Describe how you would find the solutions of the equation.

Replace n with each member of the replacement set. The of the replacement set that make the equation true are the solutions. b. Describe how you would find the solutions of the inequality.

Replace n with each member of the replacement set. The of the replacement set that make the inequality true are the solutions. c. Explain how the solution set for the equation is different from the solution set for the inequality.

The solution set for the equation contains only one number, 3. The solution set for the inequality contains the four numbers 0, 1, 2, and 3.

Glencoe Algebra 1

What You Learned 4. Look up the word solution in a dictionary. What is one meaning that relates to the way we use the word in algebra?

Sample answer: answer to a problem 20

Answers

8 b Simplify. 9

false false

8 9

The solution is .

true false

Glencoe Algebra 1

Exercises Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

3. Consider the equation 3n 6 15 and the inequality 3n 6 15. Suppose the replacement set is {0, 1, 2, 3, 4, 5}.

Chapter 1

2(4) b Add in the numerator; subtract in the denominator. 3(3)

false

1

1

Find the solution of each equation if the replacement sets are X , , 1, 2, 3 4 2 and Y {2, 4, 6, 8}. 1 2

5 2

1. x {2}

2. x 8 11 {3}

3. y 2 6 {8}

4. x2 1 8 {3}

5. y2 2 34 {6}

6. x2 5 5

12

7. 2(x 3) 7

1 4

9 4

8. ( y 1)2 {2}

1 16

14

9. y2 y 20 {4}

Solve each equation. 10. a 23 1 7 1 4

5 8

7 8

11. n 62 42 20 18 3 23

12. w 62 32 324 15 6 27 24

13. k

14. p 3

15. s 3

16. 18.4 3.2 m 15.2

17. k 9.8 5.7 15.5

18. c 3 2 5

Chapter 1

21

1 2

1 4

3 4

Glencoe Algebra 1

(Lesson 1-3)

A9

2(3 1) b Original equation 3(7 4)

Since a 9 makes the equation 3a 12 39 true, the solution is 9. The solution set is {9}.


2(3 1) 3(7 4)

Solve b.

Answers

3(6) 12 39 → 30 39 3(7) 12 39 → 33 39 3(8) 12 39 → 36 39 3(9) 12 39 → 39 39 3(10) 12 39 → 42 39

Example 2

Page A9

A mathematical sentence with one or more variables is called an open sentence. Open sentences are solved by finding replacements for the variables that result in true sentences. The set of numbers from which replacements for a variable may be chosen is called the replacement set. The set of all replacements for the variable that result in true statements is called the solution set for the variable. A sentence that contains an equal sign, , is called an equation.


Inequality Symbol

10:26 AM

Open Sentences


Lesson 1-3

1-3

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____


NAME ______________________________________________ DATE______________ PERIOD _____

1-3 1-3

(continued)

Open Sentences

→ → → → →

4 10 7 10 10 10 13 10 16 10

false false false true true

Since replacing a with 7 or 8 makes the inequality 3a 8 10 true, the solution set is {7, 8}. Exercises Find the solution set for each inequality if the replacement set is X {0, 1, 2, 3, 4, 5, 6, 7}. 2. x 3 6

{3, 4, 5, 6, 7}

3. 3x 18

{0, 1, 2, 3, 4, 5, 6, 7}

{7}

x 5. 2 5

3x 6. 2 8

7. 3x 4 5

8. 3(8 x) 1 6

9. 4(x 3) 20

{4, 5, 6, 7}

no numbers

{4, 5, 6, 7}

{7}

{0, 1, 2, 3, 4, 5} {2, 3, 4, 5, 6, 7}

Find the solution set for each inequality if the replacement sets are

14

1 2

X , , 1, 2, 3, 5, 8 and Y {2, 4, 6, 8, 10}. 10. x 3 5

11. y 3 6

{3, 5, 8}

{2, 4, 6, 8, 10}

x 2

13. 4

16. 4x 1 4

1 19. 3x 2 4

1 1 , 4 2

Chapter 1

15. 2

{8, 10}

{2, 4}

17. 3x 3 12

18. 2( y 1) 18

20. 3y 2 8

1 21. (6 2x) 2 3 2

{3, 5, 8}

{2}

{8, 10}

{2, 3, 5, 8} 22

Glencoe Algebra 1


Glencoe Algebra 1

{1, 2, 3, 5, 8}

{6, 8, 10} 2y 5

14. 2

14, 12, 1, 2, 3, 5

y 4

12. 8y 3 51


x 4. 1 3

4. 3b 15 48 11

5. 4b 12 28 10

6. 3 0 12

36 b

Find the solution of each equation using the given replacement set. 1 2

5 4

冦 12

3 4

5 4

冧

3 4

2 3

1 4

5 6

冦 23

3 5 4 4 4 3

冧

13 9

冦 49

5 2 7 9 3 9

冧

7 9

8. x ; , , ,

7. x ; , , 1,

4 3

9. (x 2) ; , , ,

10. 0.8(x 5) 5.2; {1.2, 1.3, 1.4, 1.5} 1.5

Solve each equation. 11. 10.4 6.8 x 3.6

12. y 20.1 11.9 8.2 6 18 31 25

46 15 3 28

14. c 4

2(4) 4 3(3 1)

16. n 1

13. a 1

15. b 2

6(7 2) 3(8) 6

Find the solution set for each inequality using the given replacement set. 17. a 7 13; {3, 4, 5, 6, 7} {3, 4, 5}

18. 9 y 17; {7, 8, 9, 10, 11} {7}

19. x 2 2; {2, 3, 4, 5, 6, 7} {2, 3, 4}

20. 2x 12; {0, 2, 4, 6, 8, 10} {8, 10}

21. 4b 1 12; {0, 3, 6, 9, 12, 15}

22. 2c 5 11; {8, 9, 10, 11, 12, 13} {8}

{3, 6, 9, 12, 15} y 2

23. 5; {4, 6, 8, 10, 12} {10, 12} Chapter 1

x 3

24. 2; {3, 4, 5, 6, 7, 8} {7, 8}

23

Glencoe Algebra 1

(Lesson 1-3)

A10

1. x 2 4

3. 7a 21 56 5

Answers

873946 Alg1 CH01 EP3

?

10 ?

10 ?

10 ?

10 ?

10


8 8 8 8 8

2. 4a 8 16 6

Page A10

1. 5a 9 26 7

10:26 AM

Find the solution set for 3a 8 10 if the replacement set is {4, 5, 6, 7, 8}.

Replace a in 3a 8 10 with each value in the replacement set. 3(4) 3(5) 3(6) 3(7) 3(8)

5/10/06

Open Sentences Find the solution of each equation if the replacement sets are A {4, 5, 6, 7, 8} and B {9, 10, 11, 12, 13}.

Solve Inequalities An open sentence that contains the symbol , , , or is called an inequality. Inequalities can be solved the same way that equations are solved. Example

Skills Practice

Lesson 1-3

1-3

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944


NAME ______________________________________________ DATE______________ PERIOD _____

1-3 1-3

Practice


Open Sentences

4. 7b 8 16.5 3.5

5. 120 28a 78

3 2

3. 6a 18 27

3 2

28 b

6. 9 16 4

Find the solution of each equation using the given replacement set. 7 8

17 12

冦 12

13 7 5 2 24 12 8 3

冧

13 24

3 4

冦 12

1 2

1 2

冧

1 2

10. 12(x 4) 76.8 ; {2, 2.4, 2.8, 3.2, 3.6} 2.4

Solve each equation. 11. x 18.3 4.8 13.5 97 25 41 23

14. k 4

12. w 20.2 8.95 11.25 4(22 4) 3(6) 6

37 9 18 11

13. d 4 5(22) 4(3) 4(2 4)

15. y 3

16. p 2 3

17. a 7 10; {2, 3, 4, 5, 6, 7}

18. 3y 42; {10, 12, 14, 16, 18}

{2}

{14, 16, 18} 20. 4b 4 3; {1.2, 1.4, 1.6, 1.8, 2.0}

{0.5, 1, 1.5} 3y 21. 2; {0, 2, 4, 6, 8, 10} 5

{0, 2}

{1.8, 2.0}

冦

1 1 3 1 5 3 22. 4a 3; , , , , , 8 4 8 2 8 4

3 4

冧

23. TEACHING A teacher has 15 weeks in which to teach six chapters. Write and then solve an equation that represents the number of lessons the teacher must teach per week if 6(8.5) there is an average of 8.5 lessons per chapter.

n

Nutrition Facts Serving Size 1 cup (228g) Servings Per Container 2 Amount Per Serving

Calories 250

Calories from Fat 110 % Daily Value *

Total Fat 12g Saturated Fat 3g

27

VEHICLES For Exercises 5 and 6, use the following information. Recently developed hybrid cars contain both an electric and a gasoline engine. Hybrid car batteries store extra energy, such as the energy produced by braking. Since the car can use this stored energy to power the car, the hybrid uses less gasoline per mile than cars powered only by gasoline. Suppose a new hybrid car is rated to drive 45 miles per gallon of gasoline.

18 % 15 %

5. It costs $40 to fill the gasoline tank with gas that costs $2.50 per gallon. Write and solve an equation to find the distance the hybrid car can go using one tank of gas.

Trans Fat 3g

; 3.4 15

LONG DISTANCE For Exercises 24 and 25, use the following information. Gabriel talks an average of 20 minutes per long-distance call. During one month, he makes eight in-state long-distance calls averaging $2.00 each. A 20-minute state-to-state call costs Gabriel $1.50. His long-distance budget for the month is $20.


19. 4x 2 5; {0.5, 1, 1.5, 2, 2.5}

2. FOOD Part of the Nutrition Facts label from a box of macaroni and cheese is shown below.

1161 g 202 ; 8686 gal

Cholesterol 30mg

10 %

40 (45) m; 720 mi 2.50

Write and solve an inequality to determine how many servings of this item that Alisa can have for lunch if she is restricted no more than 45 grams of cholesterol.

45 c ; 1.5 servings or less 30

3. CRAFTS You need at least 30 yards of yarn to crochet a small scarf. Cheryl bought a 100-yard ball of yarn and has already used 10 yards. Write and solve an inequality to find how many scarves she can crochet. 100 – 10 30s;

6. Write and solve an equation to find the cost of gasoline per mile for this hybrid car. Round to the nearest cent.

2.50 c; 6¢ per mi 45

3 scarves

Glencoe Algebra 1

24. Write an inequality that represents the number of 20 minute state-to-state calls Gabriel can make this month. 8(2) 1.5s 20 25. What is the maximum number of 20-minute state-to-state calls that Gabriel can make this month? 2 Chapter 1

24

Answers

Glencoe Algebra 1

Chapter 1

25

Glencoe Algebra 1

(Lesson 1-3)

A11

Find the solution set for each inequality using the given replacement set.

12 c 3; 9:00 AM

g gal in pool

Answers

9. 1.4(x 3) 5.32; {0.4, 0.6, 0.8, 1.0, 1.2}

0.8

27 8

8. (x 2) ; , 1, 1 , 2, 2 2

7. x ; , , , ,

4. POOLS There are approximately 202 gallons per cubic yard of water. Write and solve an equation for the number of gallons of water that fill a pool with a volume of 1161 cubic feet. (Hint: There are 27 cubic feet per cubic yard.)

Page A11

2. 4b 8 6 3.5


1 2

1 2

1. a 1

873946 Alg1 CH01 EP3

and B {3, 3.5, 4, 4.5, 5}.

1. TIME There are 6 time zones in the United States. The eastern part of the U.S., including New York City, is in the Eastern Time Zone. The central part of the U.S., including Dallas, is in the Central Time Zone, which is one hour behind Eastern Time. San Diego is in the Pacific Time Zone, which is 3 hours behind Eastern Time. Write and solve an equation to determine what time it is in California if it is noon in New York.

10:26 AM

Open Sentences

1 3 Find the solution of each equation if the replacement sets are A 0, , 1, , 2 2 2

Lesson 1-3

1-3

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

1-3

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-3 1-3

Enrichment

Spreadsheet Activity

Solution Sets

5/10/06

Solving Open Sentences A spreadsheet is a tool for working with and analyzing numerical data. The data is entered into a table in which each row is numbered and each column is labeled by a letter. You can use a spreadsheet to find solutions of open sentences.

Consider the following open sentence. It is the name of a month between March and July.

1. It is the name of a state beginning with the letter A.

{Alabama, Alaska, Arizona, Arkansas}

Step 2 The second column contains the formula for the left side of the open sentence. To enter a formula, enter an equals sign followed by the formula. Use the name of the cell containing each replacement value to evaluate the formula for that value. For example, in cell B2, the formula contains A2 in place of x.

2. It is a primary color.

{red, yellow, blue} 3. Its capital is Harrisburg. {Pennsylvania}

Vermont, Massachusetts, Rhode Island, Connecticut}


6. It is the name of a month that contains the letter r.

{Jan, Feb, Mar, Apr, Sept, Oct, Nov, Dec} 7. During the 1990s, she was the wife of a U.S. President.

{Barbara Bush, Hillary Clinton} 8. It is an even number between 1 and 13. {2, 4, 6, 8, 10,12} 9. 31 72 k {41} 10. It is the square of 2, 3, or 4.{4, 9, 16} Write an open sentence for each solution set. 11. {A, E, I, O, U} It is a vowel.

The solution set contains the values for which the open sentence is true. The solution set is {7, 8, 9, 10}.

13. {June, July, August} It is a summer month.

1 2 3 4 5 6 7 8

x

A

B

7 8 9 10 11 12

Sheet 1

A

C

4(x - 3) < 31 =B2<31 =B3<31 =B4<31 =B5<31 =B6<31 =B7<31

4(x - 3) =4*(A2-3) =4*(A3-3) =4*(A4-3) =4*(A5-3) =4*(A6-3) =4*(A7-3) Sheet 2

Sheet 3

B

7 8 9 10 11 12

4(x - 3)

Sheet 1

Sheet 2

C

4(x - 3) < 31 TRUE TRUE TRUE TRUE FALSE FALSE

16 20 24 28 32 36

Sheet 3

Exercises Use a spreadsheet to find the solution of each equation or inequality using the given replacement set. 1. x 7.5 18.3; {8.8, 9.8, 10.8, 11.8}

2. 6(x + 2) 18; {0, 1, 2, 3, 4, 5}

3. 4x 1 17; {0, 1, 2, 3, 4, 5}

4. 4.9 x 2.2; {2.6, 2.7, 2.8, 2.9, 3.0}

5. 2.7x 18; {6.1, 6.3, 6.5, 6.7, 6.9}

6. 12x 8 22; {2.1, 2.2, 2.3, 2.4, 2.5, 2.6}

{10.8} {4}

{6.7, 6.9}

{1}

{2.6, 2.7}

{2.1, 2.2, 2.3, 2.4, 2.5, 2.6}

14. {Atlantic, Pacific, Indian, Arctic} It is an ocean. Chapter 1

26

Glencoe Algebra 1


Glencoe Algebra 1

12. {1, 3, 5, 7, 9} It is an odd number between 0 and 10.


5. x 4 10 {6}

Step 3 The third column determines whether the open sentence is true or false for the value in the replacement set. These formulas will return TRUE or FALSE.

x

Chapter 1

27

Glencoe Algebra 1

(Lesson 1-3)

A12

4. It is a New England state. {Maine, New Hampshire,

1 2 3 4 5 6 7 8

Lesson 1-3

Step 1 Use the first column of the spreadsheet for the replacement set. Enter the numbers using the formula bar. Click on a cell of the spreadsheet, type the number and press ENTER.

Page A12

Write the solution set for each open sentence.

10:26 AM

Example Use a spreadsheet to find the solution set for 4(x 3) 31 if the replacement set is {7, 8, 9, 10, 11, 12}. You can solve the open sentence by replacing x with each value in the replacement set.

Answers

You know that a replacement for the variable It must be found in order to determine if the sentence is true or false. If It is replaced by either April, May, or June, the sentence is true. The set {April, May, June} is called the solution set of the open sentence given above. This set includes all replacements for the variable that make the sentence true.

A1-A34_CRM01-873944


1-4

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-4 1-4



Identity and Equality Properties

10:27 AM

Identity and Equality Properties Identity and Equality Properties


The identity and equality properties in the chart below can help you solve algebraic equations and evaluate mathematical expressions.


1. Write the Roman numeral of the sentence that best matches each term. 5 7

V III

b. multiplicative identity

II. 18 18

VIII

d. Multiplicative Inverse Property

I

g. Transitive Property h. Substitution Property

II

V. 6 0 6

IV

VI. If 2 4 5 1 and 5 1 6, then 2 4 6.

VI

VII. If n 2, then 5n 5 2.

VII

VIII. 4 0 0

What You Learned 2. The prefix trans- means “across” or “through.” Explain how this can help you the meaning of the Transitive Property of Equality.

Glencoe Algebra 1

Sample answer: The Transitive Property of Equality tells you that when a b and b c, you can go from a through b to get to c.

a b a b For every number , a, b 0, there is exactly one number such that 1.

Reflexive Property

For any number a, a a.

Symmetric Property

For any numbers a and b, if a b, then b a.

b

a

Transitive Property

For any numbers a, b, and c, if a b and b c, then a c.

Substitution Property

If a b, then a may be replaced by b in any expression.

a. 8n 8 Multiplicative Identity Property n 1, since 8 1 8

a 5454 Reflexive Property b. If n 12, then 4n 4 12. Substitution Property

1 3

1 3

n , since 3 1 Exercises Name the property used in each equation. Then find the value of n. 1. 6n 6

2. n 1 8

3. 6 n 6 9

4. 9 n 9

3 5. n 0 8

6. n 1

Mult. Identity; 1

Mult. Identity; 8

Add. Identity; 0

Answers

Glencoe Algebra 1

Substitution Property; 9

3 Add. Identity; 8

3 4

4 Mult. Inverse; 3

Name the property used in each equation. 7. If 4 5 9, then 9 4 5.

Symmetric Property

9. 0(15) 0 Mult. Prop. of Zero

8. 0 21 21

Add. Identity

10. (1)94 94 Mult. Identity

11. If 3 3 6 and 6 3 2, then 3 3 3 2. Transitive Property 13. (14 6) 3 8 3

Reflexive Property 28

a

Example 2 Name the property used to justify each statement.

12. 4 3 4 3

Chapter 1

b

Example 1 Name the property used in each equation. Then find the value of n.

b. n 3 1 Multiplicative Inverse Property

IV. If 12 8 4, then 8 4 12. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

f. Symmetric Property

III. 3 1 3

For any number a, a 0 0.

Multiplicative Inverse Property

(Lesson 1-4)

A13

c. Multiplicative Property of Zero

e. Reflexive Property

7 5

I. 1


a. additive identity

For any number a, a 1 a.

Multiplicative Property of 0

Answers

Read the Lesson

For any number a, a 0 a.

Multiplicative Identity

Chapter 1

Substitution Property 29

Glencoe Algebra 1

Lesson 1-4

2 r 2; Sample answer: The rank did not change for either team from week 6 to week 7.

Additive Identity

Page A13

Write an open sentence to represent the change in rank r of Auburn from week 6 to week 7. Explain why the solution is the same as the solution in the introduction.


NAME ______________________________________________ DATE______________ PERIOD _____

1-4 1-4

(continued)


1. n 0 19

Substitution; 9 3 3 Substitution; 3 3 0

Substitution; 24 8 16 Additive Identity; 16 0 16

Multiplicative Identity; 1

4. 18 1 3 2 2(6 3 2)

1 2(151 14) 4 Subst.

18 1 3 2 2(2 2) Subst.

1 2(15 14) 4 Mult. Identity

18 1 3 2 2(0) Substitution

4

1 2(1) 4 1 24

4

4

4

Substitution Mult. Identity

Mult. Prop. Zero Substitution Add. Identity

6. 3(5 5 12) 21 7

13 Subst. Substitution Substitution Substitution Additive Identity

3(5 5 1) 21 7 Subst. 3(5 5) 21 7 Mult. Identity 3(0) 21 7 Substitution 0 21 7 Mult. Prop. Zero 03 Substitution 3 Additive Identity

30

Glencoe Algebra 1

Multiplicative Prop. of Zero; 0 14. 11 (18 2) 11 n

1 Multiplicative Inverse;

Substitution Prop.; 9

3

Evaluate each expression. Name the property used in each step. 16. 2[5 (15 3)]

15. 7(16 42)

7(16 16) Substitution 7(1) Substitution 7 Multiplicative Identity 17. 4 3[7 (2 3)]

4(8 8) 1 Substitution 4(0) 1 Substitution 01 Mult. Prop. of Zero 1 Additive Identity 1 2

19. 6 9[10 2(2 3)]

20. 2(6 3 1)

6 9[10 2(5)] Substitution 6 9(10 10) Substitution 6 9(0) Substitution 60 Mult. Prop. of Zero

Chapter 1

2(5 5) Substitution 2(0) Substitution 0 Mult. Prop. of Zero

18. 4[8 (4 2)] 1

4 3(7 6) Substitution 4 3(1) Substitution 4 3 Multiplicative Identity 1 Substitution

6


Glencoe Algebra 1

10 5 4 2 2 4 2 13 2 2 13 0 13 13

Mult. Identity Substitution

18 6 0 12 0 12

Mult. Inverse Substitution

5. 10 5 22 2 13

18 3 2 2(0) 18 6 2(0)

Reflexive Prop.; 5


1 4

3. 2(3 5 1 14) 4

12. n 14 0

13. 3n 1


Substitution Mult. Inverse


11. 5 4 n 4

(Lesson 1-4)

A14

1

10. (7 3) 4 n 4


15 1 9 2(5 5) Substitution 15 1 9 2(0) Substitution 15 1 9 0 Mult. Prop. Zero 15 9 0 Mult. Identity 60 Substitution 6 Substitution

Substitution

Reflexive Prop.; 3

9. 2(9 3) 2(n)

2. 15 1 9 2(15 3 5)

8. 2 n 2 3

Additive Identity; 0

Additive Identity 31

1 2(2 1) Substitution 1 2(1) 1 2 2

1

2

2

Substitution Multiplicative Identity Multiplicative Inverse Glencoe Algebra 1

Page A14

Multiplicative Inverse; 4

Answers

冢冣冥

1 1 2 4 4 1 2 2

Chapter 1

6. n 9 9

7. 5 n 5

1 1 2 1. 2 4 2

21 1


1 4

5. n 1

Exercises

4. 0 n 22

Multiplicative Prop. of Zero; 0

Multiplicative Identity; 24 1 24 Multiplicative Property of Zero; 5(0) 0

Evaluate each expression. Name the property used in each step.

冤


3. 28 n 0

10:27 AM

1 8 5(3 3) 1 8 5(0) 8 5(0) 80 0

2. 1 n 8


Evaluate 24 1 8 5(9 3 3). Name the property used in each step.

24 1 8 5(9 3 3) 24 24 24 24 16 16

5/10/06

Identity and Equality Properties Name the property used in each equation. Then find the value of n.

Use Identity and Equality Properties The properties of identity and equality can be used to justify each step when evaluating an expression. Example

Skills Practice

Lesson 1-4

1-4

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944


1-4

NAME ______________________________________________ DATE______________ PERIOD _____

1-4 1-4

Practice



1. EXERCISE Annika goes on a walk every day in order to get the exercise her doctor recommends. If she walks at a 1 rate of 3 miles per hour for of an hour,

2. (8 7)(4) n(4)


Substitution Prop.; 15 4. n 0.5 0.1 0.5

5. 49n 0

6. 12 12 n

1 Multiplicative Inverse; 5

3

1

then she will have walked 3 miles. 3 Evaluate the expression and name the property used.

Reflexive Prop.; 0.1

Multiplicative Prop. of Zero; 0

4. PARTY PLANNING Chase is planning a dinner party for 18 guests. He needs to have the same number of place settings as guests, and the same number of water glasses as place settings. What property must be used to determine the number of water glasses he needs for the party? Explain. The Transitive Property;

1 mi; Multiplicative Inverse


Page A15

3. 5n 1

10:27 AM


Name the property used in each equation. Then find the value of n. 1. n 9 9

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

if guests settings and settings glasses, then guests glasses.

Evaluate each expression. Name the property used in each step.

2 Substitution Substitution Mult. Prop. of Zero Additive Identity Substitution

51 6

USPS First Class Mail: Standard Letter Rates

Multiplicative Inverse Substitution

Weight (ounces)

Cost

0.25

$0.39

2(15 5) 3(9 8) 2(10) 3(1) 20 3(1) 20 3 23

Substitution Substitution Multiplicative Identity Substitution

GARDENING For Exercises 11 and 12, use the following information. Mr. Katz harvested 15 tomatoes from each of four plants. Two other plants produced four tomatoes each, but Mr. Katz only harvested one fourth of the tomatoes from each of these.

1 11. Write an expression for the total number of tomatoes harvested. 4(15) 2 4 4

Glencoe Algebra 1

12. Evaluate the expression. Name the property used in each step.

1 1 4(15) 2 4 60 2 4 4 4

60 2(1) 60 2 62

Chapter 1

Substitution Multiplicative Inverse Multiplicative identity Substitution 32

Answers

Glencoe Algebra 1


10. Evaluate the expression. Name the property used in each step.


9. Write an expression that represents the profit Althea made. 2(15 5) 3(9 8)

the following information. Some toll highways assess tolls based on where a car entered and exited. The table below shows the highway tolls for a car entering and exiting at a variety of exits. Assume that the toll for the reverse direction is the same.

0.5

$0.39

Entered

Exited

Toll

0.75

$0.39

Exit 5

Exit 8

$0.50

1

$0.39

Exit 8

Exit 10

$0.25

1.25

$0.60

Exit 10

Exit 15

$1.00

1.5

$0.60

Exit 15

Exit 18

$0.50

1.75

$0.60

Exit 18

Exit 22

$0.75

SALES For Exercises 9 and 10, use the following information. Althea paid $5.00 each for two bracelets and later sold each for $15.00. She paid $8.00 each for three bracelets and sold each of them for $9.00.

TOLL ROADS For Exercises 5 and 6, use

Source: www.usps.gov

5. Running an errand, Julio travels from Exit 8 to Exit 5. What property would you use to determine the toll?

Write an equation that represents the difference between the cost of mailing a 0.5 ounce and a 1.0 ounce letter. Name the property illustrated.

Symmetric Property of Equality

$0.39 $0.39 0; Additive Inverse

6. Gordon travels from home to work and back each day. He lives at Exit 15 on the toll road and works at Exit 22. Write and evaluate an expression to find his daily toll cost. What property or properties did you use? t 2 ($0.50 $0.75);

3. CAPACITY Use the substitution and transitive properties to find how many 1-cup servings there are in 1 gallon of sports drink. 16 c

Chapter 1

t $2.50; Substitution

33

Glencoe Algebra 1

Lesson 1-4

1 5(14 13) 4 Substitution 4 1 5(1) 4 Substitution 4 1 5 4 Multiplicative Identity 4

(Lesson 1-4)

A15

2 6(9 9) 2 6(0) 2 202 22 0

2. MAIL The chart below shows the cost of mailing letters of various weight through the United States Postal Service.

8. 5(14 39 3) 4

Answers

1 4

7. 2 6(9 32) 2

1-4

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-5 1-5

Enrichment


Closure


A binary operation matches two numbers in a set to just one number. Addition is a binary operation on the set of whole numbers. It matches two numbers such as 4 and 5 to a single number, their sum.


If the result of a binary operation is always a member of the original set, the set is said to be closed under the operation. For example, the set of whole numbers is closed under addition because 4 5 is a whole number. The set of whole numbers is not closed under subtraction because 4 5 is not a whole number.

Add $14.95 and $34.95.

5/10/06

The Distributive Property

10:27 AM

How would you find the amount spent by each of the first eight customers at Instant Replay Video Games on Saturday?

1. Explain how the Distributive Property could be used to rewrite 3(1 5).

Find the sum of 3 times 1 and 3 times 5.

Tell whether each operation is binary. Write yes or no.

↵, where a ↵ b means to choose the lesser number from a and b yes 2. Explain how the Distributive Property can be used to rewrite 5(6 4).

2. the operation ©, where a © b means to cube the sum of a and b yes

Write the difference of 5 times 6 and 5 times 4, that is 5 6 5 4.

Term

5. the operation ⇑, where a ⇑ b means to match a and b to any number greater than either number no

Tell whether each set is closed under addition. Write yes or no. If your answer is no, give an example. 8. odd numbers no; 3 7 10

9. multiples of 3 yes

10. multiples of 5 yes

11. prime numbers no; 3 5 8

12. nonprime numbers no; 22 9 31

13. multiplication: a b yes

14. division: a b no; 4 3 is not a

15. exponentation: ab yes

16. squaring the sum: (a b)2 yes

Chapter 1

whole number

34

Glencoe Algebra 1


Glencoe Algebra 1

Tell whether the set of whole numbers is closed under each operation. Write yes or no. If your answer is no, give an example.


6. the operation ⇒, where a ⇒ b means to round the product of a and b up to the nearest 10 yes


A16

3. Write three examples of each type of term. Sample answers are given.

4. the operation exp, where exp(a, b) means to find the value of ab yes

7. even numbers yes

(Lessons 1-4 and 1-5)

3. the operation sq, where sq(a) means to square the number a no

Example

number

3, 17, 0.25

variable

w, t 2, x

product of a number and a variable

4y, 0.78z, r

quotient of a number and variable

x 2s 6 , , 3 7 5t

1 8

4. Tell how you can use the Distributive Property to write 12m 8m in simplest form. Use the word coefficient in your explanation.

Sample answer: Add the coefficients of the two and multiply by m.

What You Learned 5. How can the everyday meaning of the word identity help you to understand and what the additive identity is and what the multiplicative identity is?

Sample answer: When you add 0 (the additive identity) to a number, the result is the very same number you started with. The same is true if you multiply the number by 1 (the multiplicative identity).

Chapter 1

35

Glencoe Algebra 1

Lesson 1-5

1. the operation

Page A16

Answers

Read the Lesson

A1-A34_CRM01-873944


1-5

NAME ______________________________________________ DATE______________ PERIOD _____

1-5 1-5




The Distributive Property Simplify Expressions

The Distributive Property can be used to help evaluate

A term is a number, a variable, or a product or quotient of numbers and variables. Like are that contain the same variables, with corresponding variables having the same powers. The Distributive Property and properties of equalities can be used to simplify expressions. An expression is in simplest form if it is replaced by an equivalent expression with no like or parentheses.

expressions. Distributive Property

For any numbers a, b, and c, a(b c) ab ac and (b c)a ba ca and a(b c) ab ac and (b c)a ba ca.

Example

Rewrite 6(8 10) using the Distributive Property. Then evaluate.

6(8 10) 6 8 6 10 48 60 108

Multiply. Add.

Rewrite 2(3x2 5x 1) using the Distributive Property. Then simplify.

2(3x2 5x 1) 2(3x2) (2)(5x) (2)(1) 6x2 (10x) (2) 6x2 10x 2

4(a2 3ab) 1ab 4a2 12ab 1ab 4a2 (12 1)ab 4a2 11ab

Multiplicative Identity Distributive Property Distributive Property Substitution

Exercises

Distributive Property

Simplify each expression. If not possible, write simplified.

Multiply.

1. 12a a

Simplify.

11a

2. 3x 6x

3. 3x 1

9x

simplified

(Lesson 1-5)

Exercises

2. 6(12 t) 72 6t

3. 3(x 1) 3x 3

4. 6(12 5) 102

5. (x 4)3 3x 12

6. 2(x 3) 2x 6

7. 5(4x 9) 20x 45

8. 3(8 2x) 24 6x

9. 12 6 x 72 6x

冢

1 2

冣

10. 12 2 x 24 6x

Glencoe Algebra 1

13. 2(3x 2y z)

6x 4y 2z 1 4

16. (16x 12y 4z)

4x 3y z Chapter 1

1 4

11. (12 4t) 3 t

14. (x 2)y

冢

1 2

冣

12. 3(2x y) 6x 3y

15. 2(3a 2b c)

xy 2y

2g 1 7. 20a 12a 8

32a 8 1 2

10. 2p q

simplified 13. 3x 2x 2y 2y

x

5. 2x 12

6. 4x2 3x 7

simplified 8. 3x2 2x2

simplified 9. 6x 3x2 10x2

5x 2

6x 13x2

11. 10xy 4(xy xy)

12. 21c 18c 31b 3b

2xy

39c 28b

14. xy 2xy

15. 12a 12b 12c

simplified

xy

6a 4b 2c

17. (2 3x x2)3

6 9x 3x2 36

Answers

18. 2(2x2 3x 1)

4x2 6x 2 Glencoe Algebra 1

1 4

16. 4x (16x 20y)

8x 5y Chapter 1

17. 2 1 6x x2

18. 4x2 3x2 2x

1 6x x2

7x2 2x

37

Glencoe Algebra 1

Lesson 1-5

1. 2(10 5) 10

4. 12g 10g 1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Rewrite each expression using the Distributive Property. Then simplify.


A17

Answers

Example 2

Simplify 4(a2 3ab) ab.

4(a2 3ab) ab


Page A17

Example 1

(continued)

10:27 AM

Evaluate Expressions

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

1-5

NAME ______________________________________________ DATE______________ PERIOD _____

1-5 1-5

Skills Practice

Practice



1. 4(3 5) 4 3 4 5; 32

2. 2(6 10) 2 6 2 10; 32

1. 9(7 8)

2. 7(6 4)

3. 6(b 4)

3. 5(7 4) 5 7 5 4; 15

4. (6 2)8 6 8 2 8; 32

4. (9 p)3

5. (5y 3)7

6. 15 f

5. (a 7)2 a 2 7 2; 2a 14

6. 7(h 10) 7 h 7 10; 7h 70

9 7 9 8; 135

m n m 4; mn 4m

冣 1

5y 7 3 7; 35y 21 8. m(n 4)

1 3

15 f 15 ; 3 15f 5 9. (c 4)d

c d 4 d; cd 4d

Use the Distributive Property to find each product.

10. 3(a b 1)

2(x) 2(y) 2(1); 2x 2y 2

冢

Answers

8. (x y)6 x 6 y 6; 6x 6y

6 b 6 4; 6b 24

3(a) 3(b) 3(1); 3a 3b 3

10. 9 499 4491

11. 7 110 770

13. 12 2.5 30

14. 27 2 63

冢冣 1 3

12. 21 1004 21,084

冢 14 冣

15. 16 4 68

Use the Distributive Property to find each product. Simplify each expression. If not possible, write simplified.

冢 18 冣

15. 12 1 15

16. 8 3 25

Simplify each expression. If not possible, write simplified. 18. 17g g 18g

19. 16m 10m 6m

20. 12p 8p 4p

21. 2x2 6x2 8x2

22. 7a2 2a2 5a2

23. 3y2 2y simplified

24. 2(n 2n) 6n

25. 4(2b b) 4b Chapter 1

26. 3q2 q q2 2q2 q

38

17. 3(5 6h) 15 18h

18. 14(2r 3) 28r 42

19. 12b2 9b2 21b 2

20. 25t3 17t3 8t 3

21. c2 4d 2 d 2 c 2 3d 2

6a

23. 4(6p 2q 2p)

24. x x

22.

3a2

simplified

2b2

16p 8q

2x

2 3

x 3

DINING OUT For Exercises 25 and 26, use the following information. The Ross family recently dined at an Italian restaurant. Each of the four family ordered a pasta dish that cost $11.50, a drink that cost $1.50, and dessert that cost $2.75. 25. Write an expression that could be used to calculate the cost of the Ross’ dinner before adding tax and a tip. 4(11. 5 1.5 2.75) 26. What was the cost of dining out for the Ross family? $63.00

ORIENTATION For Exercises 27 and 28, use the following information. Madison College conducted a three-day orientation for incoming freshmen. Each day, an average of 110 students attended the morning session and an average of 160 students attended the afternoon session. 27. Write an expression that could be used to determine the total number of incoming freshmen who attended the orientation. 3(110 160) 28. What was the attendance for all three days of orientation? 810

Glencoe Algebra 1


Glencoe Algebra 1

17. 2x 8x 10x

16. w 14w 6w 9w

Chapter 1

39

Glencoe Algebra 1

Lesson 1-5

冢 14 冣

冢 13 冣

14. 15 2 35


13. 15 104 1560


12. 9 99 891

(Lesson 1-5)

A18

11. 5 89 445

Page A18

16 3b 16 0.25; 48b 4

7 6 7 4; 14

10:27 AM

9 3 p 3; 27 3p 7. 16(3b 0.25)

9. 2(x y 1)

5/10/06



7. 3(m n) 3 m 3 n; 3m 3n

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944


1-5

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-5


Enrichment

The Maya

4(200 3) 800 12 788

use the following information. Letisha and Noel each opened a checking , a savings , and a college fund. The chart below shows the amounts that they deposited into each . Savings

College

Letisha

$125

$75

$50

Noel

$250

$50

$50

6. If Noel used only $50 bills when he deposited the money to open his s, how many $50 bills did he deposit? 7 $50 bills

4. FENCES Demonstrate the Distributive Property by writing two equivalent expressions to represent the perimeter of the fenced dog pen below.

7. If all s earn 1.5% interest per year and no further deposits are made, how much interest will Letisha have earned one year after her s were opened? $3.75

2n 2m and 2(n m) m

Glencoe Algebra 1

Dog Pen

••

•• _____ 12 _____

3

•••

••• _____ 13 _____

4

••••

•••• _____ 14 _____

The Maya developed a system of numeration that was based on the number twenty. The basic symbols of this system are shown in the table at the right. The places in a Mayan numeral are written vertically—the bottom place represents ones, the place above represents twenties, the place above that represents 20 20, or four hundreds, and so on. For instance, this is how to write the number 997 in Mayan numerals.

5 _____

_____ 15 _____ _____ • _____ 16 _____ _____

←

2 400 800

_____ ••••

←

9

_____ •• _____ _____

← 17

n

20

180

1

• 6 _____ •• 7 _____ ••• 8 _____

_____ •• 17 _____ _____ _____ ••• 18 _____ _____

•••• 9 _____

_____ •••• 19 _____ _____

17 997

z w

1.

vwz •••• 2. _____

•••

4. vxy

3. xv

x

5. wx z

● ●

•••

6. vz xy

● ●

•••• _____ _____ 7. w(v x z) _____

8. vwz

_____ • _____ _____

• ••••

••• •••• _____ • _____ _____

••• _____ •• _____ ••

9. z(wx x)

● ● _____ • _____ _____

Tell whether each statement is true or false. • • • _____ • • ••• _____ _____ 10. _____ _____ _____

true

_____ ••• _____

_____ •

11. _____ _____ ••• _____ •

false

_____ •••

_____ ••• _____

12. _____ _____ _____

false

_____ • • • ( _____ _____ _____ ) true 13. ( • • • _____ ) _____

14. How are Exercises 10 and 11 alike? How are they different?

Both involve changing the order of the symbols. Exercise 10 involves changing the order of the addends in an addition problem. Exercise 11 involves changing the order of the digits in a numeral. Chapter 1

40

Answers

Glencoe Algebra 1

Chapter 1

41

Glencoe Algebra 1

Lesson 1-5

3

53 53 5(3) 5 5 5 5 15 3 18

Checking


3

• _____ 11 _____

2

••• ,x ••••,y ● _____ • , w _____ ● , and _____ Evaluate each expression when v _____ • • . Then write the answer in Mayan numerals. Exercise 5 is done for you. _____ z _____


of fabric. Use the Distributive Property to find the number of yards of fabric needed for 5 costumes. (Hint: a mixed number can be written as the sum of an integer and a fraction.)

3

_____ 10 _____

•

(Lesson 1-5)

A19

5

● ●

1

••

INVESTMENTS For Exercises 6 and 7, 3. COSTUMES Isabella’s ballet class is performing a spring recital for which they need butterfly costumes. Each 3 butterfly costume is made from 3 yards

0

Answers

2. LIBRARY In Cook County Library’s children’s section there are 7 shelves and 4 tables. Each shelf and table displays 12 books. Write and evaluate an expression to find how many books are in the children’s section. 12(7 4) 132

The Maya were a Native American people who lived from about 1500 B.C. to about 1500 A.D. in the region that today encomes much of Central America and southern Mexico. Their many accomplishments include exceptional architecture, pottery, painting, and sculpture, as well as significant advances in the fields of astronomy and mathematics.

Page A19

5. MENTAL MATH During a math facts speed contest, Jamal calculated the following expression faster than anyone else in his class. 197 4 When classmates asked him how he was able to answer so quickly, he told them he used the Distributive Property to think of the problem differently. Write and evaluate an expression using the Distributive Property that would help Jamal perform the calculation quickly.

$39(23 2) $975

10:27 AM

The Distributive Property 1. OPERA Mr. Delong’s drama class is planning a field trip to see Mozart’s famous opera Don Giovanni. Tickets cost $39 each, and there are 23 students and 2 teachers going on the field trip. Write and evaluate an expression to find the group’s total ticket cost.

NAME ______________________________________________ DATE______________ PERIOD _____

3-6 1-6


Get Ready for the Lesson How are the expressions 0.4 1.5 and 1.5 0.4 alike? different?

The numbers and the operation are the same; the order of the numbers is different.

Example 1

1. Write the Roman numeral of the term that best matches each equation. I. Associative Property of Addition

I

c. 2 (3 4) (2 3) 4

IV

Commutative Property

Evaluate 8.2 2.5 2.5 1.8.

8.2 2.5 2.5 1.8 8.2 1.8 2.5 2.5 (8.2 1.8) (2.5 2.5) 10 5 15

Associative Property Multiply. Multiply.

The product is 180. III. Commutative Property of Addition

Commutative Prop. Associative Prop. Add. Add.

The sum is 15. IV. Commutative Property of Multiplication Exercises

2. What property can you use to change the order of the in an expression?

Commutative Property of Addition Associative Property of Multiplication 4. What property can you use to combine two like to get a single term?

Distributive Property 5. To use the Associative Property of Addition to rewrite the sum of a group of , what is the least number of you need? three

What You Learned

Sample answer: To travel back and forth, as between a suburb and a city; in the Commutative Property of Addition, a b b a, the quantities a and b are switched back and forth. 42

Glencoe Algebra 1


Glencoe Algebra 1

6. Look up the word commute in a dictionary. Find an everyday meaning that is close to the mathematical meaning and explain how it can help you the mathematical meaning.


3. What property can you use to change the way three factors are grouped?


Evaluate each expression. 1. 12 10 8 5 35

2. 16 8 22 12 58

3. 10 7 2.5 175

4. 4 8 5 3 480

5. 12 20 10 5 47

6. 26 8 4 22 60

1 2

1 2

7. 3 4 2 3 13

1 2

1 2

10. 4 5 3 13

4 5

2 9

3 4

8. 12 4 2 72

11. 0.5 2.8 4 5.6

1 5

1 2

13. 18 25 80

14. 32 10 32

16. 3.5 8 2.5 2 16

17. 18 8 8

Chapter 1

1 2

1 9

43

9. 3.5 2.4 3.6 4.2 13.7

12. 2.5 2.4 2.5 3.6 11

1 4

1 7

15. 7 16 4

3 4

1 2

18. 10 16 60

Glencoe Algebra 1

(Lesson 1-6)

A20

d. 2 (3 4) 2 (4 3)

II

II. Associative Property of Multiplication

Example 2

Evaluate 6 2 3 5.

62356325 (6 3)(2 5) 18 10 180

Answers

III

b. 2 (3 4) (2 3) 4

Chapter 1

For any numbers a and b, a b b a and a b b a. For any numbers a, b, and c, (a b) c a (b c ) and (ab)c a(bc).

Page A20

Read the Lesson

Commutative Properties Associative Properties

10:27 AM

Commutative and Associative Properties The Commutative and Associative Properties can be used to simplify expressions. The Commutative Properties state that the order in which you add or multiply numbers does not change their sum or product. The Associative Properties state that the way you group three or more numbers when adding or multiplying does not change their sum or product.


5/10/06

Commutative and Associative Properties


a. 3 6 6 3

Study Guide and Intervention Lesson 1-6

1-6

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944



NAME ______________________________________________ DATE______________ PERIOD _____

3-6 1-6

(continued)

Commutative and Associative Properties The Commutative and Associative Properties can be used along with other properties when evaluating and simplifying expressions. Example

8y 16x 7y 8y 7y 16x (8 7)y 16x 15y 16x

Commutative () Distributive Property Substitution

1. 4x 3y x

2. 3a 4b a

5x 3y

4a 4b

16x 21y 4 3

1.7x 0.5y 1 3

11. z2 9x2 z2 x2

7 28 z 2 x 2 3 3

9. 5(0.3x 0.1y) 0.2x

12. 6(2x 4y) 2(x 9)

14x 24y 18

Write an algebraic expression for each verbal expression. Then simplify. 13. twice the sum of y and z is increased by y

3y 2z

14. four times the product of x and y decreased by 2xy

2xy

Glencoe Algebra 1

15. the product of five and the square of a, increased by the sum of eight, a2, and 4

6a 2 12

16. three times the sum of x and y increased by twice the sum of x and y

5x 5y

Chapter 1

8. 1.6 0.9 2.4 4.9

9. 4 6 5 16

1 2

1 2

44

Answers

10. 2x 5y 9x 11x 5y

11. a 9b 6a 7a 9b

12. 2p 3q 5p 2q 7p 5q

13. r 3s 5r s 6r 4s

14. 5m2 3m m2 6m2 3m

15. 6k2 6k k2 9k 7k2 15k

16. 2a 3(4 a) 5a 12

17. 5(7 2g) 3g 35 13g

Write an algebraic expression for each verbal expression. Then simplify, indicating the properties used.

Glencoe Algebra 1


1 2

7 x

7. 1.7 0.8 1.3 3.8

6. 6n 2(4n 5)

14n 10

8. 5(2x 3y) 6( y x)

4 3

2rs 2


1 2

15rs

10x 8y

10. (x 10)

6. 5 7 10 4 1400

18. three times the sum of a and b increased by a

3(a b) a 3(a) 3(b) a 3a 3b a 3a a 3b (3a a) 3b (3 1)a 3b 4a 3b

Distributive Property Multiply Commutative () Associative () Distributive Property Substitution

19. twice the sum of p and q increased by twice the sum of 2p and 3q

2(p q) 2(2p 3q) 2(p) 2(q) 2(2p) 2(3q) 2p 2q 4p 6q 2p 4p 2q 6q (2p 4p) (2q 6q) (2 4)p (2 6)q 6p 8q Chapter 1

Distributive Property Multiply Commutative () Associative () Distributive Property Substitution 45

Glencoe Algebra 1

(Lesson 1-6)

A21

2 3

3. 8rs 2rs2 7rs

5. 6(x y) 2(2x y)

7a 9b

5. 2 4 5 3 120

Answers

Exercises

7. 6(a b) a 3b

4. 5 3 4 3 180

Simplify each expression.


13a 2 4b

3. 32 14 18 11 75


The simplified expression is 15y 16x.

4. 3a2 4b 10a2

2. 36 23 14 7 80

Page A21

Simplify 8(y 2x) 7y.

1. 16 8 14 12 50

10:27 AM

Commutative and Associative Properties Evaluate each expression.

Simplify Expressions

8(y 2x) 7y

Skills Practice Lesson 1-6

1-6

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-6

Practice


Commutative and Associative Properties 2. 6 5 10 3 900

3. 7.6 3.2 9.4 1.3 21.5

4. 3.6 0.7 5 12.6

1 2 5. 7 2 1 9 9

1 10 3

3 4

1 3

6. 3 3 16 200

7. 9s2 3t s2 t 10s 2 4t

10. 2(3x y) 5(x 2y) 11x 12y

11. 3(2c d) 4(c 4d) 10c 19d

12. 6s 2(t 3s) 5(s 4t) 17s 22t

13. 5(0.6b 0.4c) b 4b 2c

1 1 1 14. q 2 q r 2 4 2

冢

Bus Route

冣 qr

15. Write an algebraic expression for four times the sum of 2a and b increased by twice the sum of 6a and 2b. Then simplify, indicating the properties used.

12 people got on

Second stop

4 people off; 15 on

Third stop

16 people off; 7 on

Fourth stop

11 people off; 14 on

following information. Kim, Doug, and Conner all run on the cross country team. In the last race Doug finished first, Kim finished 3 minutes after Doug, and Conner finished with a time that was twice Doug’s time. 5. What is the sum of their times?

SCHOOL SUPPLIES For Exercises 16 and 17, use the following information. Kristen purchased two binders that cost $1.25 each, two binders that cost $4.75 each, two packages of paper that cost $1.50 per package, four blue pens that cost $1.15 each, and four pencils that cost $.35 each. 16. Write an expression to represent the total cost of supplies before tax.

2(1.25 4.75 1.50) 4(1.15 0.35) 17. What was the total cost of supplies before tax? $21.00

GEOMETRY For Exercises 18 and 19, use the following information. 18. Using the commutative and associative properties to group the in a way that makes evaluation convenient, write an expression to represent the perimeter of the pentagon. Sample answer: (1.25 0.25) (0.9 1.1) 2.5

x (x 3) (2x) x x 2x 3 4x 3 min

3. MENTAL MATH The triangular banner has a base of 9 centimeters and a height of 6 centimeters. Using the formula for area of a triangle, the banner’s area can 1 be expressed as 9 6 . Gabrielle 2

6. What property or properties did you use?

finds it easier to write and evaluate

冢

Associative and Commutative Properties of Addition, and Distributive Property

冣

1 6 9 to find the area. Is 2

Gabrielle’s expression equivalent to the area formula? Explain.

7. Evaluate the expression if Doug ran the race in 27 minutes. 111 min

h b

Yes; the Commutative and Associative Properties of Multiplication allow it to be rewritten.

19. What is the perimeter of the pentagon? 6 in. Chapter 1

46

Glencoe Algebra 1


Glencoe Algebra 1

The lengths of the sides of a pentagon in inches are 1.25, 0.9, 2.5, 1.1, and 0.25.


Distributive Property Multiply Commutative () Associative () Distributive Property Substitution


How many people are on the bus after the fourth stop? 17

Chapter 1

47

Glencoe Algebra 1

(Lesson 1-6)

A22

4(2a b) 2(6a 2b) 4(2a) 4(b) 2(6a) 2(2b) 8a 4b 12a 4b 8a 12a 4b 4b (8a 12a) (4b 4b) (8 12)a (4 4)b 20a 8b

First stop

SPORTS For Exercises 5–7, use the

Answers

9. 6y 2(4y 6) 14y 12

8. (p 2n) 7p 8p 2n

Sample answer: (60 84) 62 84 (60 62) 206

Page A22

2. BUS STOPS Mr. McGowan drives a city bus. Occasionally he keeps track of the number of riders for market research. The chart below shows a morning route.


4. ANATOMY The human body has 60 bones in the arms and hands, 84 bones in the upper body and head, and 62 bones in the legs and feet. Use the Associative Property to write and evaluate an expression that represents the total number of bones in the human body.

10:27 AM

1. 13 23 12 7 55

1. SCHOOL SUPPLIES At a local school supply store, a highlighter costs $1.25, a ballpoint pen costs $0.80, and a spiral notebook costs $2.75. Use mental math and the Associative Property of Addition to find the total cost if one of each item is purchased. $4.80

5/10/06



Lesson 1-6

1-6

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944


1-6

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-7

Enrichment


Let’s make up a new operation and denote it by , so that a b means ba.


2 3 9 2) 3 21 3 32 9 (1

If you know the heat was not too high, what must have caused the popcorn to burn?

32

8

Read the Lesson

3. Does the operation appear to be commutative? no

1. Write hypothesis or conclusion to tell which part of the if-then statement is underlined.

4. What number is represented by (2 1) 3? 3

a. If it is Tuesday, then it is raining. conclusion

5. What number is represented by 2 (1 3)? 9

b. If our team wins this game, then they will go to the playoffs. conclusion

6. Does the operation appear to be associative? no

c. I can tell you your birthday if you tell me your height. hypothesis

2. What does the term valid conclusion mean?

8. What number is represented by 3 2? 12 9. Does the operation appear to be commutative? yes 10. What number is represented by (2 3) 4? 65 11. What number is represented by 2 (3 4)? 63 12. Does the operation appear to be associative? no 13. What number is represented by 1 (3 2)? 12 14. What number is represented by (1 3) (1 2)? 12 15. Does the operation appear to be distributive over the operation ? yes


7. What number is represented by 2 3? 12


A23

e. If x is an even number, then x 2 is an odd number. conclusion

3 2 (3 1)(2 1) 4 3 12 (1 2) 3 (2 3) 3 6 3 7 4 28

Sample answer: A valid conclusion is a statement that has to be true if you used true statements and correct reasoning to obtain the conclusion. 3. Give a counterexample for the statement If a person is famous, then that person has been on television. Tell how you know it really is a counterexample.

Sample answer: President Abraham Lincoln was and still is famous, but he was never on television. There was no television when Lincoln was alive.

What You Learned 4. Write an example of a conditional statement you would use to teach someone how to identify an hypothesis and a conclusion. See students’ work.

Glencoe Algebra 1

16. Let’s explore these operations a little further. What number is represented by (4 2)? 3375 3 17. What number is represented by (3 4) (3 2)? 585 18. Is the operation actually distributive over the operation ? no Chapter 1

48

Answers

Glencoe Algebra 1

Chapter 1

49

Glencoe Algebra 1

(Lessons 1-6 and 1-7)

d. If 3x 7 13, then x 2. hypothesis

Let’s make up another operation and denote it by , so that a b (a 1)(b 1).

Answers

9

2. What number is represented by 3 2? 23

Lesson 1-7

The kernels heated unevenly.

3? 32 1. What number is represented by 2

Page A23


10:27 AM

Logical Reasoning and Counterexamples

Properties of Operations

NAME ______________________________________________ DATE______________ PERIOD _____

1-7

Study Guide and Intervention Logical Reasoning and Counterexamples


(continued)

Example 2 Identify the hypothesis and conclusion of each statement. Then write the statement in if-then form.

Example 1 Determine a valid conclusion from the statement If two numbers are even, then their sum is even for the given conditions. If a valid conclusion does not follow, write no valid conclusion and explain why.

a. If it is Wednesday, then Jerri has aerobics class. Hypothesis: it is Wednesday Conclusion: Jerri has aerobics class

a. You and Marylynn can watch a movie on Thursday. Hypothesis: it is Thursday Conclusion: you and Marylynn can watch a movie If it is Thursday, then you and Marylynn can watch a movie.

a. The two numbers are 4 and 8. 4 and 8 are even, and 4 8 12. Conclusion: The sum of 4 and 8 is even.

b. If 2x 4 10, then x 7. Hypothesis: 2x 4 10 Conclusion: x 7

b. The sum of two numbers is 20. Consider 13 and 7. 13 7 20 However, 12 8, 19 1, and 18 2 all equal 20. There is no way to determine the two numbers. Therefore there is no valid conclusion.

b. For a number a such that 3a 2 11, a 3. Hypothesis: 3a 2 11 Conclusion: a 3 If 3a 2 11, then a 3.

Exercises Exercises

Identify the hypothesis and conclusion of each statement.

3. If 12 4x 4, then x 2. H: 12 4x 4; C: x 2 4. If it is Monday, then you are in school. H: it is Monday; C: you are in school 5. If the area of a square is 49, then the square has side length 7. H: the area of a

square is 49; C: the square has side length 7

Identify the hypothesis and conclusion of each statement. Then write the statement in if-then form. 6. A quadrilateral with equal sides is a rhombus. H: a quadrilateral has equal sides;

C: the figure is a rhombus; If a quadrilateral has equal sides, then the quadrilateral is a rhombus.

C: the number is divisible by 4; If a number is divisible by 8, then it is divisible by 4.

8. Karlyn goes to the movies when she does not have homework. H: Karlyn does not

have homework. C: Karlyn goes to the movies; If Karlyn does not have homework, then Karlyn goes to the movies.

Chapter 1

50

Glencoe Algebra 1


Glencoe Algebra 1

7. A number that is divisible by 8 is also divisible by 4. H: a number is divisible by 8;


run fast


1. If it is April, then it might rain. H: it is April; C: it might rain 2. If you are a sprinter, then you can run fast. H: you are a sprinter; C: you can

Determine a valid conclusion that follows from the statement If the last digit of a number is 0 or 5, then the number is divisible by 5 for the given conditions. If a valid conclusion does not follow, write no valid conclusion and explain why. 1. The number is 120. Conclusion: 120 is divisible by 5. 2. The number is a multiple of 4. No valid conclusion; a multiple of 4 need not

end in 0 and never ends in 5.

3. The number is 101. No valid conclusion because the number does not end

in 0 or 5

Find a counterexample for each statement. 4. If Susan is in school, then she is in math class. Susan is in school and she is in

history class.

5. If a number is a square, then it is divisible by 2. 25 is a square that is not

divisible by 2.

6. If a quadrilateral has 4 right angles, then the quadrilateral is a square. A rectangle

with 5 and w 6

7. If you were born in New York, then you live in New York. You could be born in

New York and then live in California.

8. If three times a number is greater than 15, then the number must be greater than six.

5.5; 3(5.5) is greater than 15, but 5.5 is less than 6.

9. If 3x 2 10, then x 4. 4; 3(4) 2 10, but 4 is not less than 4. Chapter 1

51

Glencoe Algebra 1

(Lesson 1-7)

A24

Example 2 Provide a counterexample to this conditional statement. If you use a calculator for a math problem, then you will get the answer correct. Counterexample: If the problem is 475 5 and you press 475 5, you will not get the correct answer.

Page A24

Example 1 Identify the hypothesis and conclusion of each statement.

10:27 AM

Deductive Reasoning and Counterexamples Deductive reasoning is the process of using facts, rules, definitions, or properties to reach a valid conclusion. To show that a conditional statement is false, use a counterexample, one example for which the conditional statement is false. You need to find only one counterexample for the statement to be false.

Answers

Conditional Statements A conditional statement is a statement of the form If A, then B. Statements in this form are called if-then statements. The part of the statement immediately following the word if is called the hypothesis. The part of the statement immediately following the word then is called the conclusion.

5/10/06


Lesson 1-7

1-7

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944


NAME ______________________________________________ DATE______________ PERIOD _____

1-7

Skills Practice Logical Reasoning and Counterexamples

10:27 AM




1. If it is Sunday, then mail is not delivered.

1. If it is raining, then the meteorologist’s prediction was accurate.

H: it is Sunday, C: mail is not delivered

H: it is raining, C: the meteorologist’s prediction was accurate

H: you are hiking in the mountains, C: you are outdoors

Identify the hypothesis and conclusion of each statement. Then write the statement in if-then form.

3. If 6n 4 58, then n 9. H: 6n 4 58, C: n 9 Identify the hypothesis and conclusion of each statement. Then write the statement in if-then form.

3. When Joseph has a fever, he stays home from school.

H: Joseph has a fever, C: he stays home from school; If Joseph has a fever, then he stays home from school.

4. Martina works at the bakery every Saturday.

4. Two congruent triangles are similar.

H: two triangles are congruent, C: they are similar; If two triangles are congruent, then they are similar.

5. Ivan only runs early in the morning.

H: Ivan is running, C: it is early in the morning; If Ivan is running, it is early in the morning. 6. A polygon that has five sides is a pentagon.

H: a polygon has five sides, C: it is a pentagon; If a polygon has five sides, then it is a pentagon.

5. The product of two numbers is 12. No valid conclusion; The product is even,

but one of the numbers could be odd, such as 4 3.

8. Hector did not earn an A in science. Hector scored less than 85 on the exam. 9. Hector scored 84 on the science exam. Hector did not earn an A in science. 10. Hector studied 10 hours for the science exam. No valid conclusion; the

conditional statement does not mention the number of hours Hector studied.

Find a counterexample for each statement. 11–14. Sample answers are given. 11. If the car will not start, then it is out of gas. The battery could be dead.

Glencoe Algebra 1

12. If the basketball team has scored 100 points, then they must be winning the game.

The other team could have scored 101 points.

13. If the Commutative Property holds for addition, then it holds for subtraction.

41 14

14. If 2n 3 17, then n 7. When n 7, 2n 3 is equal to 17, not less than 17.

Answers

Glencoe Algebra 1


7. Hector scored an 86 on his science exam. Hector earned an A in science.

6. Two numbers are 8 and 6. The product of the numbers is even. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Determine whether a valid conclusion follows from the statement If Hector scores an 85 or above on his science exam, then he will earn an A in the class for the given condition. If a valid conclusion does not follow, write no valid conclusion and explain why.

Find a counterexample for each statement. 7–8. Sample answers are given. 7. If the refrigerator stopped running, then there was a power outage.

Perhaps someone accidentally unplugged it while cleaning. 8. If 6h 7 5, then h 2.

When h 2, then 6h 7 5, and so is not less than 5.

GEOMETRY For Exercises 9 and 10, use the following information. 9–10. Sample If the perimeter of a rectangle is 14 inches, then its area is 10 square inches. 9. State a condition in which the hypothesis and conclusion are valid.

answers are given.

A rectangle has a length of 5 in. and a width of 2 in. 10. Provide a counterexample to show the statement is false. A rectangle with a length

of 6 in. and a width of 1 in. has a perimeter of 14 in. and an area of 6 in2.

11. ADVERTISING A recent television commercial for a car dealership stated that “no reasonable offer will be refused.” Identify the hypothesis and conclusion of the statement. Then write the statement in if-then form.

H: there is a reasonable offer, C: it will not be refused; If there is a reasonable offer, then it will not be refused. Chapter 1

53

Glencoe Algebra 1

(Lesson 1-7)

A25

Determine whether a valid conclusion follows from the statement If two numbers are even, then their product is even for the given condition. If a valid conclusion does not follow, write no valid conclusion and explain why.

Answers

H: it is Saturday, C: Martina works at the bakery; If it is Saturday, then Martina works at the bakery.

52

Page A25

2. If x 4, then 2x 3 11. H: x 4, C: 2x 3 11

2. If you are hiking in the mountains, then you are outdoors.

Chapter 1

Practice

Lesson 1-7

1-7

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

1-7

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-7


Enrichment

Counterexamples

4. AUTOMOBILES Is the following conclusion valid? If not, find a counterexample. If the weather is sunny, it is a good day to wear a T-shirt.

Some statements in mathematics can be proven false by counterexamples. Consider the following statement. You can prove that this statement is false in general if you can find one example for which the statement is false.

Quadrilaterals

In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample. Sample answers are given.

Trapezoids

1. a (b c) (a b) c

5. If a square is a rhombus and a square is a rectangle, then a rhombus is a rectangle.

3. PRIME NUMBERS For centuries, mathematicians have tried to develop a formula to generate prime numbers. Legendre and Euler each came up with a number of polynomial formulas that generate primes. Consider the following conditional statement and find a counterexample to show that it is not always true. If n is a whole number, 2n2 11 is a prime number.

Not valid; if a square is a rhombus and a rhombus is a parallelogram, then a square is a parallelogram.

6. If a quadrilateral is not a parallelogram, it is a trapezoid.

Not valid; if a quadrilateral is a parallelogram, it is not a trapezoid.

7. If a quadrilateral is not a square, it is not a rhombus. valid

3. a b b a

64 3 2

46

6 (4 2) (6 4) 2 1.5 6 2 2 3 0.75

4. a (b c) (a b) (a c)

6 (4 2) (6 4) (6 2) 6 6 1.5 3 1 4.5

2 3

5. a (bc) (a b)(a c)

6 (4 2) (6 4)(6 2) 6 8 (10)(8) 14 80

6. a2 a2 a4

6 2 62 64 36 36 1296 72 1296

7. Write the verbal equivalents for Exercises 1, 2, and 3.

1. Subtraction is not an associative operation. 2. Division is not an associative operation. 3. Division is not a commutative operation. 8. For the distributive property a(b c) ab ac it is said that multiplication distributes over addition. Exercises 4 and 5 prove that some operations do not distribute. Write a statement for each exercise that indicates this.

4. Division does not distribute over addition. 5. Addition does not distribute over multiplication. Chapter 1

54

Glencoe Algebra 1


Glencoe Algebra 1

When n 11, the expression yields 253, which is divisible by 11 (not prime).


State whether each statement is valid. If it is not valid, write a new statement that is valid.


The area of circle B is 7850 cm2.

2. a (b c) (a b) c

Chapter 1

55

Glencoe Algebra 1

(Lesson 1-7)

A26

6 (4 2) (6 4) 2 6222 4 0

Rhombuses

Answers

In general, for any numbers a and b, the statement a b b a is false. You can make the equivalent verbal statement: subtraction is not a commutative operation.

Page A26

7337 4 4

Parallelograms

Squares

Lesson 1-7

Let a 7 and b 3. Substitute these values in the equation above.

use the following information. The Venn diagram shows the relationships of various quadrilaterals.

Rectangles

10:27 AM

For any numbers a and b, a b b a.

Hypothesis: if she is five years No. If it is a sunny day in winter, old it will not be warm enough for a Conclusion: Helene will go to T-shirt. school If-Then: If Helene is five years old, QUADRILATERALS For Exercises 5–7, then she will go to school.

2. GEOMETRY Write a valid conclusion that follows from the statement below for the given condition. If a valid conclusion does not follow, write no valid conclusion and explain why. If the radius of a circle is multiplied by 10, its area is multiplied by 100. Circle A has a radius of 5 centimeters and an area equal to 78.5 square centimeters, while circle B has a radius of 50 centimeters.

5/10/06

Logical Reasoning and Counterexamples 1. KINDERGARTEN Identify the hypothesis and conclusion and write the statement in if-then form. Helene will go to school when she is five years old.

A1-A34_CRM01-873944


NAME ______________________________________________ DATE______________ PERIOD _____

1-8



Number Systems

Square Roots

The expression 兹3600 苶 is read, “the square root of 3600.” How would you read the expression 兹苶 64?

the square root of 64

Read the Lesson

Example 1

Complete each statement below.

.

radical sign 1. The symbol 兹苵 is called a and is used to indicate a nonnegative or principal square root of the expression under the symbol.

25 represents the negative

2. A rational approximation of an irrational number is a rational number that is close to, but not equal to, the value of the irrational number.

25 5 2 → 49 7

principal

6. The irrational numbers and rational numbers together form the set of real numbers.

What You Learned 7. Use a dictionary to look up several words that begin with “ir-”. What does the prefix “ir-” mean? How can this help you the meaning of the word irrational?

Glencoe Algebra 1

Sample answer: The prefix “ir-” means not. So an irrational number is a number that is not a rational number.

Answers

Glencoe Algebra 1


3025


1600

b. the negative square root of 729 729

56

兹0.16 苶 0.4

Find each square root.

5. Write each of the following as a mathematical expression that uses the 兹苵 symbol.

Chapter 1

冑苳

49

25 5 49 7

Exercises

square

4. A number whose positive square root is a rational number is a perfect square .

c. the principal square root of 3025

25 . square root of

冢冣

Find 0.16 .

兹0.16 苶 represents the positive and negative square roots of 0.16. 0.16 0.42 and 0.16 (0.4)2

苶 8 1. 兹64

2. 兹81 苶 9

4. 兹100 苶 10

5.

冑苳 25

6. 兹121 苶 11

8.

冑苳 54

9.

7.

5 冑苳 12 25 144

10. 兹3600 苶 60

13.

冑苳 67 144 196

Chapter 1

4 25

25 16

11. 兹6.25 苶 2.5

14.

冑苳 67 36 49

57

3. 兹16.81 苶 4.1

11 冑苳 10 121 100

12. 兹0.000 苶4苶 0.02

15. 兹1.21 苶 1.1

Glencoe Algebra 1

(Lesson 1-8)

A27

3. The positive square root of a number is called the root of the number.

Example 2

25 49

Answers

冑苳 49

Find

Page A27

A square root is one of two equal factors of a number. For example, the square roots of 36 are 6 and 6, since 6 6 or 62 is 36 and (6)(6) or (6)2 is also 36. A rational number like 36, whose square root is a rational number, is called a perfect square. The symbol 兹苵 is a radical sign. It indicates the nonnegative, or principal, square root of 36 6 and 兹36 苶 6. The symbol 兹36 苶 the number under the radical sign. So 兹苶 represents both square roots.


a. the positive square root of 1600

10:27 AM

Number Systems


Lesson 1-8

1-8

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____


NAME ______________________________________________ DATE______________ PERIOD _____

1-8

(continued)

Skills Practice

Number Systems

{1, 2, 3, 4, …}

Whole Numbers

{0, 1, 2, 3, 4, …}

Integers

{…, 3, 2, 1, 0, 1, 2, 3, …}

Rational Numbers

a {all numbers that can be expressed in the form , where a and b are integers and b 0} b

Irrational Numbers

a {all numbers that cannot be expressed in the form , where a and b are integers and b 0} b

5. 兹17 苶 4.12

6. 兹2.25 苶 1.5

冑苳

5 6

28 7

8.

integer, rational

rational

9. 兹29 苶

10. 兹196 苶

irrational

natural, whole, integer, rational

9 13

12. 兹1.8 苶

11.

Because 兹81 苶 9, this number is a natural number, a whole number, an integer, and a rational number.

c.

32

Because 兹32 苶 5.656854249…, which is not a repeating or terminating decimal, this number is irrational.

rational

irrational

Graph each solution set.


2 3. 3

6 2. 7

rational

6. 兹25 苶

5. 3.145

rational

苶 4. 兹54

rational

irrational 8. 兹22.51 苶

7. 0.62626262…


rational

irrational

Write each set of numbers in order from least to greatest. 3 4

7 4

3 5

苶, 9. , 5, 兹25

25

5 12. , 2, 兹124 苶, 3.11 4

5 4

3.11, 2, ,

124

0.3131…,

3 0.09 , 5

1 13. 兹1.44 苶, 0.35 5

1

1.2 5 , , 0.05, 4

5

1 9 14. 0.3 苶5 苶, 2 , , 兹5 苶 3 5

1 5

1.44 , 0.35, 58

1 4

11. 1.2 苶5 苶, 0.05, , 兹5 苶

9

, 0.3 5 , 5

1 5, 2 3

Glencoe Algebra 1


Glencoe Algebra 1

5, , ,

10. 兹0.09 苶, 0.3131…,


Name the set or sets of numbers to which each real number belongs.


13. x 1

Exercises

2 1

14. x 1 0

1

2

3

4

5

6

15. x 1.5

4 3 2 1

0

1

2

3

4

0

1

2

3

4

16. x 2.5

4 3 2 1

0

1

2

3

4

4 3 2 1

Replace each ● with , , or to make each sentence true. 4 9

1 90

苶 17. ● 0.4

18. 0.0 苶9 苶●

19. 6.2 苶3 苶 ● 兹39 苶

20. ●

1 8

1 8 兹苶

Write each set of numbers in order from least to greatest. 7 3

苶, 2.3 苶6 苶, 21. 兹5

7 5 , , 2.3 6 3

23. 兹12 苶, 3.4 苶8 苶, 兹11 苶

兹6 苶 3 3 6 24. 0.4 苶3 苶, , , 0.43, 5

3.4 8 , 12 , 11 Chapter 1

2 0.05 9

2 9

22. , 0.2 苶1 苶, 兹0.05 苶 0.2 1 , ,

59

7 7

5

Glencoe Algebra 1

(Lesson 1-8)

A28

81

84 1. 12

7 10

49 100

7.


b.

Chapter 1

4.


Because 4 and 11 are integers, this number is a rational number.

3 7 4 4

3. 兹0.25 苶 0.5

Page A28

4 11

a.

2. 兹36 苶 6

Answers

Example

苶 12 1. 兹144

10:27 AM

Classify and Order Numbers Numbers such as 兹2苶 and 兹3苶 are not perfect squares. Notice what happens when you find these square roots with your calculator. The numbers continue indefinitely without any pattern of repeating digits. Numbers that cannot be written as a terminating or repeating decimal are called irrational numbers. The set of real numbers consists of the set of irrational numbers and the set of rational numbers together. The chart below illustrates the various kinds of real numbers. Natural Numbers

5/10/06

Number Systems Find each square root. If necessary, round to the nearest hundredth.

Lesson 1-8

1-8

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944


NAME ______________________________________________ DATE______________ PERIOD _____

1-8

Practice


Number Systems

冑苳

6.

2

17

5

冑苳 7 12

9.17

7. 兹0.081 苶

0.76

8. 兹3.06 苶

0.28

1.75

Amanda

Boyd

Celeste

兹97 苶

兹2.56 苶

23 8

Dominic

Eve

2.56 7

兹49 苶

Name the set or sets of numbers to which each real number belongs. 8 7

10. 兹0.062 苶5 苶

irrational

rational

Eve, Boyd, Celeste, Amanda, Dominic

144 3

12.

11.

rational

integer, rational

2. SPORTS Matthew won the 100-yard dash in a photo-finish race with a time of 15.83 seconds. Brady’s time was 15.84 seconds, and he came in third place. Use a number line to graph Matthew’s time, Brady’s time, and the possible time of the person who finished in second place.

Graph each solution set. 13. x 0.5 0

1

2

3

4

4 3 2 1

0

1

2

3

4

Matthew

2 0.1 7 , 0.03 , 8

84

7

8 , , 30 8

兹35 苶 2

19 20

20. 兹8.5 苶, , 2

19

35 , 2 , 8.5 20 2

21. SIGHTSEEING The distance you can see to the horizon is given by the formula 1.5h, where d is the distance in miles and h is the height in feet above the d 兹苶 horizon line. Mt. Whitney is the highest point in the contiguous 48 states. Its elevation is 14,494 feet. The lowest elevation, at 282 feet, is located near Badwater, California. With a clear enough sky and no obstructions, could you see from the top of Mt. Whitney to Badwater if the distance between them is 135 miles? Explain. Yes; you can see

about 149 miles from the top of Mt. Whitney to an elevation of 282 feet.

Glencoe Algebra 1

22. SEISMIC WAVES A tsunami is a seismic wave caused by an earthquake on the ocean 苶, where s is the speed in meters per second and floor. You can use the formula s 3.1兹d d is the depth of the ocean in meters, to determine the speed of a tsunami. If an earthquake occurs at a depth of 200 meters, what is the speed of the tsunami generated by the earthquake? about 43.8 m/s Chapter 1

60

Answers

Glencoe Algebra 1

Pythagorean Theorem a2 b2 c2 b

15.85

3. WEATHER The table shows how the average temperature for each month varied from the normal mean temperature each month for Barrow, Alaska. Graph these values on a number. Month

Change in Temp. (°F)

Month

Change in Temp. (°F)

Jan.

–3

Jul.

5

Feb.

–2

Aug.

–1

Mar.

2

Sep.

–8

Apr.

13

Oct.

–16

May

21

Nov.

–16

c

Jun.

15

Dec.

–10

a

The length of side c can be found by using the following rearrangement of the 2 b2 苶苶. Pythagorean Theorem: c 兹a 2 b2 苶苶 have a symbol in 5. Should c 兹a front of the c? Squaring the value

of c or c will give c 2 in the original formula. However, a negative value of c is not possible because the formula uses lengths of sides of a triangle, which can only be positive numbers.

Source: World Almanac 2005, pg 185

20

Chapter 1

10

0

Jul.

兹7 苶 8

84 30

19. , 兹8 苶,

15.84

GEOMETRY For Exercises 5 and 6, use the following information. The Pythagorean Theorem is used to find the length of an unknown side of a right triangle when two side lengths are known.

Second place 15.835

Mar.

兹2 苶 8

苶, , 0.1 苶7 苶 18. 兹0.03

Brady

15.83

Jan. Feb. Aug.

Write each set of numbers in order from least to greatest.

15.82

Sept.

15.81

Oct., Nov.

16. 8.1 苶7 苶 ● 兹66 苶


5 兹5 苶 17. ● 6


苶3 苶 ● 兹0.93 苶 15. 0.9

15.80

Dec.

Replace each ● with , , or to make each sentence true.

冪莦 B

B is the brightness (in lumens per square inch). Using a light meter, a product engineer finds the brightness of a 200-watt bulb is 0.244 lumens per square inch. How far is the light meter from the bulb? 36 in.

10

6. Find the length of the hypotenuse c if a 6 centimeters and b 8 centimeters.

10 cm

20

61

Glencoe Algebra 1

(Lesson 1-8)

A29

4 3 2 1

14. x 3.5

318 given by the equation D , where

Answers

苶 9. 兹93

4. LIGHTING The brightness of a light bulb depends on the observer’s distance from the bulb. For a 200-watt bulb, the distance D (in inches) from the bulb is

Page A29

5.

7.87

4 289

4. 兹84 苶

Lesson 1-8

18

3. 兹25 苶

May

2. 兹62 苶

Jun.

苶 1. 兹324

1. MATH CLASS In Mrs. Carson’s math class, students draw numbers to determine the order in which each will solve a problem on the board. If the order is least to greatest value, list the students in order of their turn.

10:27 AM

Number Systems

Find each square root. If necessary, round to the nearest hundredth.

Apr.

1-8

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

1-8

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-8

Enrichment

Graphing Calculator Activity

Scale Drawings

Closet

Closet Bath

Bedroom

200 ft 2. On the map, measure the frontage of Lot 2 on Sylvan Road in inches. What is the actual frontage in feet?

200 ft 3. What is the scale ratio represented on the floor plan?

1:100 4. On the floor plan, measure the width of the living room in centimeters. What is the actual width in meters?

4m 5. About how many square meters of carpeting would be needed to carpet the living room?

24

m2

Answers will vary. 7. Use your scale drawing to determine how many square meters of tile would be needed to install a new floor in your classroom.

Exercises Evaluate each expression for each set of values. Express answers as fractions when possible. 1. 3x 8y 2z

115 4

2. |2a 5b|

a. x 1, y 2, and z 6

a. a 4 and b 16 72

b. x 5, y 7, and z 1 70

b. a 5 and b 20 110

4

2

3. 5x2 4x 12

b |b 4ac| 4. 2a 2

3 2

a. x 8 340

a. a 4, b 12, and c 9

b. x 5 93

b. a 3, b 7, and c 20 47

c. x 1

c. a 2, b 8, and c 5 24

3

91 9

5. Create a rational expression with three variables and an absolute value. Choose values for the variables and evaluate your expression. See students’ work.

Answers will vary. Chapter 1

62

Glencoe Algebra 1


Glencoe Algebra 1

6. Make a scale drawing of your classroom using an appropriate scale.


1. On the map, how many feet are represented by an inch?


Answer each question.

b. a 3, b 8, and c 4 To evaluate the expression again, you do not have to repeat the keystrokes from part a. Instead use the replay command, 2nd [ENTRY]. The expression appears again without the answer. Use the arrow keys to scroll to the beginning of the expression and change the values for a, b, and c. Then press ENTER to re-evaluate.

Chapter 1

63

Glencoe Algebra 1

(Lesson 1-8)

A30

Sunshine Lake

Closet

Lesson 1-8

Sylvan Road

Closet

Page A30

Kitchen

Answers

a. a 4, b 6, and c 5 Enter the values for a, b, and c using STO . Then enter the expression. Use parentheses to group the numerator and the denominator. The absolute value function can be found in the NUM menu of MATH . to add a closing parenthesis when using abs( . The Frac command from the MATH menu displays the answer as a fraction. Keystrokes: 4 STO ALPHA [A] ALPHA [:] 6 STO ALPHA [A] ALPHA [:] 5 STO ALPHA [A] ALPHA [:] ( ALPHA [C] x 2 — MATH ) ) ENTER ALPHA [A] x 2 — ALPHA [B] x 2 ( 2 ALPHA [A] ALPHA [B] ) MATH ENTER ENTER

Dining Area

Lot 2

for each set of values. Express Evaluate 2ab

your answers as fractions.

Living Room

Bedroom

c2 |a2 b2|

Example

Lot 1

10:27 AM

The map at the left below shows building lots for sale. The scale ratio is 1:2400. At the right below is the floor plan for a two-bedroom apartment. The length of the living room is 6 m. On the plan the living room is 6 cm long.

Lot 3

5/10/06

Evaluating Expressions When evaluating the same algebraic expression for different sets of rational values, it is sometimes helpful to use the store key STO and ENTRY which is the 2nd function of ENTER . ENTRY allows you to scroll up to a previous line.

A1-A34_CRM01-873944


1-9

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

NAME ______________________________________________ DATE______________ PERIOD _____

1-9



Functions and Graphs

10:27 AM



Interpret Graphs

A function is a relationship between input and output values. In a function, there is exactly one output for each input. The input values are associated with the independent variable, and the output values are associated with the dependent variable. Functions can be graphed without using a scale to show the general shape of the graph that represents the function.

Read the introduction to Lesson 1-9 in your textbook. The numbers 25%, 50% and 75% represent the

Example 1 The graph below represents the height of a football after it is kicked downfield. Identify the independent and the dependent variable. Then describe what is happening in the graph.

Read the Lesson 1. Write another name for each term. a. coordinate system coordinate plane

Page A31

percent of blood flow to the brain and the numbers 0 number of days after the concussion .

through 10 represent the

Example 2 The graph below represents the price of stock over time. Identify the independent and dependent variable. Then describe what is happening in the graph.

Height

c. vertical axis y-axis

Time Time

2. Identify each part of the coordinate system.

The independent variable is time and the dependent variable is price. The price increases steadily, then it falls, then increases, then falls again.

y-axis

3. In your own words, tell what is meant by the dependent variable and independent variable. Use the example below. dependent variable

independent variable

the distance it takes to stop a motor vehicle

is a function of

d

the speed at which the vehicle is traveling s

Sample answer: The value of the dependent variable is a result of the value of the independent variable. Since d is a result of s, d is the dependent variable and s is the independent variable.

Glencoe Algebra 1

What You Learned 4. In the alphabet, x comes before y. Use this fact to describe a method for ing how to write ordered pairs. Sample answer: Since x comes before y, when

writing ordered pairs, write the x value before the y value.

Chapter 1

64

Answers

Glencoe Algebra 1

1. The graph represents the speed of a car as it travels to the grocery store. Identify the independent and dependent variable. Then describe what is happening in the graph.

Speed

Ind: time; dep: speed. The car starts from a standstill, accelerates, then travels at a constant speed for a while. Then it slows down and stops. 2. The graph represents the balance of a savings over time. Identify the independent and the dependent variable. Then describe what is happening in the graph.

Time

Balance (dollars)

Ind: time; dep: balance. The balance has an initial value then it increases as deposits are made. It then stays the same for a while, again increases, and lastly goes to 0 as withdrawals are made. 3. The graph represents the height of a baseball after it is hit. Identify the independent and the dependent variable. Then describe what is happening in the graph.

Ind: time; dep: height. The ball is hit a certain height above the ground. The height of the ball increases until it reaches its maximum value, then the height decreases until the ball hits the ground. Chapter 1

65

Time

Height Time

Glencoe Algebra 1

Lesson 1-9


x

O


Exercises

x-axis

origin

(Lesson 1-9)

The independent variable is time, and the dependent variable is height. The football starts on the ground when it is kicked. It gains altitude until it reaches a maximum height, then it loses altitude until it falls to the ground.

y

A31

Answers

Price

b. horizontal axis x-axis


NAME ______________________________________________ DATE______________ PERIOD _____

1-9

(continued)


Skills Practice 2. The graph below represents a puppy exploring a trail. Describe what is happening in the graph. Is the function discrete or continuous? Distance from Trailhead

Height

Example

A music store s that if you buy 3 CDs at the regular price of $16, then you will receive one CD of the same or lesser value free.

1

2

3

4

5

Total Cost ($)

16

32

48

48

64

CD Cost 80 Cost ($)

1

2

3

4

20

21

23

23

24

0

Value ($)

a. Identify the independent and dependent variables.

ind: age; dep: length

1

2

3

4

20,000 18,000 16,000 14,000 13,000

a. Identify the independent and dependent variables. ind: age; dep: value

b. Write a set of ordered pairs representing the data in the table.

b. Draw a graph showing the relationship between age and value. Is the function discrete or continuous? Value (thousands of $)

(0, 20), (1, 21), (2, 23), (3, 23), (4, 24) c. Draw a graph showing the relationship between age and length. 25 24

20 18 16 14 12 0

22 21

1 2 3 4 Age (years)

Time

LAUNDRY For Exercises 4–7, use the table that shows the charges for washing and pressing shirts at a cleaners.

Time

Number of Shirts

2

4

6

8

Total Cost ($)

3

6

9

12 15 18

10 12

4. Identify the independent and dependent variables.

independent : number of shirts; dependent: total cost 5. Write the ordered pairs the table represents.

(2, 3), (4, 6), (6, 9), (8, 12), (10, 15), (12, 18) 6. Draw a graph of the data.

21 18 15 12 9 6 3

5

0

2

4 6 8 10 12 14 Number of Shirts

20 0

7. Use the data to predict the cost for washing and pressing 16 shirts. $24

1 2 3 4 5 Age (months)

66

Glencoe Algebra 1


Glencoe Algebra 1

23

The function is discrete.

22

Total Rainfall

Time

2. The table below represents the value of a car versus its age. Age (years)

Total Rainfall

Total Cost ($)

0

Length (inches)

Total Rainfall

6


Age (months)

1 2 3 4 5 Number of CDs

Chapter 1

67

Glencoe Algebra 1

(Lesson 1-9)

A32

1. The table below represents the length of a baby versus its age in months.

The function is discrete.

20 0

Exercises

Length (inches)

40

The puppy goes a distance on the trail, stays there for a while, goes ahead some more, stays there for a while, then goes back to the beginning of the trail. The function is continuous.

3. WEATHER During a storm, it rained lightly for a while, then poured heavily, and then stopped for a while. Then it rained moderately for a while before finally ending. Which graph represents this situation? C A B C

60


b. Write the data as a set of ordered pairs. (1, 16), (2, 32), (3, 48), (4, 48), (5, 64)

Chapter 1

The football is thrown upward from above the ground, reaches its maximum height, and then falls downward until it hits the ground.

c. Draw a graph that shows the relationship between the number of CDs and the total cost. Is the function discrete or continuous?

Answers

Number of CDs

Time

Time

Page A32

a. Make a table showing the cost of buying 1 to 5 CDs.

10:27 AM

Draw Graphs You can represent the graph of a function using a coordinate system. Input and output values are represented on the graph using ordered pairs of the form (x, y). The x-value, called the x-coordinate, corresponds to the x-axis, and the y-value, or y-coordinate corresponds to the y-axis. A discrete function is a function whose graph consists of points that are not connected. When a function can be graphed with a line or smooth curve, it is a continuous function.

5/10/06

Functions and Graphs 1. The graph below represents the path of a football thrown in the air. Describe what is happening in the graph.

Lesson 1-9

1-9

A1-A34_CRM01-873944

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

A1-A34_CRM01-873944


NAME ______________________________________________ DATE______________ PERIOD _____

1-9

Practice


Functions and Graphs 2. The graph below represents a student taking an exam. Describe what is happening in the graph.

1. BAKING Identify the graph that shows the relationship between the number of cookies and the equivalent number of dozens.

Time

As the tsunami approaches shore, the height of the wave increases more and more quickly.

Area Burning

x

Number of dozens

y 80 75 70 65 60 55 50 45 40

Time

Maple Syrup

35

y

4.50

9.00

3

4

5

13.50 18.00 22.50

4. Write the ordered pairs the table represents.(1, 4.5), (2, 9), (3, 13.5), (4, 18), (5, 22.5)

6. Use the data to predict the cost of subscribing for 9 months. $40.50

27.00 Total Cost ($)

5. Draw a graph of the data. Is the function discrete or continuous? The function is discrete.

22.50 18.00 13.50 9.00 4.50 0

Glencoe Algebra 1

7. SAVINGS Jennifer deposited a sum of money in her and then deposited equal amounts monthly for 5 months, nothing for 3 months, and then resumed equal monthly deposits. Sketch a reasonable graph of the history.

1 2 3 4 5 6 Number of Months

Balance ($)

1800

4

WEATHER For Exercises 5–7, use the

3

following information.

2

One way to estimate the distance of a thunderstorm is to count the number of seconds that from the sight of a flash of lightning until thunder is heard. Divide this number by 5 to get the approximate distance (in miles) of the storm.

1 0

Answers

Glencoe Algebra 1

80 120 160 200 240 x

40

gallons of sap

3. SALES TAX The graph below shows the amount of tax paid on items of a certain cost. Name the independent and dependent variables.

5. Identify the independent and dependent variables.

Independent: seconds counted Dependent: distance of storm

Sales Tax

Independent: cost of item Dependent: amount of tax

y 2.00 1.50 1.00 0.50 0

68

2006 x

Year

Time

Chapter 1

1900

5

5

10

15

20

25 x

Domain: 0 to 50 s Range: 0 to 10 mi

7. Is the function discrete or continuous?

cost of item ($)

Chapter 1

6. Suppose you can generally hear thunder up to 10 miles away. Identify an appropriate domain and range for this situation.

continuous

69

Glencoe Algebra 1

Lesson 1-9

2

amount of tax ($)

Total Cost ($)

1


Number of Months


charges for subscribing to an independent news server.

gallons of syrup

6

INTERNET NEWS SERVICE For Exercises 4–6, use the table that shows the monthly

(Lesson 1-9)

A33

Time

x

Number of dozens

2. NATURE It takes about 40 gallons of sap from maple trees to make 1 gallon of syrup. Let the number of gallons of sap be the independent variable. Draw a reasonable graph showing the number of gallons of syrup produced from a given amount of sap.

Area Burning

Time

Sales Tax

x

Number of dozens

Graph B

3. FOREST FIRES A forest fire grows slowly at first, then rapidly as the wind increases. After firefighters answer the call, the fire grows slowly for a while, but then the firefighters contain the fire before extinguishing it. Which graph represents this situation? B A B C Area Burning

y

Answers

The student steadily answers questions, then pauses, resumes answering, pauses again, then resumes answering.

Graph C

y

Life expectancy (years)

Time

Graph B Number of cookies

Number of cookies

Graph A y

4. AGING A person born in the early 1800s had a life expectancy of about 37 years. With improvements in medical care and pharmaceuticals, life expectancy has increased significantly. In 1900, it rose to 48 years and in 2006 to almost 78 years. Draw a reasonable graph showing the change in life expectancy.

Page A33

Number of Questions Answered

Height

10:27 AM


1. The graph below represents the height of a tsunami (tidal wave) as it approaches shore. Describe what is happening in the graph.

Number of cookies

1-9

5/10/06

Chapter 1

NAME ______________________________________________ DATE______________ PERIOD _____

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