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### Problem Solving Strategy you give the expression

Warm-ups

- 3(x + 1)
- a(b + 2)
- x(y + z)
- 5(x + (-2)
Use the distributive property to factor

- 2a + 2b
- ax + ay
- 3x + 3(-y)

3x + 3

ab +2a

xy + xz

5x - 10

2(a + b)

a(x + y)

3(x – y)

Today’s Goals

- Standards
- CA 1.0
- Identify and use arithmetic properties

- CA 2.0
- Find Opposites

- CA 4.0
- Simplify Expressions

- CA 10.0
- Add and Subtract Monomials (like terms)

- CA 1.0

Ch 2-7 Using the Distributive Property

The Distributive Property of Multiplication Over Addition

For any numbers a, b, c

a(b + c) = ab + ac

and

(b + c)a = ba + ca

Ch 2-7 Using the Distributive Property

The Distributive Property of Multiplication Over subtraction

For any numbers a, b, c

a(b – c) = ab – ac

and

(b – c)a = ba – ca

Ch 2-7 Using the Distributive Property

Factoring is the reverse of the distributive property

Factor 5x – 5y

5(x – y)

Multiply 5(x – y)

5x – 5y

Ch 2-7 Using the Distributive Property

Factor

- 3z -3y
- 10u – 30
- ua – ub – uc
- 5x – 35y – 10
- -6u – 4v – 8w
- 14u – 21w – 28

3(z – y)

10(u – 3)

u(a – b – c)

5(x – 7y – 2)

-2(3u + 2v +4w)

7(2u – 3w – 4)

Ch 2-7 Using the Distributive Property

What are the terms of each expression?

- 4a – 2b – 5c
The terms are 4a, -2b, -5c

- -8x + y – 7z
The terms are -8x, y, -7z

Ch 2-7 Using the Distributive Property

What are the terms of each expression?

- 4a – 2b – 5c
- 18x + y – 7z

4a, -2b, -5c

18x, y, -7z

Ch 2-7 Using the Distributive Property

Collect like terms.

- -7x + 2x – 3x
(-7 + 2 – 3)x = -8x

- 5x – 2y – 2x + 6y
(5 - 2)x + (-2 + 6)x

- 3.4a – 2.1a + 1.0a
= 2.3a

- -6a + 5b + 4a – b
= -2a + 4b

Ch 2-8 Inverse of a Sum and Simplifying

The Property of -1

For any rational number a.

-1 · a = -a

(Negative one times a is the additive inverse of a.)

Ch 2-8 Inverse of a Sum and Simplifying

- Is the additive inverse of 3 equal to additive inverse of -3?
No

- What is the additive inverse of 0?
0

- Write the additive inverse of x · y in 3 different ways
-(x · y), (-x) · (y) and (x) · (-y)

Ch 2-8 Inverse of a Sum and Simplifying

Multiply

- -1 · 12
- -1 · (-4)
- 0(-1)
Rename each additive inverse without parentheses

- -(2y + 3)
- -(a – 2)
- -(5y – 3z + 4w)

–12

4

0

– 2y – 3

– a + 2

–5y + 3z – 4w

Ch 2-8 Inverse of a Sum and Simplifying

Simplify:

- 3 - (x + 1)
- – (– 4a + 7b – 3c)
- –(4ab – 5ac + 6bc)
- 3 – (x + 1)
- x – (2x – 3y)
- 3z – 2y – (4z + 5y)
- 7u – 3(7u + v)
- -2(e – f) – (2e + 5f)

3 – x – 1 =2 - x

4a – 7b + 3c

– 4ab + 5ac – 6bc

2 – x

– x + 3y

– z – 7y

14u – 3v

4e – 3f

Ch 2-8 Inverse of a Sum and Simplifying

Simplify:

- [5 + (3 + 1)]
[5 + 4] = 9

- {6 – [3 + (5 – 2)]}
[6 – [3 + 3]} = {6 – 6} = 0

- [3(2x – 1) + 1] – (3x + 1)
[(6x – 3) + 1 = 3x – 1

=6x – 3 + 1 – 3x – 1 = 3x - 3

By inserting grouping symbols, how many different values can you give the expression9 • 2 + 5 • 4 = 38

Write an Equation

Ch 2-8 Inverse of a Sum and Simplifying you give the expression

- In two days Lupe hiked 65 km. She hiked 34.3 km the first day. How far did she hike the second day?

Let x = km that Lupe hiked the second day.

34.3 km

x

65 km

x = 65 - 34.3

65 = x + 34.3

Dan earns $3 for every lawn he mows. How many lawns must he mow to earn $54?

Let x = # of lawns mowed

$54 = x • $3

Tania sold three times as many tickets as Michele. Michele sold 16 tickets. How many did Tania sell?

Let T = tickets that Tania sold

T = 3 • 16

Michele = 16

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