Int Math 2 Section 2-6 1011
- 1. SECTION 2-6 Simplify Variable Expressions
- 2. ESSENTIAL QUESTION • How do you add, subtract, multiply, and divide to simplify variable expressions? • Where you’ll see this: • Sports, finance, photography, fashion, population
- 3. VOCABULARY 1. Order of Operations:
- 4. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers
- 5. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers G E M D A S
- 6. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers Grouping symbols E M D A S
- 7. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers Grouping symbols Exponents M D A S
- 8. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers Grouping symbols Exponents Multiplication D A S
- 9. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers Grouping symbols Exponents Multiplication Division A S
- 10. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers Grouping symbols Exponents Multiplication Division } from left to right A S
- 11. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers Grouping symbols Exponents Multiplication Division } from left to right A ddition S
- 12. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers Grouping symbols Exponents Multiplication Division } from left to right A ddition Subtraction
- 13. VOCABULARY 1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers Grouping symbols Exponents Multiplication Division } from left to right A ddition Subtraction } from left to right
- 14. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 15. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 16. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 17. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 18. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 19. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 20. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 7x +15 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 21. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x 7x +15 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 22. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x 7x +15 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 23. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x 7x +15 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 24. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x 7x +15 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 25. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x +5 7x +15 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 26. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x +5 7x +15 −3x + 5 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 27. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x +5 7x +15 −3x + 5 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n)
- 28. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x +5 7x +15 −3x + 5 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n) 4x + 4y − 7x + 7y
- 29. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x +5 7x +15 −3x + 5 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n) 4x + 4y − 7x + 7y −3x +11y
- 30. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x +5 7x +15 −3x + 5 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n) 4x + 4y − 7x + 7y −3x +11y
- 31. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x +5 7x +15 −3x + 5 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n) 4x + 4y − 7x + 7y 2mn + 2m − 5mn + 5n −3x +11y
- 32. EXAMPLE 1 Simplify. a. 5(x + 3) + 2x b. 2x − 5(x −1) 5x +15 +2x 2x −5x +5 7x +15 −3x + 5 c. 4(x + y) − 7(x − y) d. 2(mn + m) − 5(mn − n) 4x + 4y − 7x + 7y 2mn + 2m − 5mn + 5n −3x +11y 2m − 3mn + 5n
- 33. EXAMPLE 2 The ticket prices at Matt Mitarnowski’s Googolplex are $8.00 for regular admission and $5.50 for students and seniors. For Saturday’s first show, 350 tickets were sold. Write and simplify a variable expression for the total admission fees of the tickets for that show.
- 34. EXAMPLE 2 The ticket prices at Matt Mitarnowski’s Googolplex are $8.00 for regular admission and $5.50 for students and seniors. For Saturday’s first show, 350 tickets were sold. Write and simplify a variable expression for the total admission fees of the tickets for that show. How many regular admission tickets were sold?
- 35. EXAMPLE 2 The ticket prices at Matt Mitarnowski’s Googolplex are $8.00 for regular admission and $5.50 for students and seniors. For Saturday’s first show, 350 tickets were sold. Write and simplify a variable expression for the total admission fees of the tickets for that show. How many regular admission tickets were sold? r = regular tickets sold
- 36. EXAMPLE 2 The ticket prices at Matt Mitarnowski’s Googolplex are $8.00 for regular admission and $5.50 for students and seniors. For Saturday’s first show, 350 tickets were sold. Write and simplify a variable expression for the total admission fees of the tickets for that show. How many regular admission tickets were sold? r = regular tickets sold How many student and senior tickets were sold?
- 37. EXAMPLE 2 The ticket prices at Matt Mitarnowski’s Googolplex are $8.00 for regular admission and $5.50 for students and seniors. For Saturday’s first show, 350 tickets were sold. Write and simplify a variable expression for the total admission fees of the tickets for that show. How many regular admission tickets were sold? r = regular tickets sold How many student and senior tickets were sold? 350 − r
- 38. So what was the total?
- 39. So what was the total? 8.00r + 5.50(350 − r )
- 40. So what was the total? 8.00r + 5.50(350 − r ) 8.00r +1925 − 5.50r
- 41. So what was the total? 8.00r + 5.50(350 − r ) 8.00r +1925 − 5.50r 2.50r +1925
- 42. So what was the total? 8.00r + 5.50(350 − r ) 8.00r +1925 − 5.50r 2.50r +1925 The total admission fees were 2.50r + 1925 dollars
- 43. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? a. Two consecutive pages have a sum of 175. What are the pages?
- 44. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages?
- 45. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175
- 46. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175
- 47. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 −1
- 48. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 −1 −1
- 49. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 −1 −1 2n =174
- 50. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 −1 −1 2n =174 2 2
- 51. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 −1 −1 2n =174 2 2 n = 87
- 52. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 87 is the first page, 88 is the next. −1 −1 2n =174 2 2 n = 87
- 53. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 87 is the first page, 88 is the next. −1 −1 2n =174 Check: 2 2 n = 87
- 54. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 87 is the first page, 88 is the next. −1 −1 2n =174 Check: 87+88= 2 2 n = 87
- 55. EXAMPLE 3 If a page in a book is numbered n, what is the number of the next page? n+1 a. Two consecutive pages have a sum of 175. What are the pages? n + (n +1) =175 2n +1=175 87 is the first page, 88 is the next. −1 −1 2n =174 Check: 87+88=175 2 2 n = 87
- 56. EXAMPLE 3 b. Three consecutive pages have a sum of 768.
- 57. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768
- 58. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768
- 59. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3
- 60. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3 −3
- 61. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3 −3 3n = 765
- 62. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3 −3 3n = 765 3 3
- 63. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3 −3 3n = 765 3 3 n = 255
- 64. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3 −3 3n = 765 3 3 n = 255 The pages are 255, 256, and 257.
- 65. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3 −3 3n = 765 3 3 n = 255 The pages are 255, 256, and 257. Check:
- 66. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3 −3 3n = 765 3 3 n = 255 The pages are 255, 256, and 257. Check: 255+256+257=
- 67. EXAMPLE 3 b. Three consecutive pages have a sum of 768. n + (n +1) + (n + 2) = 768 3n + 3 = 768 −3 −3 3n = 765 3 3 n = 255 The pages are 255, 256, and 257. Check: 255+256+257= 768
- 68. EXAMPLE 4 Find the area of the shaded region. 3(x − 4) 3 5 6(x + 5)
- 69. EXAMPLE 4 Find the area of the shaded region. 3(x − 4) 3 5 6(x + 5) Shaded area = Larger area - smaller area
- 70. EXAMPLE 4 Find the area of the shaded region. 3(x − 4) 3 5 6(x + 5) Shaded area = Larger area - smaller area A = 5 ⎡6(x + 5)⎤ − 3 ⎡3(x − 4)⎤ ⎣ ⎦ ⎣ ⎦
- 71. EXAMPLE 4 Find the area of the shaded region. 3(x − 4) 3 5 6(x + 5) Shaded area = Larger area - smaller area A = 5 ⎡6(x + 5)⎤ − 3 ⎡3(x − 4)⎤ ⎣ ⎦ ⎣ ⎦ A = 5 ⎡6x + 30 ⎤ − 3 ⎡3x −12 ⎤ ⎣ ⎦ ⎣ ⎦
- 72. EXAMPLE 4 Find the area of the shaded region. 3(x − 4) 3 5 6(x + 5) Shaded area = Larger area - smaller area A = 5 ⎡6(x + 5)⎤ − 3 ⎡3(x − 4)⎤ ⎣ ⎦ ⎣ ⎦ A = 5 ⎡6x + 30 ⎤ − 3 ⎡3x −12 ⎤ ⎣ ⎦ ⎣ ⎦ A = 30x +150 − 9x + 36
- 73. EXAMPLE 4 Find the area of the shaded region. 3(x − 4) 3 5 6(x + 5) Shaded area = Larger area - smaller area A = 5 ⎡6(x + 5)⎤ − 3 ⎡3(x − 4)⎤ ⎣ ⎦ ⎣ ⎦ A = 5 ⎡6x + 30 ⎤ − 3 ⎡3x −12 ⎤ ⎣ ⎦ ⎣ ⎦ A = 30x +150 − 9x + 36 A = 21x +186
- 74. EXAMPLE 4 Find the area of the shaded region. 3(x − 4) 3 5 6(x + 5) Shaded area = Larger area - smaller area A = 5 ⎡6(x + 5)⎤ − 3 ⎡3(x − 4)⎤ ⎣ ⎦ ⎣ ⎦ A = 5 ⎡6x + 30 ⎤ − 3 ⎡3x −12 ⎤ ⎣ ⎦ ⎣ ⎦ A = 30x +150 − 9x + 36 A = 21x +186 units2
- 75. PROBLEM SET
- 76. PROBLEM SET p. 78 #1-37 odd “I only have good days and better days.” - Lance Armstrong