Rational Numbers & Equations

Name ________________________

Period _____

Revised 8/2014

2

3

TABLE OF CONTENTS

PA Common Core 8TH Grade Math Standards .……………………………..………… 4

Coordinate Plane …………...………………………………………………….…………… 5

Rational Numbers Directions …...……………………………………….....………….. 8

Integer Operations …………………………………………………………..…………….. 9

Rational Numbers Operations …………………………………………….………….… 18

Evaluate Algebraic Expressions ………………………………………………………. 24

Equations ………………………………………………………………………….………. 26

Applications of Equations ……………………………………………………………… 30

Pythagorean Theorem …………………………………………………………………… 33

8th Grade PSSA Formula Sheet …………………………………………….………….. 35

Vocabulary & Verbal Models …………………………..…………………….………… 36

Multiplication Table ……………………………………………………………………… 38

SPECIAL NOTE This workbook is designed for 8th grade math students at Log College Middle School. Any person who wishes to copy any part of this book must have written permission from Mrs. Gismondi, who created, designed and compiled this work.

4

PA Common Core 8th Grade Mathematics Standards in Workbook #1 REPORTING CATEGORY – M08.A-N: THE NUMBER SYSTEM ANCHOR M08.A-N.1: Demonstrate an understanding of rational and irrational numbers. Descriptor M08.A-N.1.1: Apply concepts of rational and irrational numbers.

M08.A-N.1.1.1: Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths). 2.1, 4.7 M08.A-N.1.1.2: Convert a terminating or repeating decimal to a rational number (limit repeating decimals to thousandths). 2.1 M08.A-N.1.1.3: Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144). Example: √5 is between 2 and 3 but closer to 2. 4.5, 4.6, 4.6 Lab & Tech Lab M08.A-N.1.1.4: Use rational approximations of irrational numbers to compare and order irrational numbers. 2.2, 4.6 M08.A-N.1.1.5: Locate/identify rational and irrational numbers at their approximate locations on a number line. 2.2, 4.7X

REPORTING CATEGORY – M08.B-E: EXPRESSIONS AND EQUATIONS ANCHOR M08.B-E.1: Demonstrate an understanding of expressions and equations with radicals and integer exponents. Descriptor M08.B-E.1.1: Represent and use expressions and equations to solve problems involving radicals and integer exponents.

M08.B-E.1.1.1: Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents). Properties will be provided. Example: 312 × 3 -15 = 3 -3 = 1/(33) 4.2, 4.3 M08.B-E.1.1.2: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of perfect squares (up to and including 122) and cube roots of perfect cubes (up to and including 53) without a calculator. Example: If x2 = 25 then x = ±√25. 4.5, 8.5, 8.6, 8.9

ANCHOR M08.B-E.3: Analyze and solve linear equations and pairs of simultaneous linear equations. Descriptor M08.B-E.3.1: Write, solve, graph, and interpret linear equations in one or two variables, using various methods.

M08.B-E.3.1.1: Write and identify linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 2.7, 2.8 Lab, 2.8, 8.5, 8.6, 8.9, 11.1, 11.2, 11.3, 11.3A M08.B-E.3.1.2: Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 2.7, 2.8 Lab, 2.8, 8.5, 8.6, 8.9, 11.1, 11.2, 11.3, 11.3A

REPORTING CATEGORY – M08.C-G: GEOMETRY ANCHOR M08.C-G.2: Understand and apply the Pythagorean theorem. Descriptor M08.C-G.2.1: Solve problems involving right triangles by applying the Pythagorean theorem.

M08.C-G.2.1.1: Apply the converse of the Pythagorean theorem to show a triangle is a right triangle. 4.8, 4.9, 4.9X M08.C-G.2.1.2: Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Figures provided for problems in 3-dimensions will be consistent with Eligible Content in grade 8 and below.) 4.8, 4.9 M08.C-G.2.1.3: Apply the Pythagorean theorem to find the distance between 2 points in coordinate system. 4.8, 4.9

ANCHOR M08.C-G.3: Solve real world and mathematical problems involving volume. Descriptor M08.C-G.3.1: Apply volume formulas of cones, cylinders, and spheres.

M08.C-G.3.1.1: Apply formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems. Formulas will be provided. 8.5 Lab, 8.5, 8.6 Lab, 8.6, 8.9

5

6

Scaled Sketches

0 20 40 60 80 100 120 140 160 180 0 2 4 6 8 10 12 14 16 18 20

7

x

0

y

0

x

0

y

0

8

DIR

ECTIO

NS FO

R R

ATIO

NA

L NU

MB

ERS

9

Simplify each of the following: Absolute Value & Opposites 1. – ( – 4 ) 2. – ( – 9 ) 3. – ( 7 + 3 )

4. – ( 8 / 4 ) 5. – ( 18 – 11 )

6. − −−25

7. −−48

8. −−721

9. 1 10. − 3 11. 1 6+ 12. 64 8/ 13. 7 8+ − 14. − • −4 9 15. 18 6− −

16. – ( – 8 ) 17. – ( 7 + 12 ) 18. – ( 5 + 2 ) 19. – ( 15 – 8 ) 20. – ( 5 – 4 )

21. −− 67

22. − 36

23. − −−515

24. −41 25. 9 26. 5 1− 27. 2 10+ 28. − + −3 4 29. − •7 5 30. 45 36−

31. – [– (– 6 ) ] 32. – ( 9 � 7 ) 33. – ( 36 / 6 ) 34. – ( 3 � 0 ) 35. – ( 6 – 1 )

36. −−49

37. 812−

38. −−

624

39. 2 40. − 20 41. 15 5− 42. 3 7• 43. − − −6 4

44. 3 3+ − 45. − −24 8/

46. ( )[ ]3−−− 47. – ( 6 � 2) 48. – ( 7 / 7 ) 49. – ( 9 + 6 ) 50. – ( 14 – 8 )

51. − −−317

52. − 915

53. − −−210

54. − 11 55. 5 56. 7 16+ 57. 14 9− 58. 12 13+ 59. − •7 10 60. 32 16/ −

10

Integers Worksheet 1 Demonstrate the following problems using 2-colored counters. Write answer where indicated.

1. 3 + 4 Answer = 7

9. – 1 + 1 Answer =

17. 5 + 4 Answer =

2. 6 + 5 Answer =

10. –3 + 5 Answer =

18. –4 + – 7 Answer =

3. 5 + 2 Answer =

11. 7 + – 2 Answer =

19. – 11 + 8 Answer =

4. 9 + 1 Answer =

12. –4 + 1 Answer =

20. 5 + – 12 Answer =

5. –6 + –4 Answer = – 10

13. 6 + – 8 Answer =

21. 13 + – 9 Answer =

6. –7 + – 5 Answer =

14. 9 + – 5 Answer =

22. – 10 + 3 Answer =

7. – 4 + – 3 Answer =

15. –12 + 10 Answer =

23. – 4 + – 10 Answer =

8. – 8 + – 1 Answer =

16. 9 + – 2 Answer =

24. –7 + 7 Answer =

11

Add integers 1 1. –4 + 3 2. 7 + – 9 3. 0 + 3 4. 0 + 4 5. 18 + – 11 6. –2 + – 1 7. –8 + 4 8. –4 + − 3 9. 9 + – 9 10. –6 + − 4 11. 0 + – 2 12. 8 + – 7 13. –2 + 11 14. –3 + – 3 15. 8 + – 2 16. –13 + − 5 17. 0 + – 3 18. 0 + – 6 19. –64 + – 8 20. –7 + – 8

21. 0 + 8 22. 17 + – 3 23. 5 + – 2 24. 15 + – 7 25. 2 + – 3 26. –8 + 1 27. –6 + 11 28. 2 + – 3 29. –1 + − 6 30. 0 + – 1 31. 5 + 0 32. 5 + – 6 33. 7 + – 17 34. 0 + − 7 35. 6 + – 7 36. –14 + − 7 37. 5 + 9 38. – 5 + – 1 39. –2 + − 10 40. –8 + − 5

41. –2 + − 8 42. – 9 + – 7 43. 36 + – 6 44. – 3 + 0 45. – 6 + – 0 46. – 9 + – 1 47. – 8 + 2 48. – 6 + 8 49. – 40 + – 10 50. – 5 + − 7 51. 12 + – 5 52. – 7 + – 4 53. 6 + − 6 54. – 2 + – 6 55. – 8 + – 4 56. –22 + − 8 57. 0 + − 20 58. 15 + – 5 59. – 3 + – 7 60. 0 + 10

61. – 4 + 8 62. – 6 + – 3 63. 7 + – 7 64. 9 + − 6 65. 4 + – 8 66. 6 + 5 67. 4 + − 7 68. – 4 + – 2 69. 3 + – 10 70. 63 + 7 71. – 13 + − 6 72. 0 + 2 73. – 2 + – 13 74. – 19 + – 1 75. –8 + − 9 76. – 11 + − 9 77. – 5 + 7 78. 7 + − 16 79. 4 + – 19 80. 11 + – 11

12

Add Integers 2

1. –5 + 3 2. 7 + – 8 3. 0 + 6 4. 1 + 9 5. 15 + – 12 6. –8 + – 6 7. –9 + 14 8. –24 + − 3 9. 7 + – 9 10. –4 + − 4 11. 0 + – 1 12. 13 + – 7 13. –12 + 11 14. –6 + – 3 15. 9 + – 12 16. –13 + − 5 17. 8 + – 10 18. 20 + – 6 19. –64 + – 8 20. –1 + – 1

21. 3 + 8 22. 16 + – 3 23. 5 + – 8 24. 14 + – 7 25. 2 + – 8 26. –6 + 1 27. –6 + 17 28. 31 + – 43 29. –11 + − 6 30. 7 + – 1 31. 5 + 3 32. 5 + – 6 33. 17 + – 17 34. 4 + − 7 35. 8 + – 7 36. –14 + − 7 37. 18 + 9 38. – 28 + – 3 39. –2 + − 10 40. –28 + − 5

41. –4 + – 8 42. – 9 + – 11 43. 42 + – 6 44. – 3 + 5 45. – 6 + – 7 46. – 12 + – 1 47. – 5 + 2 48. – 23 + 8 49. – 40 + – 30 50. – 12 + − 7 51. 12 + – 7 52. – 7 + – 4 53. 15 + − 6 54. – 11 + – 6 55. –18 + – 4 56. –22 + – 8 57. 21 + – 20 58. 32 + – 5 59. – 81 + – 7 60. 20 + 10

61. – 3 + 8 62. – 6 + – 12 63. 17 + – 7 64. 18 + – 6 65. 24 + – 18 66. 36 + 5 67. 54 + – 7 68. – 44 + – 22 69. 43 + – 10 70. 43 + 7 71. – 18 + – 6 72. 0 + 2 73. – 12 + – 15 74. – 19 + – 21 75. –7 + – 9 76. – 19 + – 9 77. – 25 + 27 78. 53 + – 16 79. 24 + – 19 80. 31 + – 11

13

Integers Worksheet Write the following vertically. Use P=Positive, N=Negative. Write the answers. P N 1. 5 – 4 = 1 5 P - 4 N 1

8. 12 − −4=

15. 17 – 6 =

P N 2. 6 – 3 = 3

6 P −3 N 3

9. 15 − − 7 =

16. 21 – 43 =

3. 9 – 6 =

10. –6 − − 3=

17. 3 − − 7=

4. – 4 – 2 =

11. − −−18 13=

18. – 21 – 35 =

5. – 8 – 3 =

12. 4 – 16 =

19. – 4 − − 4=

6. – 10 – 5 =

13. 19 – 11 =

20. 33 – 43 =

7. – 11 – 9 =

14. – 3 – 9 =

21. –7 − − 6=

14

Subtract Integers 1

1. – 6 – 9 2. – 26 – 11 3. 9 – 8 4. 11 – 41 5. 13 – 19 6. – 9 – 4 7. – 1 – 13 8. –2 − − 3 9. 23 + – 3 10. – 9 − − 4 11. 30 + – 6 12. 6 + – 8 13. –5 – 18 14. – 3 – 4 15. 16 + – 18 16. –13 − − 13 17. 6 + – 6 18. 9 – 0 19. –12 + – 13 20. – 36 – 16

21. 6 – 8 22. – 19 + – 13 23. 9 + – 2 24. 16 – 13 25. 5 – 12 26. – 9 + 20 27. – 8 + 16 28. 9 + – 3 29. –17 − − 6 30. 14 – 0 31. 23 – 4 32. 5 – 7 33. 6 – 37 34. 7 − − 7 35. – 3 + – 5 36. –14 − − 10 37. 5 – 11 38. – 24 – 1 39. –61 − − 10 40. – 14 − − 8

41. 13 − − 8 42. – 5 + – 11 43. 41 – 7 44. – 9 + 1 45. – 7 – 16 46. – 11 – 13 47. – 13 – 14 48. – 26 – 15 49. – 34 + – 21 50. – 35 − − 7 51. 62 – 53 52. – 25 + – 13 53. 70 − − 6 54. – 51 + – 6 55. – 72 – 52 56. –16 − − 8 57. 20 − − 20 58. 10 – 15 59. – 7 + – 23 60. 30 + 39

61. 12 – 8 62. – 7 − − 1 63. 8 − − 3 64. 4 − − 60 65. 8 – 18 66. 45 + 11 67. 14 − − 7 68. – 7 + – 2 69. 23 – 29 70. 47 – 51 71. – 14 − − 6 72. 1 – 7 73. – 39 – 16 74. – 81 – 1 75. – 45 − − 5 76. – 19 − − 9 77. – 13 + 69 78. 37 − − 16 79. 8 – 43 80. 48 + – 24

15

Subtract Integers 2 1. – 4 – 8 2. – 17 – 10 3. 0 – 8 4. 9 – 5 5. 18 – 17 6. – 2 – 6 7. – 8 – 8 8. –4 − − 3 9. 11 + – 9 10. – 8 − − 4 11. 0 + – 2 12. 9 + – 7 13. –2 – 13 14. – 9 – 9 15. 8 + – 5 16. –13 − − 5 17. 0 + – 8 18. 0 – 9 19. –69 + – 9 20. – 18 – 17

21. 0 – 7 22. – 17 + – 3 23. 5 + – 8 24. 15 – 17 25. 2 – 13 26. – 8 + 11 27. – 6 + 15 28. 2 + – 3 29. –7 − − 6 30. 0 – 16 31. 15 – 4 32. 5 – 6 33. 7 – 27 34. 5 − − 7 35. – 8 + – 7 36. –14 − − 7 37. 5 – 7 38. – 5 – 1 39. –3 − − 10 40. – 11 − − 5

41. – 6 − − 8 42. – 8 + – 7 43. 36 – 6 44. – 3 + 0 45. – 6 – 0 46. – 10 – 1 47. – 8 – 2 48. – 6 – 15 49. – 43 + – 19 50. – 21 − − 7 51. 12 – 50 52. – 9 + – 13 53. 9 − − 6 54. – 3 + – 6 55. – 8 – 4 56. –32 − − 8 57. 0 − − 20 58. 15 – 10 59. – 3 + – 23 60. 0 + 39

61. – 4 – 8 62. – 6 − − 5 63. 7 – – 4 64. 3 − − 6 65. 4 – 4 66. 61 + 15 67. 4 − − 7 68. – 9 + – 2 69. 3 – 15 70. 63 – 5 71. – 18 − − 6 72. 0 – 5 73. – 21 – 16 74. – 19 – 6 75. – 28 − − 9 76. – 11 − − 9 77. – 54 + 72 78. 70 − − 16 79. 4 – 29 80. 16 + – 21

16

Multiply & Divide Integers 1

1. –5 • 7 2. 6 • – 4 3. 9 • 3 4. 12 • 4 5. 9 • – 10 6. –6 • – 7 7. –3 • 8 8. –11 • − 3 9. 8 • – 9 10. –7 • − 4 11. 6 • – 3 12. 12 • – 10 13. –12 • 11 14. –13 • – 6 15. 33 • – 3 16. –41 • − 7 17. 8 • – 3 18. 10 • – 99 19. –61 • – 9 20. –35 • – 4

21. 12 • 8 22. 18 • – 2 23. 12 • – 5 24. 11 • – 10 25. 2 • – 31 26. –9 • 12 27. –5 • 14 28. 7 • – 9 29. –4 • − 6 30. 0 • – 41 31. 8 • 0 32. 7 • – 1 33. 4 • – 32 34. 1 • − 7 35. 0 • – 7 36. –4 • – 11 37. 15 • 6 38. – 4 • – 2 39. –30 • – 10 40. –6 • – 5

41. 3 6 4÷ − 42. 45 9÷ 43. 5 6 7− ÷ 44. 63 9÷ 45. 5 4 6− ÷ − 46. 1 6 8÷ − 47. 3 5 7÷

48. 6 48

−

49. 2 79−

50. 2 4 8− ÷ 51. 8 5 1 7÷

52. 1 83 6

−−

53. 1 4 2÷ −

54. 7 22 4−

55. 9 6 1 2− ÷ − 56. 4 2 7÷

57. 2 61 3

−−

58. 72 1

−−

59. 5 68

−−−

60. 4 9 7÷ −

61. 7 7 1 1− ÷ − 62. 7 6 4− ÷

63. 6 84

−−

64. 6 6 2 2÷ 65. 9 5 1 9− ÷ −

66. 41 8

−

67. 2 1 7− ÷ 68. 1 2 6− ÷ − 69. 0 1 1÷ 70. 8 1 2 7− ÷

71. 1 01 2 0−

72. 8 0 8− ÷ −

73. 4 11 2 3−

74. 8 4 4− ÷ −

75. 2 83 5

−−−

76. 86 4

−

77. 1 8 6− ÷ 78. 9 0 1 8÷ 79. 2 5 0 1 0− ÷

80. 4 56 0−

17

All Operations 1

1. –2 + 3 2. 8 – 9 3. 0 + 5 4. 0 / 7 5. 19 – 10 6. –4 – 1 7. –7 • 2 8. –9 − − 3 9. 9 / – 9 10. –9 − − 4 11. 0 / – 3 12. 9 + – 7 13. –2 + 6 14. –1 – 3 15. 7 • – 2 16. –14 − − 5 17. 0 • – 2 18. 0 – 8 19. –56 / – 8 20. –7 – 6

21. 0 + 5 22. 16 + – 3 23. 6 + – 3 24. 16 – 7 25. 4 – 5 26. –9 + 1 27. –9 + 10 28. 6 • – 3 29. –1 − − 8 30. 0 – 5 31. 8 – 0 32. 5 • – 3 33. 6 – 16 34. 0 − − 9 35. 4 + – 7 36. –13 − − 7 37. 2 • 2 38. – 5 – 4 39. –3 − − 10 40. –8 − − 4

41. –1 − − 8 42. – 1 • – 9 43. 30 / – 6 44. – 1 • 0 45. – 7 – 0 46. – 5 – 1 47. – 6 / 3 48. – 2 • 8 49. – 30 / – 10 50. – 5 − − 6 51. 13 – 5 52. – 7 • – 8 53. 0 − − 6 54. – 3 • – 6 55. – 1 – 4 56. –23 − − 8 57. 0 − − 10 58. 13 – 5 59. – 7 • – 7 60. 0 • 5

61. – 5 + 8 62. – 6 • – 1 63. 6 / – 6 64. 9 − − 1 65. 3 – 9 66. 6 + 8 67. 4 − − 1 68. – 4 / – 1 69. 2 – 10 70. 63 / 9 71. – 13 − − 7 72. 0 • 6 73. – 2 – 7 74. – 10 • – 1 75. –8 − − 5 76. – 11 − − 4 77. – 5 + 6 78. 7 − − 12 79. 4 – 18 80. 9 + – 9

18

Decimal Rationals Review Find the answers by showing all work.

1) – 23.35 + 34.64 2) – 36.7 • 4.62 3) – 5.8 • 6.4

4) – 0.35 ÷ ÷ – 0.5 5) 1.243 + 67.37 6) 96 ÷ – 0.24

7) – 84.5 – 73.8 8) 1.1 • – 0.63 9) – .67 – .45 10) 6.74 ÷ – 32 11) – 895 − 235.7 12) – 89 • 0.007 13) – 4.132 + 8.795 14) 23.41 ÷ – 8.5 15) –.48 • – 2.7 16) – 35.1 ÷ – 70 17) – 45.23 − 4.523 18) – 35.6 − 35.6 19) 372 ÷ – 0.6 20) – .9996 + .7358 21) – 3.02 • .23 22) – 80.11 • .55 23) 54.3 − .543 24) – .723 – .387 25) 45.2 • 6.3 26) 8.4 ÷ – .0005 27) – 5.4 + 6.93 28) .67 – 3.62 29) – 54 – 7.34 30) – .75 • – 6.8

19

Multiplication & Division of Rational Fractions Divide the following problems and write the answers in lowest terms. Show all work.

1. 34

• 23

2. − 5

12• 8

7

3. 3 35 10

÷

4. 5 17 3

− ÷ −

5. 12 77 9

− •

6. 1 43 5

•

7. 3 14 12

− ÷ −

8. 7 48

− • −

9. 1 14 10

− ÷ −

10. 6 27 7

− ÷

11. 1 72 10

÷ −

12. 3 98 8

− • −

13. 5 26

• −

14. 3 95 10

÷ −

15. 5 58 12

− ÷ −

16. 5 19 7

•

17. 243

÷

18. 2 723 6

− •

19. 3 12 14 6

− ÷ −

20. 14 32

− ÷ −

21. 2 715 8

− • −

22. 2 923 5

− •

23. 5 31 16 8

− ÷ −

24. 5 317 4

• −

25. 3 13 14 2

• −

26. 1 72 12 8

− ÷ −

27. 4 725 12• −

28. 2 12 13 6

• −

29. 13 25

• −

30. 3 34 18 4

− ÷

31. 7 73 19 10

÷

32. 3 43 110 7

÷ −

33. 1 25 63 5

• −

34. 13 34

− ÷

35. 4 95 5

− • −

36. 11 315 2

÷

20

Addition of Rational Fractions Simplify the following problems and write the answers in lowest terms. Show all work.

1. − 15

+ − 25

2. − 5

9− 1

9

3. − 9

10+ 1

10

4. −1115

+ 815

5. − 18

− − 58

6. 12

+ 13

7.

− 17

− 12

8. 14

− 23

9. − 25

− 310

10. 13

− − 56

11. 16

+ 49

12. −23

+ − 25

13. 34

− − 18

14. 5

12− 2

3

15. − 34

+ − 56

16. 13

− − 310

17. − 14

+ 49

18. − 310

+ 215

19. 1

12− 1

4

20. 3

14− 4

7

21. − 118

− 34

22. − 27

− 37

23. 9

13− 6

13

24. 89

− − 79

25. − 518

− −56

26. 3 2

3− 1 2

5

27. − 8 1

4+ 5 2

3

28. − 10 2

5− 4 3

4

29. − 6 5

8− 7 7

10

30. 6 3

7+ 2 1

3

31. 11 1

3− 4 1

2

32. 14 3

4− 2 1

2

33. −7 5

8+ 2 7

12

34. 11 1

9− 5

18

35. 79

− 2 13

36. 3 6

7− 5 1

2

37. − 12 2

9− 2 1

4

38. − 9 4

7+ 7 3

5

39. − 6

11− 4 3

5

40. − 4 5

11+ 7 1

3

41. 4 15 65 5

−

42. 4 7

8+ 8 3

8

43. 6 7

9− 7 7

9

44. − 8 9

10+ 2 1

2

45. −8 1

9− 7 5

9

21

Fractions Rationals Review

22

Change the following repeating decimals to fraction: 1. 0.6 6. 8.51 11. 2.98 16. 7.02 2. 0.5 7. 19.74 12. 11.47 17. 87.22 3. 11.3 8. 4.356 13. 31.82 18. 41.02 4. 5.9 9. 0.74 14. 541.67 19. 0.532 5. 11.3 10. 0.98 15. 837.07 20. 11.4532 Approximate the following radicals to the nearest whole number without using a calculator: 21. 8 26. 17 31. 138 36. 288

22. 24 27. 39 32. 199 37. 293

23. 1203 28. 152 33. 324 38. 219

24. 88 29. 563 34. 763 39. 126

25. 1253 30. 370 35. 603 40. 69 Use the approximation method to place the following radicals in the correct position on the number line below: 41. 200 , 563 , 87 , 111 , 1313 , 10 , 151 , 703 , 56 , 7 , 173 , 63 , 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 42. 75 , 53 , 39 , 150 , 1403 , 22 , 123 , 253 , 5 , 90 , 533 , 119 , 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14

The Number System

23

Comparing Numbers Tell whether each is a Rational or Irrational Number. Then graph each set on a number line.

1) 0.01, 0.001, 0.1, and 0.0001 2) 2.25, 0.253, 0.2485, and 2.249 3) 0.38, 1.5, 0.475, and 2.249 4) 0.006, 5.02, 0.503, 0.1483 5) 0.98, 0.89, 0.934, and 0.9 6) 0.201, – 0.19, – 1.2, and – 0.21 7) 0.465, – 0.4053, – 0.47, and – 4.5 8) 0.51, – 0.583, 0.60, and – 0.5126 9) 0.04, – 1.25, – 0.156, and – 2.3 10) 0.76, 07, – 0.076, and – 0.0710 11) 1 1 1

2 3 5, , and 12) 5 32

8 3 5, , and 13) 1 1 1

4 2 6, , and− − 14) 31 2

2 5 10, , and− − − 15) 5 311

16 8 4, , and− −

16) 52 113 12 6, , and− −

17) 2 25 3, 1 , and 0.25−

18) 7 2

8 52.41, 1 , and 2− − − 19) 9 1

10 25.46, 5 , and 5 20) 21

1000.34, 0.56, 0.13,− − 21) 4.5, 4.62, 4.72, and 4.26 22) 16 9, 37, 42 23) 2, 27, 5.6, 1.43 24) − 12, − 3.75, − 37, − 6.3

25) 24, 2 10, 8 3, 5 5 26) 4.5, 4.62, 4.72, and 21

27) – 5.3, – 6.3, – 5.27 and − 30

28) 3 1 1 147 8 7 151 , 2 , 2 , and− − −

29) 38 , 0.23, 5

9 , 12 , 0.55

30) 1

22.53, 3.6, and 2− − −

24

Algebraic Expressions Write a P for positive or N for negative above each term then simplify the following:

1. 2a + 4a + 8a

2. 5x + 6x + 9x

3. 3n + 9n + 11n

4. 5y + 6y + 14y

5. 6p + 9p – 5p

6. 2w + w + w + 4w

7. 8x + 17x + x

8. 4g + 9g + 8q

9. 3c + 9c + c + 7c

10. 8w + 8f + 8w

11. 9a + 8m + 3m

12. 5w + 15v + 8w + 6

13. 4j + 9h + 9j + 7j

14. 2x + x + 3x + 5

15. 9f + 5y + 3f + 4y

16. 4x + 7y + x + y

17. 18x + 14 + x + 17x + 6

18. 5c + 2c + 8c + c

19. 9w – 4w

20. 4x – 3x

21. 2c + –6c

22. 5x – 8x

23. 3x – –12x

24. 3w – 8w

25. –6p – 2p

26. 3k – –9k

27. 15x – 19x

28. –x – x

29. x – – 2x

30. c + 3c

31. 5x – 2x

32. 3x – – 2x

33. 4w – 9w

34. – 5p – p

35. 6k – – 5k

36. x – 9x

37. – 8x – x

38. 5 + x – 3

39. 3 – – j – 10

40. g – 5 – 3g

41. 12 – 3q – 9

42. 5 – x – 6

43. 3 + x + 6

44. 7 – –k – 4

45. 8g – 2 – 2g

46. 4q – 15

47. 7 – 5x – –6x

48. 8 + 9p – 6 – 2p

49. 4x + 7 – 4x – 7

50. 3w – 6 + 3w – 8

51. – x + 7 – 4x – 5

52. 9 + 6x + 5x – l

53. 9 – 4 + 17k – 3

54. 13r + 5s – 6r – 2s

55. 23m + 4n – –16m

56. 5y – 9 – 3y + 11

57. 5q – 2w + 9w – 8g

58. 24b – 4b – 6b – –9b

59. 6h – v + 27v + 5h

60. 13f – 7g + 18f + 4f

61. 2k – 7 + 11k – –14k

62. 7m –- 2m – 8m – m

63. –15 + 12x + 11 – 13x

64. 6c + 2h – –9c + 3h

65. 12 + 5v + 4v – 8 – 9v

66. 26 + 36b + 4b + 30b

67. 2x + 6 – x – 5

68. w – 5 – 3w – – 7

69. – x – 1 – 2x – 3

70. – 3r + 4s – 7r – s

71. 12m – n – 23m – 4n

72. 2y – 7 – 4y + 16

73. 9g – 8w + 2w – 4g

74. 14b – 3b – b – – 5b

75. 3h – v + 9v + 17v + 9h

76. 4f – 5q + 13f – 6f

77. 9k + 5 + 13k – – 18k

78. – 6m – 4m – 8m – m

79. –16 + 14x + 17 – 14x

80. 5c + 3h – – 5c + 4h

81. 4x – 6x – 7x – 9x

Fractions Rationals

Chapter 11

25

Simplify the following using the Distributive Property: 1. 2 ( x + 6 ) 2. 5 ( x – 8 ) 3. 6 ( 8 – x ) 4. 3 ( 2 – g ) 5. 8 ( 4 + 2 x ) 6. 9 ( 5 + 8x ) 7. 8 ( 9 – 3y ) 8. 4 ( 8 – 6y ) 9. –5 ( 3x + 7 ) 10. –4 ( 8x + 7 ) 11. –3 ( 2 – 3x ) 12. –3 ( 7 – 4x ) 13. –5 (–3x – 7 ) 14. –2 (–x – 5 ) 15. –3 (–9 – 11x ) 16. –11 (–2 – 5x )

17. –5 ( 3x + 2y )

18. 4 ( 2x + 5 ) + 11

19. 2 ( 4 + 8x ) + 12

20. 5 ( x + 13 ) + x

21. 4 ( 5x + 12 ) + 6x

22. 2 ( x – 11 ) + 1

23. 2m – 7( 8w + 9m )

24. 6x – 6 ( 6x – 7 )

25. 2w + 7 ( 7w – 9 )

26. 9 – 4 ( 2n + 3 )

27. 6 – 5 ( 3 – 9c )

28. 3 ( 2x + 8 ) – 6

29. 7 ( 4 + 7x ) – 5

30. 3 ( 2x – 11 ) – 10x

31. 3 ( 5 + x ) – 6

32. 5 ( 6x + 8 ) – 3x

33. 9 ( 3x – 2y ) + 9y

34. 7y – 4 ( 2y – 9 )

35. 3m – 8 ( 7w + 6m ) 36. 4x + 8 ( 3x – 4 ) 37. 6w – 5 ( 9w + 6 ) 38. 7 – 3 ( 2 + 3n ) 39. 1 + 5 ( 7n – 8 ) 40. 3 – 8 ( 11x – 4 } 41. 16 – 5 ( 4 – 8x )

42. 6 ( 4z + 8 ) + ( 11z + 4 )

43. 2 ( 6 + 5x ) + 3 ( 4 + 8x )

44. – 3 ( 5 – 3w) – ( 5w + 4 )

45. 7 ( 2c + 8 ) – ( 9 – 5c )

46. – 6( 3y – 3p ) + 2(5p + 4y )

47. 9 ( 3x + 7 ) + ( 5x – 4 )

48. – 11 ( 8 – 3h ) – 3 ( h – 2 )

49. 9 ( 5 – 8d ) + 7 ( d + 9 )

50. 8 ( k + m ) – ( k + m ) 51. 4(2x – 7) – 5 (6 – 3x)

26

Solve the following equations showing all the work. 1. a – 11 = 15 2. b – 8 = 17 3. y + 7 = 29 4. x + 18 = 31 5. – 76 + m = 92 6. – 49 + n = 63 7. c – 30 = – 19 8. d – 24 = – 15 9. p + 18 = – 32 10. s + 90 = – 55 11. 24 + t = 0 12. 0 = z – 14 13. v – 37 = – 54 14. w – 94 = – 110 15. – 7 + k = – 17 16. – 18 + h = – 38 17. 45 = x + 16 18. 39 = y + 12 19. – 19 – a = – 23 20. b – 32 = – 82 21. 1 – x = 4 22. 16 = 3 – a 23. 18 = 5 – g 24. 3 – h = – 26 25. 25 = 18 – k 26. 4 = – p – 6 27. 45 – m = 15 28. 34 – r = 34 29. 52 = 81 – z

30. 61 = – y – 4 31. 36x = 72 32. 10y = – 10 33. 3c = – 21 34. – 8a = 32 35. 12 4b= 36. 13 7t = 37. − =1

10 5r 38. − =1

9 9s 39. 0 = – 4 k 40. – 7 = – 7p 41. c4 1= − 42. d2 2= 43. – 11f = – 88 44. – 27p = – 81 45. 4 3= − u 46. − =1 13

n 47. 5x – 1 = 26 48. 4y – 2 = 14 49. 2z + 4 = 8 50. 6 + 2a = 10 51. 9z – 5 = 4 52. 4y – 7 = 21 53. 6 = – 4x – 2 54. 4 – 3y = 13

55. – 4 = – 8 – 2z 56. – 7 = 3 + 5a 57. – 6x + 25 = – 11 58. 18 = 2c + 10 59. 17 – 3y = – 10 60. 8n – 14 = – 22 61. – x + 3 = – 4 62. 5 – m = 12 63. – 6 – z = – 2 64. – 4 = 6 – 2x

65.

x5

+ 9 = 13

66. x7 93

− = +

67. y5 43

+ = −

68. 1 x 7 23

− =

69. 36 y 95

− =

70. 317 10 z4

= − +

71. 32 = 7x + 8 – 5x 72. 6y + 8 – 5y = – 11 73. – 3 = 8z + 8 – 9z 74. 5 = 5x – 7x + 25 75. 3y + 7 – 5y = – 9 76. – 7 = 3p – 9 – 7p

27.

Solve the following equations. Show all work for each. 77. 9y – 18 = 3y 78. 7c – 9 = 8c 79. 8n – 12 = 5n 80. – 11m = 14 – 9m 81. – 6x = 10 – 4x 82. – 4z = 35 – 9z 83. 8p = – 5p + 65 84. – 84 + 15r = 3r 85. 11c + 36 = 8c 86. 7z – 9 = 3z + 19 87. 6 + 10t = 8t + 12 88. 3x + 7 = 16 + 6x 89. 18 + 3y = 5y – 4 90. 11a + 8 = – 2 + 9a 91. 9x – 5 = 6x + 13 92. 5 – x = x + 9 93. 14 + 3n = n – 14 94. 7 – x = 5 + 3x 95. 4y + 2 = 2y + 4 + 3y 96. 8c – 12 = 15c – 4c 97. 5x – 3 = 7x + 7 + 3x

98. y + 11 = – 2y + 6 99. – 2y + 3 – y = 11 + y 100. 16 – x = 4x + 8 + 3x 101. 2 ( y + 7 ) = 16 102. 3 ( x – 2 ) = 18

103. – 5 ( a + 2 ) = 30

104. x + 9 = 2 ( x – 3 )

105. 2 ( y + 3 ) = 12 – y

106. 25 – 5a = 3 ( 2a + 1 )

107. – 2 ( 3 – 2c ) = 10 – 4c

108. 23 = 12 – ( 6 + c )

109. 5 ( x – 1 ) = 2x + 4 ( x – 1 )

110. 13 – ( 2x – 5 ) = 2 ( x + 2 ) + 3x

111. – ( 3 – 2n ) + 7n = 3 ( n + 3 )

112. – ( y + 8 ) – 5 = 4 ( y + 2 ) – 6y

113. 3 ( c + 4 ) – 6c = 2 ( 4 – 2c )

114. 8y – 3 ( 4 – 2y ) = 6 ( y + 1 ) – 2

115. – 2 ( 3 – 4z ) + 7z = 12z – ( z + 2 )

116. – 3 ( 6 – 2x ) + 4x = – ( 2x – 6 ) 117. 7x – ( 9 – 4x ) = 3 ( x – 11 ) 118. 7r + 3 ( 7 – r ) = – ( r + 4 )

28.

LCM Method Solve the following using the LCM Method. Show all work in your notebook.

1. 34

c = 15

3. 59

= 23

c + 6

5. 23

y − 12

= 53

7.

23

x + 7 = 94

9.

12

x + 6 = 34

x − 3

11.

32

x − 3 = 23

x + 2

13. 3a + 4

12− 5

3= 2a − 1

2

15. 3y4

− 2y − 93

= y + 15

17.

4x − 36

− 5x − 45

= 9

19.

2x − 54x + 5

= 87

2. 35

y = 215

− y

4. 14

c + 23

= 13

6. 58

= 32

a + 14

8. 9 = 1

2x + 6

10. 12

y − 4 = 15

y + 7

12. 13

r − 1 = 25

r + 2

14.

2x − 37

− x2

= x + 314

16. 3b4

− 2b − 12

= b − 76

18.

6x − 58

− 2x − 712

= 9− x3

20.

1011

= 6x − 23x + 9

29.

Decimal Method Solve the following using the Decimal Method. Show all work in notebook.

1. 0.2 x = 1.8 3. 8.2 – 3.2c = – 17.4 5. 0.02 d – 2.6 = 0.84 7. 0.006 x – 7.3 = 0.14 9. 0.7 z – 0.1071 = 0.07z 11. 0.112 y + 2 = 0.012 y – 4 13. 0.5 n + 0.02 = – 0.2 – 0.6 n 15. 1.2 x + 0.004 = 1.4 x – 2

2. 2.6 x = – 7.8 4. 0.05 m = 7.45 6. 1.3 = 0.15 x – 3.2 8. 0.08 – 0.2y = – 4.4 10. – 0.09 = 5.2 – 0.1 x 12. 0.7 x – 0.11 = – 0.03 x 14. 0.23 z + 119.7 = 0.8 z 16. 0.09 c – 5.1 = 1.5 – 0.24 c

Solve the following using the LCM and Decimal Methods. Show all work in your notebook.

17. 16

c = 85+ c

19.

x − 36

= 53

21. 79

y + 12= 5

6y − 4

3

7 3

23. 9y + 5

15− 7y + 2

3= 6

5

18.

20.

22.

24.

30.

1. Seven more than three times a number is eighteen. Find the number. 2. Jason has fourteen more than three times Evelyn’s amount. Together they have 130 marbles. How many marbles does each have? 3. Seven times the difference between eleven and twelve times a number is four. Find the number. 4. April has 9 computer games. This is seven less than twice the number of Sam’s games. How many does Sam have? 5. The quotient of nine times a number and fourteen is seven. Find the number. 6. George has twelve books more than Jonathan. Together, they have 44 books. How many books does each have? 7. Six more than half a number is forty-two. Find the number. 8. Seven is to nine as fourteen is to what number?

9. Eleven is to thirty-three as what number is to twenty?

10. What number is to forty-nine as three is to seven?

11. Fifty-one is to what number as thirty-four is to two?

12. Arthur has 23 books, which is twelve books less than Jonathan’s amount. How many books does Jonathan have? 13. One number is sixteen more than five times a second number. Their sum is 58. Find the numbers. 14. Eleven is twice the sum of eight times a number and twelve. Find the number. 15. Greg earned 4 more points than twice Matt’s amount. Together, they earned 19 points. How many points has each earned? 16. Find the radius of a sphere with the volume of 1365 feet3. 17. The measures of two angles of a triangle are 41° and 83°. Find the measure of the third angle. 18. Find the height of a prism if its volume is 2295 ft3, its width is 15 ft and its length is 17 ft. 19. A triangle has a height of 38.2 m and an area of 324.7 m2. Find its base. 20. Find the number such that the sum of twice a number and three is to six as the difference between seven times the same number and five is to eight. 21. Eleven is to a number as seven is to the sum of four times that number and five. Find the number. 22. Four plus a number is to seven as twelve minus that number is to five. Find the number. 23. What was the cost of the meal if an 18% tip was $10.99?

Word Problems: Use the verbal model format to solve the following. Show all work.

31.

Use the verbal model format to solve the following. Show all work. 24. What was the original cost of a dress during a 25% off sale if the savings was $22.50? 25. How many towels did Stephanie purchase for $89.70, if each towel cost $5.98? 26. The sum of five and six times a number divided by three is the same as the difference between seven times the same number and four divided by eight. Find the number. 27. The sum of a number divided by four and nine times a number divided by six is the same as the difference between five times a number and seven divided by two. Find the number. 28. A company reimburses $49 plus forty-five cent for each mile a salesman drives. If the reimbursement check was for $121, how many miles did the salesman drive? 29. A coat, originally marked $159, was sold for $135.15 during a sale. What was the rate of the discount? 30. A stereo cost $289. The final bill after sales tax was $306.34. What was the rate of the sales tax? 31. The check in a restaurant came to $58.45 and Mr. Jones left $69. What was the rate of the tip? 32. Marcy paid $146 for her purchases including a 7% sales tax. What was the original cost of her purchases before the sales tax? 33. Mr. Brown paid $136 for a suit during a 25% off sale. What was the original price of the suit? 34. A repairman charges a service fee of $60 plus $35 for each hour he works on the item. If the final bill was $147.50, for how many hours did he work on the item? 35. The sum of nine and a twice a number is to eight as the difference between twelve times that number and five is to three. Find that number. 36. The page of your school yearbook is 8 ½ inches by 11 inches. The left and right margins are ¾ inch and 2 78 inches, respectively. The space between pictures is 3

16 inch. How wide should you make each picture to fit three across the page? 37. You are shopping for earrings. The sales tax is 5%. You have a total of $18.37. What is your price limit for the earrings?

38. You want to include four photos on the cover of a program for the school play, two across the page. The cover is 6 1/2 inches wide, and the left and right margins are ¾ inches each. The space between the pictures is ½ inches. How wide would you make the pictures? 39. Find the missing radius of the cylinder: 40. Find the missing side of this cube:

25 feet Volume =

Volume = 4505 ft3 5177.72 m3

32.

33.

Pyth

agor

ean

The

orem

:

c2 =

a2 +

b2

Pyth

agor

ean

Tri

ples

:

34.

Which of the following are right triangles? 1. 3, 4, 5 2. 6, 4, 5 3. 13, 12, 5 4. 11, 6, 9 5. 7, 24, 25 6. 13, 10, 8 7. 6, 11, 157 8. 9, 14, 115 9. 9, 7, 32 10. 12, 20, 24 11. 9, 40, 41 12. 2, 2.5, 1.5 13. 16, 356 , 10 14. 150 , 4, 13 15. 15 , 7 , 8 16. 1.7, 1.5, 0.8 Solve the following using the Pythagorean Theorem. Draw a picture. 1. A fire truck parks 15 ft away from a building. The fire truck extends its ladder 39 ft. How far up the building from the

truck's roof does the extension ladder reach?

2. Carson found an old tent in the attic of his house and decided to set it up in the back yard. However, the support sticks for the tent are missing. If the tent is 60 inches across on the bottom and 34 inches on each side, how tall of a stick does he need to set up the tent?

3. A television has a rectangular screen with a diagonal measurement of 30 inches. If the screen has a height of 18 inches, what is the width of the screen?

4. Two ships leave port at the same time. Ship X is heading due north and Ship Y is heading due east. Six hours later they are 300 miles apart. If the Ship X had traveled 240 miles from the port, how many miles had Ship Y traveled?

5. Jeremy goes to White Water Amusement Park. While there he decides to go down the park's huge waterslide called Lightning. If the slide is 120 feet high and the base of the slide is 90 feet from the pool, then what is the length of the slide?

In #6 to 10, find the length of each of these lines in a coordinate plane. 7 6 9 1 0 8

35.

36.

VO

CA

BU

LA

RY

& D

EFIN

ITIO

NS

37.

VE

RB

AL

MO

DE

LS

38.