Simplify Variable Expressions

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
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