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The real number, in mathematics, is a quantity that can be expressed as an infinite decimal expansion. The real numbers include the positive and negative integers and fractions (or rational numbers) and also the irrational numbers.

1.
Intermediate Algebra – 1.3

• Operations with

Real Numbers

• Operations with

Real Numbers

2.
Three people

were at work on a

construction site. All

were doing the same

job, but when each was

asked what the job was,

were at work on a

construction site. All

were doing the same

job, but when each was

asked what the job was,

3.
“Breaking rocks,” the first

replied. “Earning my living,” the

second said.”Helping to build a

cathedral,” said the third.” – Peter

Schultz, German businessman

replied. “Earning my living,” the

second said.”Helping to build a

cathedral,” said the third.” – Peter

Schultz, German businessman

4.
Procedure - Addition

• Adding numbers with the same sign

• To add two numbers that have

the same sign, add their

absolute values and keep the

same sign

• Adding numbers with the same sign

• To add two numbers that have

the same sign, add their

absolute values and keep the

same sign

5.
Procedure - Addition

• Adding numbers with different

signs

• To add two numbers that have

different signs, subtract their

absolute values and keep the sign

of the number with the greater

absolute value.

• Adding numbers with different

signs

• To add two numbers that have

different signs, subtract their

absolute values and keep the sign

of the number with the greater

absolute value.

6.
Procedure - Subtraction

• For any real number a

• a – b = a + (-b)

• For any real number a

• a – b = a + (-b)

7.
Distance on number line

• The distance between two

points a and b is

• d = |a – b| = |b – a|

• The distance between two

points a and b is

• d = |a – b| = |b – a|

8.
Procedure - Multiplying

• When multiplying two real

numbers that have different

signs, the product is negative

• When multiplying two real

numbers that have different

signs, the product is negative

9.
Procedure - Multiplying

• When multiplying two numbers that have

the same sign, the product is positive

• When multiplying two numbers that have

the same sign, the product is positive

10.
Procedure - multiplying

• The product of an even number

of negative factors is positive,

• The product of an odd number

of negative factors is negative.

• The product of an even number

of negative factors is positive,

• The product of an odd number

of negative factors is negative.

11.
Division

• Division by Zero is undefined.

• 4/0 is undefined

• 0/4 = 0

• Division by Zero is undefined.

• 4/0 is undefined

• 0/4 = 0

12.
a

Procedure - Division

a a a

b b b

Procedure - Division

a a a

b b b

13.
Definition Square Root

• For all real numbers a and b, if

2

b a then b is a

square root of a

• For all real numbers a and b, if

2

b a then b is a

square root of a

14.
Def: radicand

• The number or expression

under the radical symbol

2 x 3

• The number or expression

under the radical symbol

2 x 3

15.
Def: Index of radical

• The index is n

n

a

3

x

b

• The index is n

n

a

3

x

b

16.
Calculator Keys

• [+], [*], [/], [-], [^]

• [ENTER] [2ND][ENTRY]

• [2ND] [QUIT] [x,t,n]

• [MODE]

• [MATH][NUM][1:abs( ]

• [+], [*], [/], [-], [^]

• [ENTER] [2ND][ENTRY]

• [2ND] [QUIT] [x,t,n]

• [MODE]

• [MATH][NUM][1:abs( ]

17.
Norman Vincent Peale:

• “What seems impossible one

minute becomes, …, possible

the next.

• “What seems impossible one

minute becomes, …, possible

the next.

18.
Section 1.4

• Intermediate Algebra

• Properties of Real numbers

(9)

• Intermediate Algebra

• Properties of Real numbers

(9)

19.
Commutative for Addition

• a+b=b+a

• 2+3=3+2

• a+b=b+a

• 2+3=3+2

20.
Commutative for

Multiplication

• ab = ba

Multiplication

• ab = ba

21.
Associative for Addition

• a + (b + c) = (a + b) + c

• 2 + (3 + 4) = (2 + 3) + 4

• a + (b + c) = (a + b) + c

• 2 + (3 + 4) = (2 + 3) + 4

22.
Associative for Multiplication

• (ab)c = a(bc)

• (2 x 3) x 4 = 2 x (3 x 4)

• (ab)c = a(bc)

• (2 x 3) x 4 = 2 x (3 x 4)

23.
Distributive

multiplication over addition

• a(b + c) = ab + ac

• 2(3 + 4) = 2 x 3 + 2 x 4

• X(Y + Z) = XY +XZ

multiplication over addition

• a(b + c) = ab + ac

• 2(3 + 4) = 2 x 3 + 2 x 4

• X(Y + Z) = XY +XZ

24.
Additive Identity

25.
Multiplicative Identity

26.
Additive Inverse

• a(1/a) = 1 where a not equal to 0

• 3(1/3) = 1

• a(1/a) = 1 where a not equal to 0

• 3(1/3) = 1

27.
George Simmel - Sociologist

• “He is educated who

knows how to find out

what he doesn’t know.”

• “He is educated who

knows how to find out

what he doesn’t know.”

28.
Section 1.4

Intermediate Algebra

• Apply order of operations

• Please Excuse My Dear Aunt

Sally.

• P – E – M – D – A- S

Intermediate Algebra

• Apply order of operations

• Please Excuse My Dear Aunt

Sally.

• P – E – M – D – A- S

29.
The order of operations

• Perform within grouping symbols – work

innermost group first and then outward.

• Evaluate exponents and roots.

• Perform multiplication and division left to

right.

• Perform addition and subtraction left to

right.

• Perform within grouping symbols – work

innermost group first and then outward.

• Evaluate exponents and roots.

• Perform multiplication and division left to

right.

• Perform addition and subtraction left to

right.

30.
Grouping Symbols

• Parentheses

• Brackets

• Braces

• Radical symbols

• Fraction symbols – fraction bar

• Absolute value

• Parentheses

• Brackets

• Braces

• Radical symbols

• Fraction symbols – fraction bar

• Absolute value

31.
Algebraic Expression

• Any combination of numbers, variables,

grouping symbols, and operation symbols.

• To evaluate an algebraic expression, replace

each variable with a specific value and then

perform all indicated operations.

• Any combination of numbers, variables,

grouping symbols, and operation symbols.

• To evaluate an algebraic expression, replace

each variable with a specific value and then

perform all indicated operations.

32.
Evaluate Expression by

Calculator

• Plug in

• Use store feature

• Use Alpha key for formulas

• Table

• Program - evaluate

Calculator

• Plug in

• Use store feature

• Use Alpha key for formulas

• Table

• Program - evaluate

33.
The Pythagorean Theorem

• In a right triangle, the sum of the square of

the legs is equal to the square of the

hypotenuse.

2 2 2

a b c

• In a right triangle, the sum of the square of

the legs is equal to the square of the

hypotenuse.

2 2 2

a b c

34.
Equation

• A statement that two expression

have the same value

• A statement that two expression

have the same value

35.
Intermediate Algebra – 1.5

• Walt Whitman – American Poet

• “Seeing, hearing, and

feeling are miracles,

and each part and tag

of me is a miracle.”

• Walt Whitman – American Poet

• “Seeing, hearing, and

feeling are miracles,

and each part and tag

of me is a miracle.”

36.
1.5 – Simplifying Expressions

• Term – An expression that is separated by

addition

• Numerical coefficient – the numerical

factor in a term

• Like Terms – Variable terms that have the

same variable(s) raised to the same

exponential value

• Term – An expression that is separated by

addition

• Numerical coefficient – the numerical

factor in a term

• Like Terms – Variable terms that have the

same variable(s) raised to the same

exponential value

37.
Combining Like Terms

• To combine like terms, add or

subtract the coefficients and

keep the variables and their

exponents the same.

• To combine like terms, add or

subtract the coefficients and

keep the variables and their

exponents the same.

38.
example

7 3 4 x 2 7 3 4 x 2

11 3x

7 3 4 x 2 7 3 4 x 2

11 3x

39.
H. Jackson Brown Jr. Author

• “Let your

performance do the

thinking.”

• “Let your

performance do the

thinking.”

40.
Integer Exponents

• For any real number b and any natural

number n, the nth power of b o if found by

multiplying b as a factor n times.

n

b b b b b

N times

• For any real number b and any natural

number n, the nth power of b o if found by

multiplying b as a factor n times.

n

b b b b b

N times

41.
Exponential Expression – an

expression that involves

exponents

• Base – the number being multiplied

• Exponent – the number of factors of the

base.

expression that involves

exponents

• Base – the number being multiplied

• Exponent – the number of factors of the

base.

42.
Calculator Key

^

• Exponent Key

^

• Exponent Key

43.
Sydney Harris:

• “When I hear somebody

sigh,’Life is hard”, I am

always tempted to ask,

“Compared to what?”

• “When I hear somebody

sigh,’Life is hard”, I am

always tempted to ask,

“Compared to what?”

44.
Intermediate Algebra 1.5

• Introduction

• To

• Linear Equations

• Introduction

• To

• Linear Equations

45.
Def: Equation

• An equation is a

statement that two

algebraic expressions

have the same value.

• An equation is a

statement that two

algebraic expressions

have the same value.

46.
Def: Solution

• Solution: A replacement for the

variable that makes the equation

true.

• Root of the equation

• Satisfies the Equation

• Zero of the equation

• Solution: A replacement for the

variable that makes the equation

true.

• Root of the equation

• Satisfies the Equation

• Zero of the equation

47.
Def: Solution Set

• A set containing all the

solutions for the given

equation.

• Could have one, two, or many elements.

• Could be the empty set

• Could be all Real numbers

• A set containing all the

solutions for the given

equation.

• Could have one, two, or many elements.

• Could be the empty set

• Could be all Real numbers

48.
Def: Linear Equation in One

Variable

• An equation that can be written in

the form ax + b = c where a,b,c are

real numbers and a is not equal to

zero

Variable

• An equation that can be written in

the form ax + b = c where a,b,c are

real numbers and a is not equal to

zero

49.
Linear function

• A function of form

• f(x) = ax + b where a and b

are real numbers and a is not

equal to zero.

• A function of form

• f(x) = ax + b where a and b

are real numbers and a is not

equal to zero.

50.
Def: Identity

• An equation is an identity if every

permissible replacement for the variable is a

solution.

• The graphs of left and right sides coincide.

• The solution set is R

R

• An equation is an identity if every

permissible replacement for the variable is a

solution.

• The graphs of left and right sides coincide.

• The solution set is R

R

51.
Def: Inconsistent equation

• An equation with no solution is an

inconsistent equation.

• Also called a contradiction.

• The graphs of left and right sides never

intersect.

• The solution set is the empty set.

• An equation with no solution is an

inconsistent equation.

• Also called a contradiction.

• The graphs of left and right sides never

intersect.

• The solution set is the empty set.

52.
Def: Equivalent Equations

• Equivalent equations are equations that

have exactly the same solutions sets.

• Examples:

• 5 – 3x = 17

• -3x= 12

• x = -4

• Equivalent equations are equations that

have exactly the same solutions sets.

• Examples:

• 5 – 3x = 17

• -3x= 12

• x = -4

53.
Addition Property of Equality

• If a = b, then a + c = b + c

• For all real numbers a,b, and c.

• Equals plus equals are equal.

• If a = b, then a + c = b + c

• For all real numbers a,b, and c.

• Equals plus equals are equal.

54.
Multiplication Property of

Equality

• If a = b, then ac = bc is true

• For all real numbers a,b, and c

where c is not equal to 0.

• Equals times equals are equal.

Equality

• If a = b, then ac = bc is true

• For all real numbers a,b, and c

where c is not equal to 0.

• Equals times equals are equal.

55.
Solving Linear Equations

• Simplify both sides of the equation as

needed.

– Distribute to Clear parentheses

– Clear fractions by multiplying by the LCD

– Clear decimals by multiplying by a power of 10

determined by the decimal number with the

most places

– Combine like terms

• Simplify both sides of the equation as

needed.

– Distribute to Clear parentheses

– Clear fractions by multiplying by the LCD

– Clear decimals by multiplying by a power of 10

determined by the decimal number with the

most places

– Combine like terms

56.
Solving Linear Equations Cont:

• Use the addition property so that all variable

terms are on one side of the equation and all

constants are on the other side.

• Combine like terms.

• Use the multiplication property to isolate the

variable

• Verify the solution

• Use the addition property so that all variable

terms are on one side of the equation and all

constants are on the other side.

• Combine like terms.

• Use the multiplication property to isolate the

variable

• Verify the solution

57.
Ralph Waldo Emerson – American essayist,

poet, and philosopher (1803-1882)

• “The world looks like a

multiplication table or a

mathematical equation,

which, turn it how you

will, balances itself.”

poet, and philosopher (1803-1882)

• “The world looks like a

multiplication table or a

mathematical equation,

which, turn it how you

will, balances itself.”

58.
Problem Solving 1.6

• 1. Understand the Problem

• 2. Devise a Plan

– Use Definition statements

• 3. Carry out a Plan

• 4. Look Back

– Check units

• 1. Understand the Problem

• 2. Devise a Plan

– Use Definition statements

• 3. Carry out a Plan

• 4. Look Back

– Check units

59.
Types of Problems

• Number Problems

• Angles of a Triangle

• Rectangles

• Things of Value

• Number Problems

• Angles of a Triangle

• Rectangles

• Things of Value

60.
Les Brown

• “If you view all the things

that happen to you, both

good and bad, as

opportunities, then you

operate out of a higher level

of consciousness.”

• “If you view all the things

that happen to you, both

good and bad, as

opportunities, then you

operate out of a higher level

of consciousness.”

61.
Types of Problems Cont.

• Percentages

• Interest

• Mixture

• Liquid Solutions

• Distance, Rate, and Time

• Percentages

• Interest

• Mixture

• Liquid Solutions

• Distance, Rate, and Time

62.
Albert Einstein

• “In the middle of

difficulty lies

opportunity.”

• “In the middle of

difficulty lies

opportunity.”

63.
Ralph Waldo Emerson – American essayist,

poet, and philosopher (1803-1882)

• “The world looks like a

multiplication table or a

mathematical equation,

which, turn it how you

will, balances itself.”

poet, and philosopher (1803-1882)

• “The world looks like a

multiplication table or a

mathematical equation,

which, turn it how you

will, balances itself.”

64.
Section 1.8

• Solve Formulas

• Isolate a particular variable in a formula

• Treat all other variables like constants

• Isolate the desired variable using the outline

for solving equations.

• Solve Formulas

• Isolate a particular variable in a formula

• Treat all other variables like constants

• Isolate the desired variable using the outline

for solving equations.

65.
Know Formulas

• Area of a rectangle

A = LW

• Perimeter of a rectangle

• P = 2L + 2W

• Area of a rectangle

A = LW

• Perimeter of a rectangle

• P = 2L + 2W

66.
Formulas continued

• Area of a square

2

A s

• Perimeter of a square

P 4 s

• Area of a square

2

A s

• Perimeter of a square

P 4 s

67.
Formulas continued

• Area of Parallelogram

•A = bh

• Area of Parallelogram

•A = bh

68.
Formulas continued

• Trapezoid

1

A b1 b2 h

2

• Trapezoid

1

A b1 b2 h

2

69.
Formulas continued

• Area of Circle

2

A r

• Circumference of Circle

C 2 r C d

• Area of Circle

2

A r

• Circumference of Circle

C 2 r C d

70.
Formulas continued:

• Area of Triangle

1

A bh

2

• Area of Triangle

1

A bh

2

71.
Formulas continued

• Sum of measures of a triangle

o

m1 m2 m3 180

• Sum of measures of a triangle

o

m1 m2 m3 180

72.
Formulas continued

• Perimeter of a Triangle

P s1 s2 s3

• Perimeter of a Triangle

P s1 s2 s3

73.
Formulas continued

• Pythagorean Theorem

2 2 2

a b c

• Pythagorean Theorem

2 2 2

a b c

74.
Formulas continued:

• Volume of a Cube – all sides are equal

3

V s

• Volume of a Cube – all sides are equal

3

V s

75.
Formulas continued

• Rectangular solid

V lwh

• Area of Base x height

• Rectangular solid

V lwh

• Area of Base x height

76.
Formulas continued

• Volume Right Circular Cylinder

2

V r h

• Volume Right Circular Cylinder

2

V r h

77.
Formulas continued:

• Surface are of right circular cylinder

2

S 2 rh 2 r

• Surface are of right circular cylinder

2

S 2 rh 2 r

78.
Formulas continued:

• Volume of Right Circular Cone

• V=(1/3) area base x height

1 2

V r h

3

• Volume of Right Circular Cone

• V=(1/3) area base x height

1 2

V r h

3

79.
Formulas continued:

• Volume Sphere

4 3

V r

3

• Volume Sphere

4 3

V r

3

80.
Formulas continued:

• General Formula surface area right solid

• SA = 2(area base) + Lateral surface area

• SA=2(area base) + LSA

• Lateral Surface Area = LSA =

• (perimeter)*(height)

• General Formula surface area right solid

• SA = 2(area base) + Lateral surface area

• SA=2(area base) + LSA

• Lateral Surface Area = LSA =

• (perimeter)*(height)

81.
Formulas continued:

• Distance, rate and Time

d = rt

Interest

I = PRT

• Distance, rate and Time

d = rt

Interest

I = PRT

82.
Useful Calculator Programs

• CIRCLE

• CIRCUM

• CONE

• CYLINDER

• PRISM

• PYRAMID

• TRAPEZOI

• APPS-AreaForm

• CIRCLE

• CIRCUM

• CONE

• CYLINDER

• PRISM

• PYRAMID

• TRAPEZOI

• APPS-AreaForm

83.
Robert Schuller – religious leader

• “Spectacular achievement

is always preceded by

spectacular preparation.”

• “Spectacular achievement

is always preceded by

spectacular preparation.”