Equations and Inequalities

Equations and Inequalities

• Remember to think about your clue words!– Less than : Subtraction– Per : multiplication– More than : addition– No more than : <– No less than : >– More than : >– Less than : <

Equations and Inequalities

• If you can set up an expression you can set up an equation, or inequality.

• For example, • 73 – 14 = x is the same as 73 = x + 14

• All you have to do is use your inverse operations!

Equations and Inequalities

• Mary’s CD player cost $113 less than her DVD player. Her CD player cost $78. How much was her DVD player?

Equations and Inequalities

• Maria wants to pack and ship a total of 288 pens. If each box holds 12 pens, how many boxes are needed to ship the pens?

Equations and Inequalities

• An airline lets you check no more than 65 pound so luggage. One suitcase weighs 37 pounds. How much can the other suitcase weigh?

Equations and Inequalities

• A 1-ton truck has the ability to haul 1 ton, or 2000 pounds. At most, how many televisions sets can the truck carry if each TV set weighs 225lb?

Write an equation or inequality

• Susan’s CD player cost $120 less than her DVD player. Her CD player cost $64. How much was her DVD player?

• Jack wants to pack and ship 208 pens. If each box holds 8 pens

Getting Started

• s + s + s + s + 4

• 4s + 4– Are the two expressions equivalent?– How could you convince a student that the two

expressions are equivalent?

Getting Started

• 4s + 4

• 4(s + 1) – Are the two expressions equivalent?– How could you convince a student that the two

expressions are equivalent?

Objective - To use the distributive property to simplify numerical and variable expressions.

Distributive Propertya(b c) a b a c

a(b c) a b a c or

3(4 5) 3(9) 27Order of Operations

Distributive Property

3(4) 3(5)12 15

27It works!

Why use the distributive property?3(x 2) 3(x) 3(2) 3x 6

Simplify using the distributive property.1)

2)

3)

4)

5)

6)

5(x 3)5 3

5x 15

6(y 7)6 7

6y 42

3(m 8)3 8

3m 24

4(3 y)4 3 4 y

12 4y

10(x 7)10 x 10 7

10x 70

4(k 2)4 k 4 2

4k 8

5 x

6 y

3 m

Use the distributive property to write an equivalent expression. Then simplify both to show they have the same value.

1)

2)

3)

5(8 3) 5(8) 5(3) 5(11)

5540 15

55same

6(10 3) 6(10) 6(3) 6(7)42

60 1842

same

8(11 5) 8(11) 8(5) 8(6)48

88 4048same

Simplify using the distributive property.1)

2)

3)

4)

5)

6)

4(x 7)4 7

4x 28

y(y 3)y 3

2y 3y

x(2x 9)x 9

22x 9x

7(8 x)7 8 7 x

56 7x

x(a b c) x a x b x c

ax bx cx

4(3 m k) 4 3 4 m 4 k

12 4m 4k

4 x

y y

x 2x

Geometric Model for Distributive Property

Two ways to find the area of the rectangle.

4

5 2

As a whole As two parts

A w l

4 5 2

Geometric Model for Distributive Property

Two ways to find the area of the rectangle.

4

5 2

As a whole As two parts

A w l

4 5 2 4 5 4 2

4 5 4 2

same

4 5 2 4 5 4 2

Find the area of the rectangle in terms of x, y and z in two different ways.

x

y z

As a whole As two parts

A w l

x y z

Find the area of the rectangle in terms of x, y and z in two different ways.

x

y z

As a whole As two parts

A w l

x y z x y x z

x y x z

same

Use the distributive property to write anequivalent variable expression. Then simplify.

1)

2)

3)

4)

5)

6)

8(x 4)8 4

8x 32

7(3x 2)7 2

21x 14

4(5 3x)4 3x

20 12x

y(5) y(8)y(5 8)

13y

a 3 a 4 a(3 4)

7a

6m 11mm(6 11)

17m

8 x

7 3x

4 5

Use the distributive property to help simplify thefollowing without a calculator.

1) 2)5(9.96)

5(10) 5(0.04)5(10 0.04)

50 0.20

49.80

7(8.2)

7(8) 7(0.2)7(8 0.2)

56 1.4

57.4

Use the distributive property to help simplify thefollowing without a calculator.

3) 4)8($11.30)

8($11) 8($0.30)8($11 $0.30)

$88 $2.40

$90.40

7 5.95

7(6) 7(0.05)7(6 0.05)

42 0.35

41.65

TOD

• 5(x +y)

• 33 + 8(x – 4)

• 5(x + 3) + 4(y + 4)