do now you work 8 hours and earn $60. what is your earning rate? (important to include units.) you...

Do Now

You work 8 hours and earn $60. What is your earning rate? (Important to include units.)

You buy 14 gallons of gasoline at $ 3.65 per gallon. What is your total cost?

1.)

2.)

A baseball travels at 90 kilometers per hour. What is the speed in meters per second?

3.)

9 – 7 - 20119 – 7 - 2011

$7.50 per hr

$51.10

25 m / s

hr

km

1

90

sec

25m

km

m

1

1000

min60

1hr

sec60

min1

Do Now

Use your calculator to evaluate the expression.

1.)

2.)

3.)

9 – 8 - 20119 – 8 - 2011

9526

95)2( 6

2)9144(

4.)3

2675

DoSimplify the expression.

1.) 2.)

9 – 9 - 20119 – 9 - 2011

)23(2)4(6 yxyx yxyx 5234

y2x6yx 246 yx 46

y28

CAN USE THEM ON YOUR TESTS!

HELPS YOU WITH YOUR ASSIGNMENT!

1.11.1 Real Numbers and Number Operations

What you should learn:GoalGoal 11

GoalGoal 22

Use a number line to graph and order real numbers.

Identify properties of and use operations with real numbers.

Whole numbers: 0, 1, 2, 3, …

Integers: …-3, -2, -1, 0, 1, 2, 3, …Integers: …-3, -2, -1, 0, 1, 2, 3, …

Real numbers: include fractions, decimals, whole numbers, and Integers.

0-1-2-3 1 2 3

originpositive #’snegative #’s

Graph the numbers on the number line.

Ex 1)

0-1-2-3 1 2 3

0, -3, 1

Ex 2) 2, -1, -2

Write two inequalities that compare the numbers.

Ex 1) 0, -3, 1

Ex 2) 2, -1, -2

-3 < 0 <1

-2< -1< 2

Write numbers in increasing order.

Ex 1) 0.34, -3.3, 1.12

Ex 2) 2.23, 2.2, -2.23

-2.23< 2.2 < 2.23

-3.3 < 0.34 < 1.12

Properties of Addition and MultiplicationProperties of Addition and MultiplicationProperties of Addition and MultiplicationProperties of Addition and Multiplication

1. a + b = b + a Commutative property

2. (a + b) + c = a +(b + c) Associative property

3. a + (-a ) = 0 Inverse property

a b = b a

(a b) c = a ( b c )

0,11

aa

a

4. a(b + c) = ab + ac

Distributive property

Identity property5. a + 0 = a

aa 1

Using Unit Using Unit AnalysisAnalysisPerform the given operation. Give the answer with the appropriate unit of measure.

You ride in a train for 175 miles at an average speed of 50 miles per hour. How many hours does the trip take?

example

175 50mileshour

miles

Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section

When converting units with unit analysis, how do you choose whether to use a particular conversion factor or its reciprocal?

assignmentassignment

DoDoPerform indicated conversion.

1.)

2.)

3.)

9 – 12 - 20119 – 12 - 2011

4.)

116

350 feet to yards

3

2

5 hours to minutes

300 minutes

6800 seconds to hours

2.2 kilograms to grams

2200 grams

1 9

8hoursyards

1.21.2 Evaluate and Simplify Algebraic Expressions


GoalGoal 22

Evaluate algebraic expressions.

Simplify algebraic expressions by combining like terms.

Numerical expressionNumerical expression consists of numbers, operations, and grouping symbols.

Expressions Containing Exponents. Expressions Containing Exponents.

4444445 Example:

The number 4 is the BASE,the number 5 is the EXPONENT, and

54 is the POWER.

Order of Operations

ParenthesesExponents

Multiplication and Division

Addition and Subtraction

3. Then do multiplications and divisions from left to right

1. First do operations that occur within symbols of grouping. 2. Then evaluate powers

4. Finally do additions and subtractions from left to right.

Variable is a letter that represents a number.

Values of the variable are the numbers.

Algebraic expression is a collection of numbers, variables, operations, and grouping symbols.

Value of the expression is the answer after the expression is evaluated.

Evaluate is to make a substitution, do the work, and determine the value.

Definitions:

362554 2 yyxyyx

Terms: are the number.

Coefficient: is the constant in front of the variable.

Like Terms:

Constant term

Ex)4

)2( )2)(2)(2)(2( 16

Ex) 4

2 )2)(2)(2)(2( 16

Evaluate the power.

EvaluateEvaluate the expression when x = 4 and y = 8.

ex) yx 94

)8(9)4(4 substitute

7216

56

Do the work

Get the value


What does it mean to EVALUATE?


Do NowSolve for x.

1.)

9 – 13 - 20119 – 13 - 2011

63 x

3x

Perimeter = 45

63 x

3x

63 x 3x63 x3x + + + = 45

1.31.3 Solving Linear Equations


GoalGoal 22

Solve linear equations

Use linear equations to solve real-life problems.

Using Addition or Subtraction

The key to success: Whatever operation is done on one side of the equal sign, the same operation must be done on the other side.

Inverse operations undo each other. Examples are addition and subtraction.

Solving Linear EquationsSolving Linear Equations

Generalization:

If a number has been added to the variable, subtract that number from both sides of the equal sign.

If a number has been subtracted from the variable, add that number to both sides of the equal sign.

ex) 63 x3 3

ex) 64 x4 4


Generalization:

If a variable has been multiplied by a nonzero number, divide both sides by that number.

example: 4x = - 12

4 4


Generalization:

If a variable has been divided by a number, multiply both sides by that number.

example:2

6

x6 6

Hint: you always start looking at the side of the equal sign that has the Variable.

Hint: you always start looking at the side of the equal sign that has the Variable.


ex)x

5

24

ex) 43

2x

2

3

2

3

2

5

2

5

2

12x

x2

20

6

10


Generalization:

First undo the addition or subtraction, using the inverse operation.

Second undo the multiplication or division, using the inverse operation.


20105

w

example:

10 10

105

w

5 5

example: 1024 c2 2

124 c

4 4

3c

subtract 10

multiple by 5

Solving Multi-Step Equations

example:

4 4186 x

6 6

3x

14442 xx

1446 x


Example) 8325 xxx5 x5

822 xMove the smaller

#.8 8

x210 22

x5



How does Solving a linear equation differ from Simplifying a linear expression?



Page 21 # 3 – 13 odd, 21- 25 odd, 68

(Day 1 of 2)

example: )5(4)2(2 xx

4 4246 x

6 6

4x

20442 xx

2046 x


x4 x4

DoDoPerform indicated conversion.

1.)

9 – 14 - 20119 – 14 - 2011

A salesperson at Taylor Ford has a base salary of $20,000 per year and earns a 5% commission on total sales. How much must the salesperson sell to earn $40,000 in one year?

$400,000

example: )4(6)12(3 xx

x6 x6

24369 x

9 9

33333333.1x

246363 xx


(Day 2 of 2)

36 36

129 x3

4Book Answer

Are they the same?

example: 281

51 xx

15384615.6x

24013 x13

40


1340

1326

Book Answer

Are they the same?

1380


86

No Solution



77

All Solutions orAll Real Numbers work



What do you do when you have variables on both sides of the equal sign?


Page 22 # 33 – 65 odd, 70, 71

Do Now

1.)

9 – 15 - 20119 – 15 - 2011

You can estimate the diameter of a tree without boring through it by measuring its circumference. Solve the formula for d.C = d

1.41.4 Rewriting Formulas and Equations


GoalGoal 22

Rewrite equations with more than one variable

Rewrite common formulas.

example:

1042 y

2 2

124 y

4 4

3y

Rewriting Formulas and Equations

Solve for y

example:


Solve for y

y - =

+ +

+y =

Like Terms

example:


y + =- -

+y = -

+y = -

Solve for y

example:

y

y

y


Solve for y

cayb Solve this equation for y.

b bcbay

Ex 1)

a a

a

cby

(3.7) Formulas

1st Solve this equation for y

x12 x12

15123 xy

Ex ) 15312 yx

-3 -354 xy

Formulas

5)2(4 y

= 3

2; xthen, find the value of y for the given value of x. (substitute)

-32nd


What does it mean to solve for a variable in an equation?

Page 30 # 7 – 14 ALLPage 22 # 56 – 66 EVEN

First, solve this equation for y, then substitute.

# 7 pg 30 ) 263 yx

Formulas

7; x

cbx )(2

Solve this equation for x.

b2 b2

cbx 22

Ex 2)

cbx 22

2 2

2

2 cbx

(3.7) Formulas

bhA2

1

Solve this equation for b.

Ex 3)

h h

bh

A

2

1

1

2

1

2

bh

A

2

(3.7) Formulas

Example)

Solve the investment-at-simple-interest formula A = P + Prt for t.

A = P + Prt-P -P

A – P = Prt

Pr

Pr= tA - P

Pr

(3.7) Formulas

xx 24 How do you solve for x?

now solve this one for C?

rCCs

(3.7) Formulas

First, substitute the given value for x, then solve this equation for y.

24 2493 y

Ex ) 2;15312 xyx

-3 -3

3y

Formulas

153)2(12 y

If not NOW, when?

You have budgeted $100 to improve your swimming. At your local pool, it costs $50 to join and $5 each visit. Find the number of visits you can have within your budget.

9 – 16 - 20119 – 16 - 2011 do

# 1 – 27 ALLPage 65 # 1 – 17 ALL

Go to STAT

1: Edit

*CLEAR the dataBy highlighting the numbers underneath L1, L2, L3 (not the L(not the L11))

No “DO NOW” today

BUT,…get a calculator

1.51.5 Problem Solving Using Algebraic Models

What you should learn:GoalGoal 11 Use general problem solving plan to

solve real-life problems

Go to STAT

1: Edit

*fill in the data

Go to STAT again

over to CALC

4: LinReg(ax+b)

Enter

12

Looking for a Pattern

The table gives the heights to the top of the first few stories of a tall building. Determine the height to the top of the 15th story.

After the lobby, the height increases by 12 feet per story.

SOLUTION

Look at the differences in the heights given in the table.

Story

Height to topof story (feet)

Lobby 1 2 3 4

20 32 44 56 68

12

20 3232 44

12 12

44 5656 68

Go to STAT

1: Edit

*fill in the data

Go to STAT again

over to CALC

4: LinReg(ax+b)

Look at our data

x 0 1 2 3 4

y 20 32 44 56 68

y = ax+b

a = 12b = 20

That means your equation is

y = 12x + 20

Enter

On your screen

Go to STAT

1: Edit

*fill in the data

Go to STAT again

over to CALC

4: LinReg(ax+b)

Page 37 # 11

x 0 1 2 3

y 11 15 19 23

y = ax+b

a = 4b = 11

That means your equation is

y = 4x + 11

On your screen

Enter

# 11 – 15, and 22, 23

This word equation is called a verbal model.

USING A PROBLEM SOLVING PLAN

The verbal model is then used to write a mathematical statement, which is called an algebraic model.

WRITE AVERBAL MODEL.

ASSIGN LABELS.

WRITE AN ALGEBRAIC MODEL.

It is helpful when solving real-life problems to first write an equation in words before you write it in mathematical symbols.

SOLVE THEALGEBRAIC MODEL.

ANSWER THE QUESTION.

Writing and Using a Formula

The Bullet Train runs between the Japanese cities of Osaka and Fukuoka, a distance of 550 kilometers. When it makes no stops, it takes 2 hours and 15 minutes to make the trip. What is the average speed of the Bullet Train?

r550

2.25=

Write algebraic model.

Divide each side by 2.25.

Use a calculator.r244


LABELS

VERBAL MODEL Distance = Rate • Time

550Distance = (kilometers)

2.25Time = (hours)

rRate = (kilometers per hour)

ALGEBRAIC MODEL

You can use the formula d = r t to write a verbal model.

The Bullet Train’s average speed is about 244 kilometers per hour.

d = r t•r550 (2.25)=


You can use unit analysis to check your verbal model.

550 kilometers 244 kilometers

hour• 2.25 hours

UNIT ANALYSIS

USING OTHER PROBLEM SOLVING STRATEGIES

When you are writing a verbal model to represent a real-life problem, remember that you can use otherproblem solving strategies, such as draw a diagram, look for a pattern, or guess, check and revise, to helpcreate a verbal model.

Drawing a Diagram

RAILROADS In 1862, two companies were given the rights to build a railroad from Omaha, Nebraska to Sacramento, California. The Central Pacific Company began from Sacramento in 1863. Twenty-four months later, the Union Pacific company began from Omaha. The Central Pacific Company averaged 8.75 miles of track per month. The Union Pacific Company averaged 20 miles of track per month.

The companies met in Promontory, Utah, as the 1590 miles of track were completed. In what year did they meet? How many miles of track did each company build?

Write algebraic model.1590 = 8.75 + (t – 24)20t

Union Pacific time = (months)t – 24

Union Pacific rate = 20 (miles per month)

Central Pacific time =

(months)t

Central Pacific rate = 8.75 (miles per month)

Total miles of track = 1590 (miles)

Drawing a Diagram

ALGEBRAIC MODEL

LABELS

VERBAL MODEL

Total miles

of track= +• Number of

monthsMiles per

month

Central Pacific

•Number of

monthsMiles per

month

Union Pacific

Divide each side by 28.75.72 = t

The construction took 72 months (6 years) from the time theCentral Pacific Company began in 1863. They met in 1869.

1590 = 8.75 t + 20 (t – 24)ALGEBRAIC MODEL


Drawing a Diagram

1590 = 8.75 t + 20 t – 480

2070 = 28.75 t

Distributive property

Simplify.

Drawing a Diagram

The number of miles of track built by each company is as follows:

Central Pacific:

Union Pacific:

• 72 months

• (72 – 24) months

8.75 miles

20 miles

month

month

= 630 miles

= 960 miles

The construction took 72 months (6 years) from the time The Central Pacific Company began in 1863.

You can use the observed pattern to write a model for the height.

Substitute 15 for n.


Simplify.

= + h 20 12 n

Height to top of a story = h (feet)

Height per story =

12 (feet per story)

Height of lobby = 20 (feet)

ALGEBRAIC MODEL

= 200

Height to topof a story = Height per

story • Storynumber

Height oflobby +

= 20 + 12 (15)

Story number = n (stories)

LABELS

VERBAL MODEL

The height to the top of the 15th story is 200 feet.

Looking for a Pattern


After you have set up and solved an algebraic model for problem description, what remains to be done?


1.61.6 Solving Linear Inequalities


GoalGoal 22

Solve simple inequalities

Solve compound inequalities.

Verbal Phrase

All real numbers less than 3

Inequality

x < 3

Graph

0 1 2 3-1-2

Let’s describe the Let’s describe the inequality in different inequality in different

ways.ways.

Verbal Phrase

All real numbers greater than or equal to 0

Inequality

Graph

0 1 2 3-1-2

0x

What about this one…What about this one…

Solving Linear Inequalities

Ex 1)

You solve these just like you solved other linear equations.

35 x5 5

2x

Subtract 5

Solving Linear Inequalities

Ex 2) 74 x4 4

3x

Add 4

Solving 2-Step Linear Inequalities

ex) 283 xBeware….

8 863 x

Watch this… 3 32x

Reverse the inequality!Because you divided by a negative.

xx 5344

Solving Linear Inequalities with Variables on both sides

x5x5

349 x4 4

19 x9 9

9

1x

3)42(4 x

Solving Linear Inequalities using the Distributive Property

88

516 x16 16

16

5x

3168 x

xx 383494

Solving Linear Inequalities using Combing Like terms and variables on both sides of the equal sign.

55

0x

55 x

xx 3554 x3x3

Solving Compound Inequalities Involving “And”

A Compound Inequality consists of two inequalities connected by the word andand or the word oror.


40 x

All real numbers that are

greater than or equal to zero and less than 4.


greater than or equal to zero and less than 4.

0 1 2 3 4-1

10832 x

Solve for x.

8 8

1836 x3 3

62 x

8

3

61 2 3 4 5

122 x

Solve for x.

2 2

30 x1 1

30 x

1

0-3 -2 -1

2

62 x

Write an inequality that represents the statement.

ex 1) x is less than 6 and greater than 2.

103 x

ex 2) x is less than or equal to 10 and greater than -3.

20 x

ex 3) x is greater than or equal to 0 and less than or equal to 2.

Write an inequality that represents the statement.

ex 4) The frequency of a human voice is measured in hertz and has a range of 85 hertz to 1100 hertz.

110085 x

3x

What if...

and 6x

3 6

What numbers make bothboth statements true?

3x

What if...

and 6x

3 6

What numbers make bothboth statements true?

No, just the 6x

5x

What if...

and 5x

-5 5

Can this happen??

A number can’t be both….

Solving Compound Inequalities Involving “Or”

Remember…


Remember…



Less than -1 oror greater than 2.


Less than -1 oror greater than 2.

1x 2xoror

0 1 2 3-2 -1

413 x

Solve for x and graph.

752 xoror

61

1 133 x

3 3

1x

5 5

122 x2 2

6x

37 x


242 xoror

1210

7 710x

2 2

12x

14 x


155 xoror

-3-5

4 45x

5 5

3x

Is x = -4 a solution?

3x

What if...

oror 6x

3 6

What numbers make the statement true?

3x

What if...

oror 6x

3 6

What numbers make the statement true?All numbers greater than 3


Compare solving linear inequalities with solving linear equalities.


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

9 –- 20119 –- 2011 Do

Today

We will take the last section of notes when the Quiz C is complete. If you want to retake Quiz B, get one from Mrs. S.

I will be checking your Do NowsDo Nows and NotesNotes

Get Ready!

(8) Do Nows Notes: 1.1, 1.2, 1.3, 1.4, 1.5, 1.7

1.71.7 Solving Absolute Value Equations and Inequalities


GoalGoal 22

Solve absolute value equations and inequalities

Use absolute value equations and inequalities to solve real-life problems.

An open sentence involving absolute absolute valuevalue should be interpreted, solved, and graphed as a compoundcompound sentence.

Study the examples:…

Taking Notes

Last section this chapter

Test Tomorrow!

(8) Do Nows Notes: 1.1, 1.2, 1.3, 1.4, 1.5, 1.7

An open sentence involving absolute valueabsolute value should be interpreted, solved, and graphed as a compoundcompound sentence.

Study the examples:…

For cbax 0c

cbax cbax

, x is a solution of

or

For 0c , x has no solution

cbax

example 1a) 2x

2x 2xor

2-2

8xexample 1b)

What can x be?

What can x be?

Nothing…, no solution

example 2) 43 x

43 x 43 xor

-3 -3 -3 -3

1x 7x

1-7

example 3) 725 y

725 y 725 yor

+2 +2 +2 +2

95 y 55 y

5 5 55

0 1 2 3-1-2

5

41y 1y

example 4) 275 x

95 x 95 xor

-5 -5 -5 -5

4x 14x

4-14

7 7

95 x

1ST get absolute valueabsolute value

by itself.


The equation


Does the equation

45105 x has two solutions.

also have two solutions?

45105 x

Page 55 # 9 – 31 oddPage 62 # 17 – 22

Solving Absolute Value Inequalities

An absolute-value inequality is an inequality that has one of these forms:

cbax

cbax

cbax

cbax

example 2) 1043 x

1043 x 1043 xand

-4 -4 -4 -4

63 x 143 x

3 3 33

0 1 2-3 -1-2

2x 3

24x

-4-5

example 3) 514 x

514 x 514 xor

-1 -1 -1 -1

44 x 64 x

4 4 44

0 1 2-3 -1-2

1x2

11x

example 4) 743 x

Solve each open sentence.

example 5) 743 x

No solution

All numbers work.

example 6) 063 x No solution

example 7) 033 x 1

Graph each on a number line.

2x

2x

2x-1 1 20-2-3 3

-1 1 20-2-3 3

-1 1 20-2-3 3

Let’s do some examples…..

example 8) 523 x

523 x 523 xand

-3 -3 -3 -3

22 x 82 x

-2 -2 -2-2

0 1 2-1 3

1x 4x

4

example 9) 712 x

712 x 712 xor

+1 +1 +1 +1

82 x 62 x

2 2 22

4-3

4x 3x


How are absolute value inequalities containing a


symbol solved differently from those containing a or

or symbol?

do now you work 8 hours and earn $60. what is your earning rate? (important to include units.) you...

Documents

collection of numbers

order real numbers

number line

number operationswhat

numerical expression

b cassociative property3

inverse propertya b

ab c