-6a – 13a = -19a

-8n + 14n = 6n

10r - 19r = -9r

5xy + 3xy = 8xy

2s2

10ac + 19ac = 9ac

9s2 + 11s2 =

2n x 8 = 16n

24a2

5e x 9e = 45e2

6a x 4a =

28b ÷ 7 = 4b

52

24 ÷ 4d = 6/d

53 ÷ 5 =

x - 3.5 = 8.9 – 3x

4x = 12.4

x+ -3.5 = 8.93x4x = 8.9 + 3.5

x = 3.1

2x + 20 = 12

X =

2x = -8

-4

2/5 = 4y + 16 2/5

= y

= 4y-16

-4

24 + 4c = -2c

= -24

4c = -2c -

c =

24

6c

-4

12123 = ____10

22 = ______2

50

10110

4 2 4 4 1256

6 8 - 10

=4 2 =4 4 =

1

= 62=

6 10

6 8

= 36

Rename 2.025 as a mixed number

Let x = .025 (x) = (.025)

10x = 0.25

10 10

(10x) = (0.25)100 100

1000x = 25.25

Rename 2.025 as a mixed number

10x = .25 1000x = 25.25

=990x 25

x = 25/900 or 1/ 36

Rename 2.025 as a mixed number

x = 25/900 or 1/ 36

2.025 = 2 + .025

.025 = 1/362.025 = 2 1/36

Two-Step Inequalities

OBJECTIVE: Solve, graph, and check inequalities that call for two steps to simplify

2x + 20 < 12

x < -4

2x < 12 -202x < -8

Graph the solution.

-1 -2 -3 -4 -5 -6 0

Check. Substitute -4 for x.

2(-4) + 20 < 12-8 + 20 < 12

12 < 12; FalseTherefore, -4 is not a solution.

Solve. Graph and check the solution.

-1 -2 -3 -4 -5 -6 0

Check another value.2(-6) + 20 < 12

-12 + 20 < 128 < 12; True

Therefore, -6 is a solution.

Substitute -6 for x.

Try -10.

2(-10) + 20 < 12-20 + 20 < 12

0 < 12; TrueTherefore, -10 is also a solution.

3a < 16 + 11a

a -2

3a – 11a < 16-8a < 16

Graph the solution.

-1 -2 -3 -4 -5 -6 0

Check. Substitute -2 for a. 3(-2) < 16 + 11(-2)

-6 < 16 -22-6 < -6; False

Solve. Graph and check the solution.

-8 -8>

Graph the solution.

-1 -2 -3 -4 -5 -6 0

Check. Substitute -2 for a. 3(-2) < 16 + 11(-2)

-6 < 16 -22-6 < -6; False

Therefore -2 is not a solution.Substituting 0 for a.

3(0) < 16 + 11(0)

0 < 16 +0 0 < 16 True

Therefore 0 is a solution.

Homework. PB, p119-120Class work. PB, p119

Multistep Inequalities with Grouping symbols

OBJECTIVE: solve, graph, and check the solution of an inequality having a grouping symbols

4(x + 3) -2

Graph the solution.

-6 -7 -8 -9 -10 -11 -5

Solve. Graph and check the solution.

≤ 16 Multiply both sides by -2.

-2 -2

4(x + 3)

- 32≥ Apply the DPMoA.

4x + 12

≥ - 32 Subtract 12 from both sides.- 12 - 12

- 11

4x ≥ Divide both sides by 4 4 4

x ≥

- 44

-4

Graph the solution.

-6 -7 -8 -9 -10 -11 -5 -4

Check the solution. 4(x + 3)

-2≤ 16 Try -11for x.

4(-11 + 3) -2

≤ 16 Combine like terms.

4(-8) -2

≤

16 Multiply.

-32-2

16

≤

Divide

1616 ≤ True, so -11 is a solution.

Graph the solution.

-6 -7 -8 -9 -10 -11 -5 -4

Check the solution. 4(x + 3)

-2≤ 16 Try x = -5.

4(-5 + 3) -2

≤ 16 Combine like terms.

4(-2) -2

≤

16 Multiply.

-8-2

16

≤

Divide

164 ≤ True, so -5 is a solution.

HW: PB, p121-122

Class work PB, p121

Multistep Inequalities :fractions and decimals

solve, graph, and check the solution of an inequality having

fractions and decimals

objective

Pp 114-115, text

(0.12x + 0.36)

-1 -2-3 4 0

Example 1.Solve. Graph and check the solution.

0.6 Multiply both sides by 100.

12

- 36

≥

1

100100

12x + 36 ≥ 60 Subtract 36 from both sides.- 36

12x ≥ 24 Divide both sides by 12.12

x 2≥ Graph.

2 3

0.12x + 0.36 ≥ 0.6 Substitute -2 for x, then evaluate.

0.12x + 0.36 ≥ 0.6 Substitute -2 for x, then evaluate.

0.12(2) + 0.36

≥ 0.6 Multiply.

0.24 + 0.36

≥ 0.6 Add.

0.60

≥ 0.6 True, so 2 is a solution.

0.12x + 0.36

Try 4 for x, then evaluate.

≥ 0.6

0.12(4) + 0.36

≥ 0.6

0.48 + 0.36

≥0.6

Multiply.

Add.

0.84

≥ 0.6 True, so 4 is also a solution.

HW: PB, p125-126

Class work PB, p125

Compound inequalities

OBJECTIVE:

graph and find the solution of compound inequalities

pp 116-117, text

1

2

3

4

5

6 0

Graph: x > 3 and x < 7.

x > 3 Graph on the number line.

8

9

10

7 11

x < 7 Graph on the same number line.

Solution.

The solution set of the compound inequality in shortened form is:

{x | 3 < x < 7}

-2

-1

0

1

2

3 -3

Graph: z ≤ -2 or z ≥ 4.

z ≤ -2 Graph on the number line.

5

6

7

4 8

z ≥ 4 Graph on the same number line.

The solution set of the compound inequality in shortened form is:

{z | z ≤ -2 or z ≥ 4}

Homework. PB, p127-128Class work. PB, p127

Polynomials

OBJECTIVE:

define a polynomial classify a polynomial by the number of its terms simplify polynomials

pp 124-125, text

Do You Remember?

A symbol, usually a letter, used to represent a number

variable

Expressions that contain variables, numbers, and operation symbols

Algebraic Expressions

A term that doesn’t have variables

constant

Do You Remember?

It tells how many times a number or variable called the base is used as a factor.

exponent

A __ of an algebraic expression is a number, a variable, or the product of a number and one or more vaeiables.

term

Remember…

A monomial is an expression that is a number, a variable, or the product of a number and one or more variables with nonnegative exponents. examples:19, m, 7a2, 13xy, 1/4 abc10 Monomials that are real numbers are called constants.

Remember…

A polynomial is a monomial or the sums and/or differences of two or more monomials.

Each monomial in a polynomial is called a termterm..

Polynomials can be classified by their number of termsnumber of terms when they are in simplest formsimplest form.

Remember…

Types ofTypes of PolynomialPolynomial

Name Number of Terms Examples

Monomial

Binomial

Trinomial

1(mono means one)

2(bi means two)

3(tri means three)

2n, 4x3, r, 7, 6x2y5

2x + 8; 3b + p; 2a2 – 8b2

4n + b + c; 2x2 + 8x - 3

Remember…

ClassifyingClassifying Polynomials; Polynomials; before classifying a a polynomial make before classifying a a polynomial make sure it is in its simplest form.sure it is in its simplest form.

Example.Simplify: x2 + 2x + 1 + 3x2 – 4x. Then classify it.

Remember…Example. What kind of a polynomial is

x2 + 2x + 1 + 3x2 – 4x?

x2 + 2x + 1 + 3x2 – 4x

Combine like terms.

4x2 – 2x + 1

Classify.

4x2 – 2x + 1 is a trinomial because it has 3 terms.

Homework. PB, p139-140Class work. text, p125

Modeling Polynomials

OBJECTIVE:

use Algebra tiles to model polynomials

pp 128-129, text

Algebra Tiles

= x2

= -x2

= x

= -x

= 1

= -1

Examples of polynomials and their models.

x2 - 4 -3x2 + 2x +1

Write the polynomials modeled by each set of Algebra tiles.

4x2 + 7x -2x2 – 2x + 9

3x2 + 3x - 5 -3x2 + 2x - 1

If a polynomial is not in simple form, model it with Algebra tiles then combine like tiles.

Example. Simplify 3x2 – 2x – 4 + x2 + 3x.Model the polynomial.

Create zero pairs, (an x tile and a –x tile, and other opposites).

The simple form is 4x2 + x – 4.

Then rearrange the tiles so the like ones are next to each other.

Homework. PB, p143-144Class work. text, p129

Add Polynomials

OBJECTIVES: model the addition of polynomials

add polynomials algebraically

pp 130-131, text

Algebra Tiles

= x2

= -x2

= x

= -x

= 1

= -1

Example. Add 3x2 – 4x + 5 and 2x2 – x – 3.

3x2 - 4x + 5 -2x2 - x - 3

Step 1. Model each polynomial.

+

2x2 - x - 3

Step 1. Model each polynomial.

3x2 - 4x + 3

Step 2. Put the same tiles next to each other.

Step 2. Put the same tiles next to each other.

Step 3. Create zero pairs from opposite tiles.

Step 3. Create zero pairs from opposite tiles.

Step 4. Name the remaining tiles for the answer.

x2 - 5x + 2

Example. Add 2x2 + 11x + 9 and 3x2 – 6x

(2x2 + 11x + 9) (3x2 - 6x)

Polynomials can be added algebraically, in either horizontal or vertical form.

+

To add polynomials horizontally,use the Commutative and Associative properties to group and combine like terms

Remove parentheses.

2x2 + 11x + 9 + 3x2 - 6x Use the APA and CPA to group and combine like terms 2x2 + 3x2 + 11x – 6x + 9

5x2 + 5x + 9 Answer.

6x2 - 7y2

Example. Add 4x2 + 3xy – 9y2 and 6x2 – 7y2

4x2 + 3xy – 9y2

+

To add polynomials vertically, arrange like terms in columns and add the columns separately.

10x2 + - 16y2

Arrange like terms in columns

3xy Answer.