Variables and Expressions -...

transcript

Variables and Expressions

Section 1-1

Vocabulary

• Quantity

• Variable

• Algebraic expression

• Numerical expression

Definition

• Quantity – A mathematical quantity is

anything that can be measured or counted.

– How much there is of something.

– A single group, generally represented in an

expression using parenthesis () or brackets [].

• Examples:

– numbers, number systems, volume, mass,

length, people, apples, chairs.

– (2x + 3), (3 – n), [2 + 5y].

Definition

• Variable – anything that can vary or change in

value.

– In algebra, x is often used to denote a variable.

– Other letters, generally letters at the end of the alphabet

(p, q, r, s, t, u, v, w, x, y, and z) are used to represent

variables

– A variable is “just a number” that can change in value.

• Examples:

– A child’s height

– Outdoor temperature

– The price of gold

Definition

• Constant – anything that does not vary or change

in value (a number).

– In algebra, the numbers from arithmetic are constants.

– Generally, letters at the beginning of the alphabet (a, b,

c, d)used to represent constants.

• Examples:

– The speed of light

– The number of minutes in a hour

– The number of cents in a dollar

– π.

Definition

• Algebraic Expression – a mathematical

phrase that may contain variables,

constants, and/or operations.

• Examples: 5x + 3, y/2 – 4, xy – 2x + y,

3(2x + 7), 2𝑎+𝑏

Definition

• Term – any number that is added subtracted.

– In the algebraic expression x + y, x and y are

terms.

• Example:

– The expression x + y – 7 has 3 terms, x, y, and

7. x and y are variable terms; their values vary

as x and y vary. 7 is a constant term; 7 is always

Definition

• Factor – any number that is multiplied.

– In the algebraic expression 3x, x and 3 are

factors.

• Example:

– 5xy has three factors; 5 is a constant factor, x

and y are variable factors.

Example: Terms and Factors

• The algebraic expression 5x + 3;

– has two terms 5x and 3.

– the term 5x has two factors, 5 and x.

Definition

• Numerical Expression – a mathematical

phrase that contains only constants and/or

operations.

• Examples: 2 + 3, 5 ∙ 3 – 4, 4 + 20 – 7, (2 +

3) – 7, (6 × 2) ÷ 20, 5 ÷ (20 × 3)

Multiplication Notation

In expressions, there are many different ways to

write multiplication.

2) a • b

3) a(b) or (a)b

4) (a)(b)

5) a ⤫ b

We are not going to use the multiplication symbol (⤫) any

more. Why?

Can be confused with the variable x.

Division Notation

Division, on the other hand, is written as:

1) 𝑥

2) x ÷ 3

In algebra, normally write division as a fraction.

Translate Words into

Expressions

• To Translate word phrases into algebraic

expressions, look for words that describe

mathematical operations (addition,

subtraction, multiplication, division).

What words indicate a particular

operation?

Addition

• Sum

• Plus

• More than

• Increase(d) by

• Perimeter

• Deposit

• Gain

• Greater (than)

• Total

Subtraction

• Minus

• Take away

• Difference

• Reduce(d) by

• Decrease(d) by

• Withdrawal

• Less than

• Fewer (than)

• Loss of

Words for Operations - Examples

What words indicate a particular

operation?

Multiply

• Times

• Product

• Multiplied by

• Of

• Twice (×2), double (×2),

triple (×3), etc.

• Half (×½), Third (×⅓),

Quarter (×¼)

• Percent (of)

Divide

• Quotient

• Divided by

• Half (÷2), Third (÷3),

Quarter (÷4)

• Into

• Per

• Percent (out of 100)

• Split into __ parts

Words for Operations - Examples

Writing an algebraic expression with addition.

Two plus a number n

Writing an Algebraic

Expression for a Verbal Phrase.

of the

wording

Matters

Writing an algebraic expression with addition.

Two more than a number

of the

wording

Matters

Writing an algebraic expression with subtraction.

The difference of seven and a number n

7 – n

of the

wording

Matters

Writing an algebraic expression with subtraction.

Eight less than a number

y – 8

of the

wording

Matters

Writing an algebraic expression with multiplication.

one-third of a number n.

of the

wording

Matters

Writing an algebraic expression with division.

The quotient of a number n and 3

of the

wording

Matters

Example

“Translating” a phrase into an algebraic

expression:

Nine more than a number y

Can you identify the operation?

“more than” means add

Answer: y + 9

Example

expression:

4 less than a number n

Identify the operation?

“less than” means subtract

Answer: n – 4.

Why not 4 – n?????

Determine the order of the variables and constants.

Example

expression:

A quotient of a number x and12

Can you identify the operation?

“quotient” means divide

Determine the order of the variables and constants.

Answer: .

Why not ?????

Example

expression, this one is harder……

5 times the quantity 4 plus a number c

Can you identify the operation(s)?

What does the word quantity mean?

“times” means multiple and “plus” means add

that “4 plus a number c” is grouped using

parenthesis

Answer: 5(4 + c)

Your turn:

1) m increased by 5.

2) 7 times the product

of x and t.

3) 11 less than 4 times a

number.

4) two more than 6

times a number.

5) the quotient of a

number and 12.

1) m + 5

2) 7xt

3) 4n - 11

4) 6n + 2

Your Turn:

a. 7x + 13

b. 13 - 7x

c. 13 + 7x

d. 7x - 13

Which of the following expressions represents

7 times a number decreased by 13?

Your Turn:

1. 28 - 3x

2. 3x - 28

3. 28 + 3x

4. 3x + 28

Which one of the following expressions represents 28

less than three times a number?

Your Turn:

1. Twice the sum of x and three

2. Two less than the product of 3 and x

3. Three times the difference of x and two

4. Three less than twice a number x

5. Two more than three times a number x

A. 2x – 3

B. 3(x – 2)

C. 3x + 2

D. 2(x + 3)

E. 3x – 2

Match the verbal phrase and the expression

Translate an Algebraic

Expression into Words

• We can also start with an algebraic

expression and then translate it into a word

phrase using the same techniques, but in

reverse.

• Is there only one way to write a given

algebraic expression in words?

– No, because the operations in the expression

can be described by several different words and

phrases.

Give two ways to write each algebra expression in words.

A. 9 + r B. q – 3

the sum of 9 and r

9 increased by r

the product of m and 7

m times 7

the difference of q and 3

3 less than q

the quotient of j and 6

j divided by 6

C. 7m D.

Example: Translating from

Algebra to Words

j ÷ 6

a. 4 - n b.

c. 9 + q d. 3(h)

4 decreased by n

the sum of 9 and q

the quotient of t and 5

the product of 3 and h

Give two ways to write each algebra expression in words.

Your Turn:

n less than 4 t divided by 5

q added to 9 3 times h

Your Turn:

1. 9 increased by twice a number

2. a number increased by nine

3. twice a number decreased by 9

4. 9 less than twice a number

Which of the following verbal expressions

represents 2x + 9?

Your Turn:

1. 5x - 16

2. 16x + 5

3. 16 + 5x

4. 16 - 5x

Which of the following expressions represents the

sum of 16 and five times a number?

Your Turn:

• 4(x + 5) – 2 • Four times the sum of x and 5 minus two

• 7 – 2(x ÷ 3)• Seven minus twice the quotient of x and three

• m ÷ 9 – 4• The quotient of m and nine, minus four

CHALLENGE

Write a verbal phrase that describes the expression

Your Turn:

• Six miles more than yesterday• Let x be the number of miles for yesterday

• x + 6

• Three runs fewer than the other team scored• Let x = the amount of runs the other team scored

• x - 3

• Two years younger than twice the age of your

cousin• Let x = the age of your cousin

• 2x – 2

Define a variable to represent the unknown and write the

phrase as an expression.

Patterns

Mathematicians …

• look for patterns

• find patterns in physical or

pictorial models

• look for ways to create

different models for

patterns

• use mathematical models

to solve problems

Number Patterns

2 + 2 + 2

2 + 2 + 2 + 2 4(2)

n? __(2)

Number

Term Expression

Number Patterns

6(5) + 4

5(5) + 4

4(5) + 4

3(5) + 41

n? _____(5) + 4(n + 2)

How does

different

part relate

to the

number?

What’s

different?

Number Term Expression

Number Patterns

3 - 2(3)

3 - 2(2)

3 - 2(1)

3 - 2(0)1

n? 3 - 2(____)n - 1

How does

different

part relate

to the

number?

What’s

different?

Number Term Expression

Writing a Rule to Describe

a Pattern

• Now lets try a real-life problem.

Bonjouro! My name is Fernando

I am preparing to cook a GIGANTIC

home-cooked Italian meal for my family.

The only problem is I don’t know yet how

many people are coming. The more

people that come, the more spaghetti I

will need to buy.

From all the meals I have

cooked before I know:

For 1 guest I will need 2 bags of spaghetti,

For 2 guests I will need 5 bags of spaghetti,

For 3 guests I will need 8 bags of spaghetti,

For 4 guests I will need 11 bags of spaghetti.

Here is the table of how

many bags of spaghetti I

will need to buy:

Number

of Guests

Bags of

Spaghetti

The numbers in the ‘spaghetti’ column

make a pattern:

2 5 8 11

What do we need to add on each time to

get to the next number?

+ 3 + 3 + 3

We say there is a

COMMON

DIFFERENCE

between the numbers.

We need to add on the same

number every time.

What is the common difference

for this sequence?

Now we know the common

difference we can start to work out

the MATHEMATICAL RULE.

The mathematical rule is the

algebraic expression that lets us

find any value in our pattern.

We can use our common difference to help

us find the mathematical rule.

We always multiply the common

difference by the TERM NUMBER to give us

the first step of our mathematical rule.

What are the term numbers in my case

NUMBER OF GUESTS

So if we know that step one of finding the

mathematical rule is:

Common Term

Difference Numbers

then what calculations will we do in this

example?

Common Difference Term NumbersX

X3 Number of Guests

We will add a column to our original

table to do these calculations:

Number

of Guests

Bags of

Spaghetti

We are trying to find a

mathematical rule that will

take us from:

Number of Guests

Number of Bags of Spaghetti

At the moment we have:

Does this get us the answer

we want?

3n gives us: Bags of Spaghetti

What is the difference between all

the numbers on the left and all

the numbers on the right?

We will now add another column to

our table to do these calculations:

Number

of Guests

Bags of

Spaghetti

3n 3n – 1

Does this new column get us to

where we are trying to go?

So now we know our mathematical

3n –1

Your Turn:

• The table shows how the cost of renting a scooter

depends on how long the scooter is rented. What is

a rule for the total cost? Give the rule in words and

as an algebraic expression.

Hours Cost

1 $17.50

2 $25.00

3 $32.50

4 $40.00

5 $47.50

Answer:Multiply the number of

hours by 7.5 and add 10.

7.5n + 10

Practice!

Pg. 6-7 # 1-19 odd, 21-24

Order of Operations and

Evaluating Expressions

Section 1-2 Part 1

Vocabulary

• Power

• Exponent

• Base

• Simplify

Definition

A power expression has two parts, a base and

an exponent.

Power expression

ExponentBase

In the power expression 103, 10 is called the base

and 3 is called the exponent or power.

103 means 10 • 10 • 10

103 = 1000

The base, 10, is the number

that is used as a factor. 10 3 The exponent, 3, tells

how many times the

base, 10, is used as a

factor.

Definition

• Base – In a power expression, the base is

the number that is multiplied repeatedly.

• Example:

– In x3, x is the base. The exponent says to

multiply the base by itself 3 times; x3 = x ⋅ x ⋅ x.

Definition

• Exponent – In a power expression, the

exponent tells the number of times the base

is used as a factor.

• Example:

– 24 equals 2 ⋅ 2 ⋅ 2 ⋅ 2.

– If a number has an exponent of 2, the number is

often called squared. For example, 42 is read “4

squared.”

– Similarly, a number with an exponent of is

called “cubed.”

When a number is raised to the second power, we usually

say it is “squared.” The area of a square is s s = s2,

where s is the side length.

When a number is raised to the third power, we usually say

it is “cubed.” The volume of a cube is s s s = s3, where s

is the side length.s

Powers

There are no easy geometric models for numbers raised to exponents greater than 3,

but you can still write them using repeated multiplication or with a base and

exponent.

3 to the second power, or 3

squared

3 3 3 3 3

Multiplication Power ValueWords

3 3 3 3

33 to the first power

3 to the third power, or 3

3 to the fourth power

3 to the fifth power

Reading Exponents

Powers

Caution!In the expression –5², 5 is the base because the

negative sign is not in parentheses. In the

expression (–2)³, –2 is the base because of the

parentheses.

Definition

• Simplify – a numerical expression is

simplified when it is replaced with its single

numerical value.

• Example:

– The simplest form of 2 • 8 is 16.

– To simplify a power, you replace it with its

simplest name. The simplest form of 23 is 8.

Example: Evaluating Powers

Simplify each expression.

A. (–6)3

(–6)(–6)(–6)

–216

Use –6 as a factor 3 times.

B. –102

–1 • 10 • 10

–100

Think of a negative sign in front of a power as multiplying by a –1.

Find the product of –1 and

two 10’s.

Example: Evaluating Powers

Simplify the expression.

81 29 2

Use as a factor 2 times.2 9

Your Turn:

Evaluate each expression.

a. (–5)3

(–5)(–5)(–5)

–125

Use –5 as a factor 3 times.

b. –62

–1 6 6

Think of a negative sign in front of a power as multiplying by –1.

Find the product of –1 andtwo 6’s.

Your Turn:

Evaluate the expression.

Use as a factor 3 times.34

Example: Writing Powers

Write each number as a power of the given base.

A. 64; base 8

The product of two 8’s is 64.

B. 81; base –3

(–3)(–3)(–3)(–3)

(–3)4

The product of four –3’s is 81.

Your Turn:

Write each number as a power of a given base.

a. 64; base 4

The product of three 4’s is 64.

b. –27; base –3

(–3)(–3)(–3)

The product of three (–3)’s is –27.

Order of Operations

Rules for arithmetic and algebra

expressions that describe what

sequence to follow to evaluate an

expression involving more than

one operation.

Order of Operations

Is your answer 33 or 19?

You can get 2 different answers depending

on which operation you did first. We want

everyone to get the same answer so we

must follow the order of operations.

Evaluate 7 + 4 • 3.

Remember the phrase“Please Excuse My Dear Aunt Sally”

or PEMDAS.

ORDER OF OPERATIONS

1. Parentheses - ( ) or [ ]

2. Exponents or Powers

3. Multiply and Divide (from left to right)

4. Add and Subtract (from left to right)

The Rules

Step 1: First perform operations that are within grouping

symbols such as parenthesis (), brackets [], and braces {},

and as indicated by fraction bars. Parenthesis within

parenthesis are called nested parenthesis (( )). If an

expression contains more than one set of grouping symbols,

evaluate the expression from the innermost set first.

Step 2: Evaluate Powers (exponents) or roots.

Step 3: Perform multiplication or division operations in order

by reading the problem from left to right.

Step 4: Perform addition or subtraction operations in order by

reading the problem from left to right.

53621 53621

Performing operations left to right onlyPerforming operations using order of

operations

The rules for

order of

operations exist

so that everyone

can perform the

same consistent

operations and

achieve the same

results. Method 2

is the correct

method.

Can you imagine

what it would be like

if calculations were

performed differently

by various financial

institutions or what if

doctors prescribed

different doses of

medicine using the

same formulas and

achieving different

results?

Order of Operations

218654 Follow the left to right rule: First solve any

multiplication or division parts left to right. Then solve

any addition or subtraction parts left to right. 218654

A good habit to develop while learning order of operations is to

underline the parts of the expression that you want to solve first.

Then rewrite the expression in order from left to right and solve

the underlined part(s).

The order of operations must be followed each

time you rewrite the expression. 45

Divide

Multiply

Order of Operations: Example 1

Evaluate without grouping symbols

Exponents (powers)

Multiply

Subtract

Follow the left to right rule: First solve

exponent/(powers). Second solve multiplication or

division parts left to right. Then solve any addition or

subtraction parts left to right.

A good habit to develop while learning order of operations is to

underline the parts of the expression that you want to solve first.

Then rewrite the expression in order from left to right and solve

the underlined part(s).

time you rewrite the expression.

Expressions with powers

Exponents (powers)

Multiply

Subtract

Follow the left to right rule: First solve parts inside

grouping symbols according to the order of

operations. Solve any exponent/(Powers). Then solve

multiplication or division parts left to right. Then

solve any addition or subtraction parts left to right.

A good habit to develop while learning order

of operations is to underline the parts of the

expression that you want to solve first. Then

rewrite the expression in order from left to

right and solve the underlined part(s).

Grouping

symbols

8Divide

Evaluate with grouping symbols

Exponents (powers)

Multiply

Subtract

Follow the left to right rule: Follow the order of

operations by working to solve the problem above the

fraction bar. Then follow the order of operations by

working to solve the problem below the fraction bar.

Finally, recall that fractions are also division

problems – simplify the fraction.

A good habit to develop while learning order of operations is to underline the parts of the expression that

you want to solve first. Then rewrite the expression in order from left to right and solve the underlined

part(s).

)418(2

Work above the

fraction bar

Simplify:

Divide

)418(2

Work below the

fraction bar Grouping symbols

)14(2 Add

Expressions with fraction bars

Your Turn:

Simplify the expression.

8 ÷ · 3 1 2

16 · 3

There are no groupingsymbols.

Divide.

Multiply.

Your Turn:Simplify the expression.

5.4 – 32 + 6.2

5.4 – 9 + 6.2

–3.6 + 6.2

There are no groupingsymbols.

Simplify powers.

Subtract

Your Turn:Simplify the expression.

–20 ÷ [–2(4 + 1)]

–20 ÷ [–2(5)]

–20 ÷ –10

There are two sets of groupingsymbols.

Perform the operations in theinnermost set.

Perform the operation insidethe brackets.

Divide.

Your Turn:

1. -3,236

3. 107

4. 16,996

Which of the following represents 112 + 18 - 33 · 5 in

simplified form?

Your Turn:

Simplify 16 - 2(10 - 3)

Your Turn:

Simplify 24 – 6 · 4 ÷ 2

Caution!Fraction bars, radical symbols, and absolute-

value symbols can also be used as grouping

symbols. Remember that a fraction bar indicates

division.

Your Turn:Simplify.

5 + 2(–8)

(–2) – 3 3

5 + 2(–8)

(–2) – 3 3

5 + 2(–8)

–8 – 3

5 + (–16)

– 8 – 3

The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing.

Evaluate the power in the denominator.

Multiply to simplify the numerator.

Divide.

Your Turn:

Simplify.

2(–4) + 22

42 – 9

2(–4) + 22

42 – 9

–8 + 22

42 – 9

–8 + 22

16 – 9

The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing.

Multiply to simplify the numerator.

Evaluate the power in the denominator.

Add to simplify the numerator. Subtract to simplify the denominator.

Divide.

Practice

Pg. 12-13 # 1-20 all

Order of Operations and

Section 1-2 Part 2

Vocabulary

• Evaluate

• In Part 1 of this lesson, we simplified

numerical expressions with exponents and

learned the order of operations.

• Now, we will evaluate algebraic

expressions for given values of the variable.

Definition

• Evaluate – To evaluate an expression is to

find its value.

• To evaluate an algebraic expression,

substitute numbers for the variables in the

expression and then simplify the expression.

Example: Evaluating Algebraic

Expressions

Evaluate each expression for a = 4, b =7, and

c = 2.

A. b – c

b – c = 7 – 2

ac = 4 ·2

Substitute 7 for b and 2 for c.

Simplify.

Substitute 4 for a and 2 for c.

Simplify.

Your Turn:Evaluate each expression for m = 3, n = 2, and

p = 9.

b. p – n

c. p ÷ m

Substitute 3 for m and 2 for n.mn = 3 · 2Simplify.= 6

Substitute 9 for p and 2 for n.p – n = 9 – 2

Simplify.= 7

Substitute 9 for p and 3 for m.p ÷ m = 9 ÷ 3

Simplify.

Example: Evaluating Algebraic

Expressions

Evaluate the expression for the given value of x.

10 – x · 6 for x = 3

First substitute 3 for x.10 – x · 6

10 – 3 · 6 Multiply.

10 – 18 Subtract.

Example: Evaluating

Algebraic ExpressionsEvaluate the expression for the given value of x.

42(x + 3) for x = –2

First substitute –2 for x.42(x + 3)

42(–2 + 3)Perform the operation inside the parentheses.42(1)

Evaluate powers.16(1)

Multiply.16

Your Turn:Evaluate the expression for the given value of x.

14 + x2 ÷ 4 for x = 2

14 + x2 ÷ 4

First substitute 2 for x.14 + 22 ÷ 4

Square 2.14 + 4 ÷ 4

Divide.14 + 1

Add.15

Your Turn:Evaluate the expression for the given value of x.

(x · 22) ÷ (2 + 6) for x = 6

(x · 22) ÷ (2 + 6)

First substitute 6 for x.(6 · 22) ÷ (2 + 6)

Square two.(6 · 4) ÷ (2 + 6)

Perform the operations inside the parentheses.

(24) ÷ (8)

Divide.3

Your Turn:

1. -62

2. -42

What is the value of

-10 – 4x if x = -13?

Your Turn:

1. -8000

2. -320

3. -60

4. 320

5k3 if k = -4?

Your Turn:

2. -10

if n = -8, m = 4, and t = 2 ?

Example: Application

A shop offers gift-wrapping services at three price levels.

The amount of money collected for wrapping gifts on a

given day can be found by using the expression 2B + 4S +

7D. On Friday the shop wrapped 10 Basic packages B, 6

Super packages S, and 5 Deluxe packages D. Use the

expression to find the amount of money collected for gift

wrapping on Friday.

Example - Solution:

2B + 4S + 7D

First substitute the value for

each variable.2(10) + 4(6) + 7(5)

Multiply.20 + 24 + 35

Add from left to right.44 + 35

Add.79

The shop collected $79 for gift wrapping on Friday.

Your Turn:Another formula for a player's total number of bases is Hits + D + 2T + 3H.

Use this expression to find Hank Aaron's total bases for 1959, when he had 223

hits, 46 doubles, 7 triples, and 39 home runs.

Hits + D + 2T + 3H = total number of bases

First substitute values for each variable.

223 + 46 + 2(7) + 3(39)

Multiply.223 + 46 + 14 + 117

Add.400

Hank Aaron’s total number of bases for 1959 was 400.

USING A VERBAL MODEL

Use three steps to write a mathematical model.

WRITE A

VERBAL MODEL.

ASSIGN

LABELS.

WRITE AN

ALGEBRAIC MODEL.

Writing algebraic expressions that represent real-life

situations is called modeling.

The expression is a mathematical model.

A PROBLEM SOLVING PLAN USING MODELS

Writing an Algebraic Model

Ask yourself what you need to know to solve the

problem. Then write a verbal model that will give

you what you need to know.

Assign labels to each part of your verbal problem.

Use the labels to write an algebraic model based on

your verbal model.

VERBAL MODEL

your verbal model.

ALGEBRAIC

LABELS

Write an expression for the number of bottles needed to make s sleeping bags.

The expression 85s models the number of

bottles to make s sleeping bags.

Approximately eighty-five 20-ounce plastic

bottles must be recycled to produce the fiberfill

for a sleeping bag.

ContinuedApproximately eighty-five 20-ounce plastic

bottles must be recycled to produce the fiberfill

for a sleeping bag.

Find the number of bottles needed to make

20, 50, and 325 sleeping bags.

Evaluate 85s for s = 20, 50, and 325.

85(20) = 1700

To make 20 sleeping bags 1700 bottles are needed.

85(50) = 4250

To make 50 sleeping bags 4250 bottles are needed.

85(325) = 27,625To make 325 sleeping bags 27,625 bottles are needed.

Your Turn:

Write an expression for the number of bottles needed to make s sweaters.

The expression 63s models the number of

bottles to make s sweaters.

To make one sweater, 63 twenty ounce

plastic drink bottles must be recycled.

Your Turn: Continued

To make one sweater, 63 twenty ounce

plastic drink bottles must be recycled.

Find the number of bottles needed to make

12, 25 and 50 sweaters.

Evaluate 63s for s = 12, 25, and 50.

63(12) = 756

To make 12 sweaters 756 bottles are needed.

63(25) = 1575

To make 25 sweaters 1575 bottles are needed.

63(50) = 3150To make 50 sweaters 3150 bottles are needed.

Practice!

Pg. # 21-55 odd

Properties of Real Numbers

Section 1-3

Vocabulary

• Equivalent Expression

• Deductive reasoning

• Counterexample

Definition

• Equivalent Expression – Two algebraic

expressions are equivalent if they have the

same value for all values of the variable(s).

– Expressions that look difference, but are equal.

– The Properties of Real Numbers can be used to

show expressions that are equivalent for all real

numbers.

Mathematical Properties

• Properties refer to rules that indicate a standard procedure

or method to be followed.

• A proof is a demonstration of the truth of a statement in

mathematics.

• Properties or rules in mathematics are the result from

testing the truth or validity of something by experiment or

trial to establish a proof.

• Therefore every mathematical problem from the easiest to

the more complex can be solved by following step by step

procedures that are identified as mathematical properties.

Commutative and Associative

Properties

• Commutative Property – changing the order in which you

add or multiply numbers does not change the sum or

product.

• Associative Property – changing the grouping of numbers

when adding or multiplying does not change their sum or

product.

• Grouping symbols are typically parentheses (),but can

include brackets [] or Braces {}.

Commutative

Property of

Addition - (Order)

Commutative

Property of

Multiplication -

(Order)

For any numbers a and b , a + b = b + a

For any numbers a and b , a b = b a

45 + 5 = 5 + 45

6 8 = 8 6

50 = 50

48 = 48

Commutative Properties

Associative Property

of Addition -

(grouping symbols)

Associative Property

of Multiplication -

(grouping symbols)

For any numbers a, b, and c,

(a + b) + c = a + (b + c)

(ab)c = a (bc)

(2 + 4) + 5 = 2 + (4 + 5)

(2 3) 5 = 2 (3 5)

(6) + 5 = 2 + (9)

11 = 11

(6) 5 = 2 (15)

30 = 30

Associative Properties

Name the property that is illustrated in each equation.

A. 7(mn) = (7m)n

Associative Property of Multiplication

The grouping is different.

B. (a + 3) + b = a + (3 + b)

Associative Property of Addition

The grouping is different.

C. x + (y + z) = x + (z + y)

Commutative Property of Addition

The order is different.

Example: Identifying

Properties

Name the property that is illustrated in each equation.

a. n + (–7) = –7 + n

b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3

c. (xy)z = (yx)z

Commutative Property of Multiplication

The order is

different.

The grouping is

different.

The order is

different.

Your Turn:

Note!The Commutative and Associative

Properties of Addition and Multiplication

allow you to rearrange an expression.

Commutative and associative properties are very

helpful to solve problems using mental math strategies.

Solve: 18 + 13 + 16 + 27 + 22 + 24 Rewrite the problem by grouping numbers that

can be formed easily. (Associative property)

This process may change the order in which the

original problem was introduced. (Commutative

property)

(18 + 22) + (16 + 24) + (13 + 27)

(40) + (40) + (40) = 120

Properties

Commutative and associative properties are very

helpful to solve problems using mental math strategies.

Solve: 4 7 25

Rewrite the problem by changing the order in

which the original problem was introduced.

(Commutative property)

4 25 7

(4 25) 7

(100) 7 = 700

Group numbers that can be formed easily.

(Associative property)

Properties

Identity and Inverse

Properties

• Additive Identity Property

• Multiplicative Identity Property

• Multiplicative Property of Zero

• Multiplicative Inverse Property

Additive Identity Property

For any number a, a + 0 = a.

The sum of any number and zero is equal to that

number.

The number zero is called the additive identity.

If a = 5 then 5 + 0 = 5

Multiplicative Identity Property

For any number a, a 1 = a.

The product of any number and one is equal to

that number.

The number one is called the multiplicative

identity.

If a = 6 then 6 1 = 6

Multiplicative Property of

For any number a, a 0 = 0.

The product of any number and zero is equal to zero.

If a = 6, then 6 0 = 0

1such that number one

exactly is there,0 , where, number nonzeroevery For

3 3 4 3 4 12 Given the fraction ; then 1;

4 4 3 4 3 12

4the fraction is the reciprocal.

Together the two fractions are multiplicative

inverses that are equal to the product 1.

Multiplicative Inverse Property

Two numbers whose product is 1 are called multiplicative

inverses or reciprocals.

Zero has no reciprocal because any number times 0 is 0.

Identity and Inverse

Properties

Property Words Algebra Numbers

The sum of a number and 0, the additive

identity, is the original number.

n + 0 = n 3 + 0 = 0

The product of a number and 1, the

multiplicative identity, is the

original number.

n 1 = n

Additive Inverse Property

The sum of a number and its opposite, or

additive inverse, is 0.n + (–n) = 0

5 + (–5) = 0

The product of a nonzero number and

its reciprocal, or multiplicative inverse,

Example: Writing Equivalent

Expressions

A. 4(6y)

Use the Associative Property of

Multiplication4(6y) = (4•6)y

Simplify=24y

B. 6 + (4z + 3)

6 + (4z + 3) = 6 + (3 + 4z)

= (6 + 3) + 4z

= 9 + 4z

Use the Commutative

Property of Addition

Use the Associative

Property of Addition

Simplify

Example: Writing Equivalent

Expressions

mn m n

Rewrite the numerator using the

Identity Property of Multiplication

Use the rule for multiplying fractionsa c ac

b d bd

Simplify the fractions

Simplify

Your Turn:

Simplify each expression.

A. 4(8n)

B. (3 + 5x) + 7

A. 32n

B. 10 + 5b

Identify which property

that justifies each of the

following.

4 (8 2) = (4 8) 2

following.

4 (8 2) = (4 8) 2

Associative Property of Multiplication

following.

6 + 8 = 8 + 6

following.

6 + 8 = 8 + 6

following.

12 + 0 = 12

following.

12 + 0 = 12

following.

5 + (2 + 8) = (5 + 2) + 8

following.

5 + (2 + 8) = (5 + 2) + 8

following.

5 24 = 24 5

following.

5 24 = 24 5

Commutative Property of Multiplication

following.

-34 1 = -34

following.

-34 1 = -34

Deductive Reasoning

Deductive Reasoning – a form of argument in

which facts, rules, definitions, or properties are

used to reach a logical conclusion (i.e. think

Sherlock Holmes).

Counterexample

• The Commutative and Associative Properties are

true for addition and multiplication. They may not

be true for other operations.

• A counterexample is an example that disproves a

statement, or shows that it is false.

• One counterexample is enough to disprove a

statement.

Caution!One counterexample is enough to disprove

a statement, but one example is not

enough to prove a statement.

Statement Counterexample

No month has fewer than 30 days.February has fewer than 30 days, so

the statement is false.

Every integer that is divisible by 2 is

also divisible by 4.The integer 18 is divisible by 2 but is

not by 4, so the statement is false.

Example: Counterexample

Find a counterexample to disprove the statement “The Commutative

Property is true for raising to a power.”

Find four real numbers a, b, c, and d such that

a³ = b and c² = d, so a³ ≠ c².

Try a³ = 2³, and c² = 3².

a³ = b

2³ = 8

c² = d

3² = 9

Since 2³ ≠ 3², this is a counterexample. The statement is false.

Example: Counterexample

Find a counterexample to disprove the statement “The

Commutative Property is true for division.”

Find two real numbers a and b, such that

Try a = 4 and b = 8.

Since , this is a counterexample.

The statement is false.

Your Turn:

Practice!

Adding and Subtracting Real

Numbers

Section 1-4, 1-5

Vocabulary

• Absolute value

• Opposite

• Additive inverses

The set of all numbers that can be represented on a

number line are called real numbers. You can use a

number line to model addition and subtraction of real

numbers.

Addition

To model addition of a positive number, move right.

To model addition of a negative number, move left.

Subtraction

To model subtraction of a positive number, move

left. To model subtraction of a negative number,

move right.

Real Numbers

Add or subtract using a number line.

Start at 0. Move left to –4.

11 10 9 8 7 6 5 4 3 2 1 0

+ (–7)

–4 + (–7) = –11

To add –7, move left 7 units.

–4 + (–7)

Example: Adding &

Subtracting on a Number Line

Start at 0. Move right to 3.

To subtract –6, move right 6 units.

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

3 – (–6) = 9

3 – (–6)

–(–6)

Example: Adding &

Subtracting on a Number Line

–3 + 7 Start at 0. Move left to –3.

To add 7, move right 7 units.

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

–3 + 7 = 4

Your Turn:

–3 – 7 Start at 0. Move left to –3.

To subtract 7, move left 7 units.

11 10 9 8 7 6 5 4 3 2 1 0

–3 – 7 = –10

Your Turn:

–5 – (–6.5)Start at 0. Move left to –5.

To subtract –6.5, move right 6.5 units.

8 7 6 5 4 3 2 1 0

–5 – (–6.5) = 1.5

– (–6.5)

Your Turn:

Definition

• Absolute Value – The distance between a

number and zero on the number line.

– Absolute value is always nonnegative since

distance is always nonnegative.

– The symbol used for absolute value is | |.

• Example:

– The |-2| is 2 and the |2| is 2.

The absolute value of a number is the distance from

zero on a number line. The absolute value of 5 is

written as |5|.

5 units 5 units

210123456 6543- - - - - -

|5| = 5|–5| = 5

Absolute Value on the

Number Line

Rules For Adding

Use the sign of the number with the

greater absolute value.

Different signs: subtract the

absolute values.

B. –6 + (–2)

(6 + 2 = 8)

–8 Both numbers are negative, so the sum is negative.

Same signs: add the absolute values.

Example: Adding Real Numbers

–5 + (–7)

–12 Both numbers are negative, so the

sum is negative.

(5 + 7 = 12)

–13.5 + (–22.3)b.

(13.5 + 22.3 = 35.8)

–35.8 Both numbers are negative, so the

sum is negative.

Your Turn:

c. 52 + (–68)

(68 – 52 = 16)

–16Use the sign of the number with the

greater absolute value.

Different signs: subtract the

absolute values.

Your Turn:

Definition

• Additive Inverse – The negative of a

designated quantity.

– The additive inverse is created by multiplying

the quantity by -1.

• Example:

– The additive inverse of 4 is -1 ∙ 4 = -4.

Opposites

• Two numbers are opposites if their sum is 0.

• A number and its opposite are additive

inverses and are the same distance from

• They have the same absolute value.

Additive Inverse Property

Subtracting Real Numbers

• To subtract signed numbers, you can use

additive inverses.

• Subtracting a number is the same as adding

the opposite of the number.

• Example:

– The expressions 3 – 5 and 3 + (-5) are

equivalent.

A number and its opposite are additive inverses.

To subtract signed numbers, you can use additive

inverses.

11 – 6 = 5 11 + (–6) = 5

Additive inverses

Subtracting 6 is the same

as adding the inverse of 6.

Subtracting a number is the same as adding the

opposite of the number.

Subtracting Real Numbers

Subtracting Real NumbersRules For Subtracting

Subtract.

–6.7 – 4.1

–6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1.

Same signs: add absolute values.

–10.8 Both numbers are negative, so the sum

is negative.

(6.7 + 4.1 = 10.8)

Example: Subtracting Real

Numbers

Subtract.

5 – (–4)

5 − (–4) = 5 + 4

To subtract –4, add 4.

Same signs: add absolute values.(5 + 4 = 9)

Both numbers are positive, so the sum

is positive.

Example: Subtracting Real

Numbers

On many scientific and graphing calculators, there is

one button to express the opposite of a number and a

different button to express subtraction.

Helpful Hint

Subtract.

13 – 21

13 – 21 To subtract 21, add –21.

Different signs: subtract absolute values.

Use the sign of the number with the greater absolute value.–8

= 13 + (–21)

(21 – 13 = 8)

Your Turn:

–14 – (–12)

Subtract.

–14 – (–12) = –14 + 12

(14 – 12 = 2)

Use the sign of the number with the greater absolute value.

Different signs: subtract absolute values.

Your Turn:

An iceberg extends 75 feet above the sea. The

bottom of the iceberg is at an elevation of –247

feet. What is the height of the iceberg?

Find the difference in the elevations of the top of the iceberg and

the bottom of the iceberg.

elevation at top of

icebergminus

elevation at bottom

of iceberg

75 – (–247)

75 – (–247) = 75 + 247

–75 –247

The height of the iceberg is 322 feet.

What if…? The tallest known iceberg in the

North Atlantic rose 550 feet above the ocean's

surface. How many feet would it be from the top

of the tallest iceberg to the wreckage of the

Titanic, which is at an elevation of –12,468 feet?

elevation at top of

icebergminus

elevation of the

Titanic

550 – (–12,468)

550 – (–12,468) = 550 + 12,468To subtract –12,468,

add 12,468.

= 13,018

550 –12,468

Your Turn:

Distance from the top of the iceberg to the Titanic is 13,018 feet.

Practice!

Multiplying and Dividing Real

Numbers

Section 1-6

Vocabulary

• Multiplicative Inverse

• Reciprocal

When you multiply two numbers, the signs of thenumbers you are multiplying determine whetherthe product is positive or negative.

Factors Product

3(5) Both positive

3(–5) One negative

–3(–5) Both negative

15 Positive

–15 Negative

15 Positive

This is true for division also.

Multiplying Real Numbers

Rules for Multiplying and

Dividing

Find the value of each expression.

–5The product of two numbers

with different signs is negative.

12The quotient of two numbers

with the same sign is positive.

Example: Multiplying and

Dividing Real Numbers

The quotient of two numbers

Multiply.

Example: Multiplying and

Dividing Real Numbers

–7The quotient of two numbers

a. 35 (–5)

44The product of two numbers

with the same sign is positive.

b. –11(–4)

c. –6(7)

–42The product of two numbers with different

signs is negative.

Your Turn:

Reciprocals

• Two numbers are reciprocals if their product is 1.

• A number and its reciprocal are called

multiplicative inverses. To divide by a number,

you can multiply by its multiplicative inverse.

• Dividing by a nonzero number is the same as

Multiplying by the reciprocal of the number.

10 ÷ 5 = 2 10 ∙ = = 215

Multiplicative inverses

Dividing by 5 is the same as multiplying by the

reciprocal of 5, .

Reciprocals

You can write the reciprocal of a number by

switching the numerator and denominator. A whole

number has a denominator of 1.

Helpful Hint

Example 2 Dividing by Fractions

Divide.

Example: Dividing with

Fractions

To divide by , multiply by .

Multiply the numerators and

multiply the denominators.

and have the same sign,

so the quotient is positive.

Divide.

Write as an improper fraction.

and have different signs,

so the quotient is negative.

Example: Dividing with

Fractions

Divide.

and –9 have the same signs,

so the quotient is positive.

Your Turn:

Divide.

and have different signs,

so the quotient is negative.

Your Turn:

Check It Out! Example 2c

Divide.

To divide by multiply by .

The signs are different, so the

quotient is negative.

• No number can be multiplied by 0 to give a

product of 1, so 0 has no reciprocal.

• Because 0 has no reciprocal, division by 0 is

not possible. We say that division by zero is

undefined.

• The number 0 has special properties for

multiplication and division.

Multiply or divide if possible.

B. –22 0

undefined

C. –8.45(0)

Zero is divided by a nonzero number.

The quotient of zero and any nonzero

number is 0.

A number is divided by zero.

Division by zero is undefined.

A number is multiplied by zero.

The product of any number and 0 is 0.

Example: Multiplying &

Dividing with Zero

Multiply or divide.

Zero is divided by a nonzero number.

The quotient of zero and any nonzero

number is 0.

b. 0 ÷ 0

undefined A number divided by 0 is undefined.

c. (–12.350)(0)

0The product of any number and 0 is

A number is divided by zero.

A number is multiplied by zero.

Your Turn:

Find the distance traveled at a rate of 3 mi/h for 1 hour.

To find distance, multiply rate by time.

The speed of a hot-air balloon is 3 mi/h. It

travels in a straight line for 1 hours before

landing. How many miles away from the liftoff

site will the balloon land?

4• 1

3Write and as improper fractions.

43 1 1

4and have the same sign, so

the quotient is positive.

The hot-air balloon lands 5 miles from the liftoff site.

Example: Continued

What if…? On another hot-air balloon trip, the

wind speed is 5.25 mi/h. The trip is planned for 1.5

hours. The balloon travels in a straight line parallel

to the ground. How many miles away from the

liftoff site will the balloon land?

5.25(1.5) Rate times time equals distance.

= 7.875 mi Distance traveled.

Your Turn:

Practice!

The Distributive Property

Section 1-7 Part 1

Vocabulary

• Distributive Property

Distributive Property

• To solve problems in mathematics, it is

often useful to rewrite expressions in

simpler form.

• The Distributive Property, illustrated by the

area model on the next slide, is another

property of real numbers that helps you to

simplify expressions.

You can use algebra tiles to model algebraic expressions.

1 1-tile

This 1-by-1 square tile has

an area of 1 square unit.

x-tile

This 1-by-x square tile has

an area of x square units.

Area = 3(x + 2)

Area = 3(x ) + 3(2)

Model the Distributive Property using Algebra Tiles

MODELING THE DISTRIBUTIVE PROPERTY

The Distributive Property is used with Addition to Simplify

Expressions.

The Distributive Property also works with subtraction because

subtraction is the same as adding the opposite.

THE DISTRIBUTIVE PROPERTY

a(b + c) = ab + ac

(b + c)a = ba + ca

2(x + 5) 2(x) + 2(5) 2x + 10

(x + 5)2 (x)2 + (5)2 2x + 10

(1 + 5x)2 (1)2 + (5x)2 2 + 10x

y(1 – y) y(1) – y(y) y – y 2

USING THE DISTRIBUTIVE PROPERTY

The product of a and (b + c):

Distributive

Property

a(b + c) = ab + ac and (b + c)a = ba + bc;

a(b - c) = ab - ac and (b - c)a = ba - bc;

Find the sum (add) or

difference (subtract) of the

distributed products.

(y – 5)(–2) = (y)(–2) + (–5)(–2)

= –2y + 10

–(7 – 3x) = (–1)(7) + (–1)(–3x)

= –7 + 3x

= –3 – 3x

(–3)(1 + x) = (–3)(1) + (–3)(x)

Simplify.

Distribute the –3.

Simplify.

–a = –1 • a

USING THE DISTRIBUTIVE PROPERTY

Remember that a factor must multiply each term of an expression.

Forgetting to distribute the negative sign when multiplying by a negative

factor is a common error.

5(x 3)

6(y 7)

3(m 8)

4(3 y)

4 3 4 y

10(x 7)

10 x 10 7

10x 70

4( k 2)

4 k 4 2

Your Turn: Simplify

Your turn:

1. 2(x + 5) = 5. (x - 4)x =

2. (15+6x) x = 6. y(2 - 6y) =

3. -3(x + 4) = 7. (y + 5)(-4) =

4. -(6 - 3x) = 8.

3 9x x

2 10x 24x x

25 2x x

3 12x 4 20y

3 6x 2

2 6x x

Practice!

Section 1-7 Part 2

Vocabulary

• Term

• Constant

• Coefficient

• Like Terms

The process of distributing the number on the

outside of the parentheses to each term on

the inside.

a(b + c) = ab + ac and (b + c) a = ba + ca

a(b - c) = ab - ac and (b - c) a = ba - ca

Example

5(x + 7)

5 ∙ x + 5 ∙ 7

5x + 35

Two ways to find the area of the rectangle.

As a whole As two parts

Geometric Model for Distributive

Property

Geometric Model for Distributive

Property

Two ways to find the area of the rectangle.

4 5 2 4 5 4 2

4 5 4 2

4 5 2 4 5 4 2

Find the area of the rectangle in terms

of x, y and z in two different ways.

Your Turn: Find the area of the rectangle in

terms of x, y and z in two

different ways.

x y z x y x z

x y x z

xy + xz

Write the product using the Distributive Property. Then simplify.

5(50 + 9)

5(50) + 5(9)

250 + 45

Rewrite 59 as 50 + 9.

Use the Distributive Property.

Multiply.

Example: Distributive

Property with Mental MathYou can use the distributive property and mental math to make

calculations easier.

9(50) - 9(2)

9(50 - 2)

450 - 18

Rewrite 48 as 50 - 2.

Multiply.

Subtract.

Write the product using the Distributive Property. Then

simplify.

Example: Distributive

Property with Mental Math

8(30 + 3)

8(30) + 8(3)

240 + 24

Multiply.

simplify.

Your Turn:

12(98)

Rewrite 98 as 100 – 2.

Multiply.

Subtract.

12(100 – 2)

1200 – 24

12(100) – 12(2)

simplify.

Your Turn:

7(30 + 4)

7(30) + 7(4)

210 + 28

Multiply.

simplify.

Your Turn:

Find the difference mentally.

Find the products mentally.

The mental math is easier if you

think of $11.95 as $12.00 – $.05.

Write 11.95 as a difference.

You are shopping for CDs.

You want to buy six CDs

for $11.95 each.

Use the distributive property

to calculate the total cost

mentally.

6(11.95) = 6(12 – 0.05)

Use the distributive property.= 6(12) – 6(0.05)

= 72 – 0.30

= 71.70

The total cost of 6 CDs at $11.95 each is $71.70.

MENTAL MATH CALCULATIONS

SOLUTION

Definition

• Term – any number that is added or

subtracted.

– In the algebraic expression x + y, x and y are

terms.

• Example:

– The expression x + y – 7 has 3 terms, x, y, and

7. x and y are variable terms; their values vary

as x and y vary. 7 is a constant term; 7 is always

Definition

• Coefficient – The numerical factor of a

• Example:

– The coefficient of 3x2 is 3.

Definition

• Like Terms – terms in which the variables

and the exponents of the variables are

identical.

– The coefficients of like terms may be different.

• Example:

– 3x2 and 6x2 are like terms.

– ab and 3ab are like terms.

– 2x and 2x3 are not like terms.

Definition

• Constant – anything that does not vary or change

in value (a number).

– In algebra, the numbers from arithmetic are constants.

– Constants are like terms.

The terms of an expression are the parts to be added

or subtracted. Like terms are terms that contain the

same variables raised to the same powers. Constants

are also like terms.

4x – 3x + 2

Like terms Constant

Example:

A coefficient is a number multiplied by a variable.

Like terms can have different coefficients. A variable

written without a coefficient has a coefficient of 1.

1x2 + 3x

Coefficients

Example:

Like terms can be combined. To combine like

terms, use the Distributive Property.

Notice that you can combine like terms by adding

or subtracting the coefficients. Keep the variables

and exponents the same.

ax – bx = (a – b)x

Example

7x – 4x = (7 – 4)x

Combining Like Terms

Simplify the expression by combining like terms.

72p – 25p

72p and 25p are like terms.

Subtract the coefficients.

Example: Combining Like

A variable without a coefficient has a

coefficient of 1.

Write 1 as .

Add the coefficients.

and are like terms.

0.5m + 2.5n

0.5m and 2.5n are not like terms.

Do not combine the terms.

Caution!Add or subtract only the coefficients.

6.8y² – y² ≠ 6.8

2 2 26.8 5.8y y y

Simplify by combining like terms.

3a. 16p + 84p

16p + 84p

16p + 84p are like terms.

Add the coefficients.

3b. –20t – 8.5t2

–20t – 8.5t2 20t and 8.5t2 are not like terms.

–20t – 8.5t2 Do not combine the terms.

3m2 + m3 3m2 and m3 are not like terms.

3c. 3m2 + m3

Do not combine the terms.3m2 + m3

Your Turn:

SIMPLIFYING BY COMBINING LIKE TERMS

Each of these terms is the product of a number and a variable.terms

+– 3y2x +– 3y2x

number

+– 3y2x

variable.

+– 3y2x

–1 is the coefficient of x.

3 is the coefficient of y2.

x is the variable.

y is the variable.

Each of these terms is the product of a number and a variable.

x2 x2y3 y3

Like terms have the same variable raised to the same power.

y2 – x2 + 3y3 – 5 + 3 – 3x2 + 4y3 + y

variable power.Like terms

The constant terms –5 and 3 are also like terms.

Combine like terms.

SIMPLIFYING BY COMBINING LIKE TERMS

4x2 + 2 – x2 =

(8 + 3)x Use the distributive property.

= 11x Add coefficients.

8x + 3x =

Group like terms.

Rewrite as addition expression.

Multiply.

Combine like terms

and simplify.

4x2 – x2 + 2

= 3x2 + 2

3 – 2(4 + x) = 3 + (–2)(4 + x)

= 3 + [(–2)(4) + (–2)(x)]

= 3 + (–8) + (–2x)

= –5 + (–2x)

= –5 – 2x

–12x – 5x + x + 3a Commutative Property

Combine like terms.–16x + 3a

–12x – 5x + 3a + x1.

Procedure Justification

Simplify −12x – 5x + 3a + x. Justify each step.

Your Turn:

Simplify 14x + 4(2 + x). Justify each step.

14x + 4(2) + 4(x) Distributive Property

Multiply.

Commutative Property of AdditionAssociative Property of AdditionCombine like terms.

14x + 8 + 4x

(14x + 4x) + 8

14x + 4x + 8

18x + 8

14x + 4(2 + x)1.

Statements Justification

Your Turn:

Practice!

An Introduction to Equations

Section 1-8

Vocabulary

• Equation

• Open sentence

• Solution of an equation

Definition

• Equation – A mathematical sentence that states one

expression is equal to a second expression.

• mathematical sentence that uses an equal sign (=).

• (value of left side) = (value of right side)

• An equation is true if the expressions on either side of the

equal sign are equal.

• An equation is false if the expressions on either side of the

equal sign are not equal.

• Examples:

• 4x + 3 = 10 is an equation, while 4x + 3 is an expression.

• 5 + 4 = 9 True Statement

• 5 + 3 = 9 False Statement

Equation or Expression

In Mathematics there is a difference between a phrase

and a sentence. Phrases translate into expressions;

sentences translate into equations or inequalities.

ExpressionsPhrases

Equations or InequalitiesSentences

Definition

• Open Sentence – an equation that contains

one or more variables.

– An open sentence is neither true nor false until

the variable is filled in with a value.

• Examples:

– Open sentence: 3x + 4 = 19.

– Not an open sentence: 3(5) + 4 = 19.

Example: Classifying

Equations

Is the equation true, false, or open? Explain.

A. 3y + 6 = 5y – 8

Open, because there is a variable.

B. 16 – 7 = 4 + 5

True, because both sides equal 9.

C. 32 ÷ 8 = 2 ∙ 3

False, because both sides are not equal, 4 ≠ 6.

Your Turn:

Is the equation true, false, or open? Explain.

A. 17 + 9 = 19 + 6

False, because both sides are not equal, 26 ≠ 25.

B. 4 ∙ 11 = 44

True, because both sides equal 44.

C. 3x – 1 = 17

Open, because there is a variable.

Definition

• Solution of an Equation – is a value of the

variable that makes the equation true.

– A solution set is the set of all solutions.

– Finding the solutions of an equation is called

solving the equation.

• Examples:

– x = 5 is a solution of the equation 3x + 4 = 19, because 3(5) + 4 = 19 is a true statement.

Example: Identifying

Solutions of an Equation

Is m = 2 a solution of the equation

6m – 16 = -4?

6m – 16 = -4

6(2) – 16 = -4

12 – 16 = -4

-4 = -4 True statement, m = 2 is a solution.

Your Turn:

Is x = 5 a solution of the equation

15 = 4x – 4?

No, 15 ≠ 16. False statement, x = 5 is not a

solution.

A PROBLEM SOLVING PLAN USING MODELS

Procedure for Writing an Equation

your verbal model.

VERBAL MODEL

your verbal model.

ALGEBRAIC

LABELS

Writing an Equation

You and three friends are having a dim sum lunch at a Chinese

restaurant that charges $2 per plate. You order lots of plates.

The waiter gives you a bill for $25.20, which includes tax of

$1.20. Write an equation for how many plates your group

ordered.

Understand the problem situation

before you begin. For example,

notice that tax is added after the total

cost of the dim sum plates is figured.

SOLUTION

LABELS

VERBAL MODEL

Writing an Equation

Cost perplate •

Number of plates = Bill Tax–

Cost per plate = 2

Number of plates = p

Amount of bill = 25.20

Tax = 1.20

(dollars)

(plates)

25.20 1.20–2 =p

2p = 24.00

The equation is 2p = 24.

ALGEBRAIC

Your Turn:

JET PILOT A jet pilot is flying from Los Angeles, CA to Chicago, IL at a

speed of 500 miles per hour. When the plane is 600 miles from Chicago,

an air traffic controller tells the pilot that it will be 2 hours before the

plane can get clearance to land. The pilot knows the speed of the jet must

be greater then 322 miles per hour or the jet could stall.

Write an equation to find at what

speed would the jet have to fly to

arrive in Chicago in 2 hours?

LABELS

VERBAL MODEL

Solution

Speed ofjet • Time =

Distance totravel

Speed of jet = x

Time = 2

Distance to travel = 600

(miles per hour)

(miles)

(hours)

2x = 600

ALGEBRAIC

At what speed would the jet have to fly to arrive in Chicago in 2 hours?

SOLUTION You can use the formula (rate)(time) = (distance) to write a verbal model.

Example: Use Mental Math to

Find Solutions

• What is the solution to the equation? Use

mental math.

• 12 – y = 3

– Think: What number subtracted from 12 equals 3.

– Solution: 9.

– Check: 12 – (9) = 3, 3 = 3 is a true statement,

therefore 9 is a solution.

Your Turn:

What is the solution to the equation? Use mental

A. x + 7 = 13

B. x/6 = 12

Practice!

Patterns, Equations, and Graphs

Section 1-9

Vocabulary

• Solution of an equation

• Inductive reasoning

The coordinate plane is formed by

the intersection of two

perpendicular number lines called

axes. The point of intersection,

called the origin, is at 0 on each

number line. The horizontal

number line is called the x-axis,

and the vertical number line is

called the y-axis.

Review: Graphing in the

Coordinate Plane

Points on the coordinate plane are described using ordered

pairs. An ordered pair consists of an x-coordinate and a

y-coordinate and is written (x, y). Points are often named

by a capital letter.

The x-coordinate tells how many units to move left or right from

the origin. The y-coordinate tells how many units to move up or

Reading Math

Graphing in the Coordinate

Graph each point.

A. T(–4, 4)

Start at the origin.

Move 4 units left and 4 units up.

B. U(0, –5)

Move 5 units down.

•T(–4, 4)

• U(0, –5)

C. V (–2, –3)

Move 2 units left and 3 units down.

•V(–2, −3)

Example: Graphing in the

Coordinate Plane

Graph each point.

A. R(2, –3)

B. S(0, 2)

Move 2 units right and 3 units down.

Move 2 units up.

C. T(–2, 6)

Move 2 units left and6 units up.

•R(2, –3)

S(0,2)

T(–2,6)

Your Turn:

The axes divide the

coordinate plane into

four quadrants. Points

that lie on an axis are

not in any quadrant.

Graphing in the Coordinate

Name the quadrant in which each point lies.

Quadrant ll

no quadrant (y-axis)

Quadrant l

D. HQuadrant lll

Example: Locating Points

Name the quadrant in which each point lies.

no quadrant (y-axis)

Quadrant l

Quadrant lll

D. WQuadrant ll

Your Turn:

The Rectangular Coordinate System

SUMMARY: The Rectangular Coordinate System

• Composed of two real number lines – one horizontal (the x-axis) and

one vertical (the y-axis). The x- and y-axes intersect at the origin.

• Also called the Cartesian plane or xy-plane.

• Points in the rectangular coordinate system are denoted (x, y) and are

called the coordinates of the point. We call the x the x-coordinate and

the y the y-coordinate.

• If both x and y are positive, the point lies in quadrant I; if x is

negative, but y is positive, the point lies in quadrant II; if x is

negative and y is negative, the point lies in quadrant III; if x is

positive and y is negative, the point lies in quadrant IV.

• Points on the x-axis have a y-coordinate of 0; points on the y-axis

have an x-coordinate of 0.

Equation in Two Variables

An equation in two variables, x and y, is a statement in which the

algebraic expressions involving x and y are equal. The expressions

are called sides of the equation.

Any values of the variables that make the equation a true statement

are said to be solutions of the equation.

x + y = 15 x2 – 2y2 = 4 y = 1 + 4x

x + y = 15

The ordered pair (5, 10) is a solution of the equation.

5 + 10 = 15

15 = 15

Solutions to Equations

2x + y = 5

2(2) + (1) = 5

4 + 1 = 5

Example:

Determine if the following ordered pairs satisfy the

equation 2x + y = 5.

a.) (2, 1) b.) (3, – 4)

(2, 1) is a solution.

2x + y = 5

2(3) + (– 4) = 5

6 + (– 4) = 5

(3, – 4) is not a solution.

An equation that contains two variables can be used as

a rule to generate ordered pairs. When you substitute a

value for x, you generate a value for y. The value

substituted for x is called the input, and the value

generated for y is called the output.

y = 10x + 5

Output Input

Equation in Two Variables

Table of Values

Use the equation y = 6x + 5 to complete the table and list

the ordered pairs that are solutions to the equation.

x y (x, y)

y = 6x + 5

x = – 2

y = 6(– 2) + 5

y = – 12 + 5

y = – 7

(– 2, – 7)

y = 6x + 5

y = 6(0) + 5

y = 0 + 5

y = 55

(0, 5)

y = 6x + 5

y = 6(2) + 5

y = 12 + 5

y = 1717

(2, 17)

An engraver charges a setup fee of $10 plus $2 for every

word engraved. Write a rule for the engraver’s fee. Write

ordered pairs for the engraver’s fee when there are 5, 10,

15, and 20 words engraved.

Let y represent the engraver’s fee and x represent the

number of words engraved.

Engraver’s fee is $10 plus $2 for each word

y = 10 + 2 · x

y = 10 + 2x

The engraver’s fee is determined by the number

of words in the engraving. So the number of

words is the input and the engraver’s fee is the

output.

Writing Math

Number ofWords

EngravedRule Charges

Ordered Pair

x (input) y = 10 + 2x y (output) (x, y)

y = 10 + 2(5)5 20 (5, 20)

y = 10 + 2(10)10 30 (10, 30)

y = 10 + 2(15)15 40 (15, 40)

y = 10 + 2(20)20 50 (20, 50)

Example: Solution

What if…? The caricature artist increased his fees. He now

charges a $10 set up fee plus $20 for each person in the

picture. Write a rule for the artist’s new fee. Find the

artist’s fee when there are 1, 2, 3 and 4 people in the picture.

y = 10 + 20x

Let y represent the artist’s fee and x represent the number of

people in the picture.

Artist’s fee is $10 plus $20 for each person

y = 10 + 20 · x

Your Turn:

Number of People in Picture

Rule ChargesOrdered

x (input) y = 10 + 20x y (output) (x, y)

y = 10 + 20(1)1 30 (1, 30)

y = 10 + 20(2)2 50 (2, 50)

y = 10 + 20(3)3 70 (3, 70)

y = 10 + 20(4)4 90 (4, 90)

Solution:

When you graph ordered pairs generated by

a function, they may create a pattern.

Graphing Ordered Pairs

Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern.

y = 2x + 1; x = –2, –1, 0, 1, 2

2(–2) + 1 = –3 (–2, –3)

(–1, –1)

(0, 1)

(1, 3)

(2, 5)

2(–1) + 1 = –1

2(0) + 1 = 1

2(1) + 1 = 3

2(2) + 1 = 5

Input OutputOrdered

x y (x, y)

The points form a line.

Example: Graphing Ordered Pairs

–2 – 4 = –6 (–4, –6)

(–2, –5)

(0, –4)

(2, –3)

(4, –2)

–1 – 4 = –5

0 – 4 = –4

1 – 4 = –3

2 – 4 = –2

Input OutputOrdered

x y (x, y)

The points form a line.

y = x – 4; x = –4, –2, 0, 2, 41

Your Turn: Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern.

Definition

• Inductive Reasoning – is the process of

reaching a conclusion based on an observed

pattern.

– Can be used to predict values based on a

pattern.

Inductive Reasoning

• Moves from specific observations to broader

generalizations or predictions from a pattern.

• Steps:

1. Observing data.

2. Detect and recognizing patterns.

3. Make generalizations or predictions from those patterns.

Observation

Pattern

Predict

Make a prediction about the next number based on the pattern.

2, 4, 12, 48, 240

Answer: 1440

Find a pattern:

2 4 12 48 240

The numbers are multiplied by 2, 3, 4, and 5.

Prediction: The next number will be multiplied by 6. So, it will be (6)(240) or 1440.

×3 ×4 ×5

Example: Inductive Reasoning

Make a prediction about the next number based on the pattern.

Answer: The next number will be

Your Turn:

1 1 1 1, ,

4 9 16, 251,

1 1 or

Practice!

Variables and Expressions -...

Documents