SECTION 2-6Simplify Variable Expressions

ESSENTIAL QUESTION

• How do you add, subtract, multiply, and divide to simplify variable expressions?

•Where you’ll see this:

• Sports, finance, photography, fashion, population

VOCABULARY

1. Order of Operations:

VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

rouping symbols

VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

rouping symbolsxponents

VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

rouping symbolsxponentsultiplication

VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

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VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

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VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

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} from left to right

VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

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} from left to right

VOCABULARY

1. Order of Operations: Allows for us to solve problems to consistently achieve the same answers

GEMDAS

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} from left to right

}from left to right

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)

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)

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)

5x

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)

5x

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)

5x +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)

5x +15 +2x

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)

5x +15 +2x

7x +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)

5x +15 +2x

7x +15 2x

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)

5x +15 +2x

7x +15 2x

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)

5x +15 +2x

7x +15 2x −5x

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)

5x +15 +2x

7x +15 2x −5x

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)

5x +15 +2x

7x +15 2x −5x +5

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)

5x +15 +2x

7x +15 2x −5x +5

−3x + 5

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)

5x +15 +2x

7x +15 2x −5x +5

−3x + 5

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)

5x +15 +2x

7x +15 2x −5x +5

−3x + 5

4x + 4y − 7x + 7y

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)

5x +15 +2x

7x +15 2x −5x +5

−3x + 5

4x + 4y − 7x + 7y

−3x +11y

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)

5x +15 +2x

7x +15 2x −5x +5

−3x + 5

4x + 4y − 7x + 7y

−3x +11y

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)

5x +15 +2x

7x +15 2x −5x +5

−3x + 5

4x + 4y − 7x + 7y

−3x +11y

2mn + 2m − 5mn + 5n

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)

5x +15 +2x

7x +15 2x −5x +5

−3x + 5

4x + 4y − 7x + 7y

−3x +11y

2mn + 2m − 5mn + 5n

2m − 3mn + 5n

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.

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?

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

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?

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

So what was the total?

So what was the total?

8.00r + 5.50(350 − r )

So what was the total?

8.00r + 5.50(350 − r )

8.00r +1925− 5.50r

So what was the total?

8.00r + 5.50(350 − r )

8.00r +1925− 5.50r

2.50r +1925

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

EXAMPLE 3If 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?

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1 −1

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1 −1 2n =174

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1 −1 2n =1742 2

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1 −1 2n =1742 2 n = 87

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1 −1 2n =1742 2 n = 87

87 is the first page, 88 is the next.

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1 −1 2n =1742 2 n = 87

87 is the first page, 88 is the next.

Check:

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1 −1 2n =1742 2 n = 87

87 is the first page, 88 is the next.

Check: 87+88=

EXAMPLE 3If a page in a book is numbered n, what is the

number of the next page?

n + 1a. Two consecutive pages have a sum of 175. What

are the pages?

n + (n +1) =175

2n +1=175 −1 −1 2n =1742 2 n = 87

87 is the first page, 88 is the next.

Check: 87+88=175

EXAMPLE 3b. Three consecutive pages have a sum of 768.

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3 −3

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3 −3 3n = 765

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3 −3 3n = 7653 3

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3 −3 3n = 7653 3 n = 255

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3 −3 3n = 7653 3 n = 255

The pages are 255, 256, and 257.

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3 −3 3n = 7653 3 n = 255

The pages are 255, 256, and 257.

Check:

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3 −3 3n = 7653 3 n = 255

The pages are 255, 256, and 257.

Check: 255+256+257=

EXAMPLE 3b. Three consecutive pages have a sum of 768.

n + (n +1)+ (n + 2) = 768

3n + 3 = 768 −3 −3 3n = 7653 3 n = 255

The pages are 255, 256, and 257.

Check: 255+256+257=768

EXAMPLE 4Find the area of the shaded region.

3(x − 4)

3 5

6(x + 5)

EXAMPLE 4Find the area of the shaded region.

3(x − 4)

3 5

6(x + 5)

Shaded area = Larger area - smaller area

EXAMPLE 4Find 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)⎡⎣ ⎤⎦

EXAMPLE 4Find 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⎡⎣ ⎤⎦

EXAMPLE 4Find 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

EXAMPLE 4Find 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

EXAMPLE 4Find 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

PROBLEM SET

PROBLEM SET

p. 78 #1-37 odd

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