This document discusses simplifying variable expressions. It begins with an essential question about adding, subtracting, multiplying, and dividing variable expressions. It then provides the vocabulary for order of operations, including grouping symbols, exponents, multiplication, division, addition, and subtraction. Examples are provided to demonstrate simplifying expressions using order of operations. The examples include solving for variables and writing expressions in terms of variables.

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 Googleplex 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 Googleplex 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 Googleplex 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 Googleplex 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 Googleplex 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)
Larger area - smaller area

70. EXAMPLE 4
Find the area of the shaded region.
3(x − 4)
3 5
6(x + 5)
Larger area - smaller area
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)
Larger area - smaller area
5 6(x + 5) − 3 3(x − 4)
   
= 5 6x + 30  − 3 3x −12 
   

72. EXAMPLE 4
Find the area of the shaded region.
3(x − 4)
3 5
6(x + 5)
Larger area - smaller area
5 6(x + 5) − 3 3(x − 4)
   
= 5 6x + 30  − 3 3x −12 
   
= 30x +150 − 9x + 36

73. EXAMPLE 4
Find the area of the shaded region.
3(x − 4)
3 5
6(x + 5)
Larger area - smaller area
5 6(x + 5) − 3 3(x − 4)
   
= 5 6x + 30  − 3 3x −12 
   
= 30x +150 − 9x + 36
= 21x +186

74. EXAMPLE 4
Find the area of the shaded region.
3(x − 4)
3 5
6(x + 5)
Larger area - smaller area
5 6(x + 5) − 3 3(x − 4)
   
= 5 6x + 30  − 3 3x −12 
   
= 30x +150 − 9x + 36
= 21x +186 units2

75. HOMEWORK

76. HOMEWORK
p. 78 #1-11 all, 12-46 even
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