hma13_Chapter00

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INTRODUCTORY MATHEMATICAL ANALYSISFor Business, Economics, and the Life and Social Sciences

2011 Pearson Education, Inc.

Chapter 0Review of Algebra


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To be familiar with sets, real numbers, real-number line. To relate properties of real numbers in terms of their

operations.

To review the procedure of rationalizing thedenominator.

To perform operations of algebraic expressions.

To state basic rules for factoring.

To rationalize the denominator of a fraction.

To solve linear equations.

To solve quadratic equations.

Chapter 0: Review of Algebra

Chapter Objectives


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Sets of Real Numbers

Some Properties of Real Numbers

Exponents and Radicals

Operations with Algebraic Expressions

Factoring

Fractions

Equations, in Particular Linear Equations

Quadratic Equations


Chapter Outline

0.1)

0.2)

0.3)

0.4)

0.5)

0.6)

0.7)

0.8)


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A set is a collection of objects.

An object in a set is called an element of thatset.

Different type of integers:

The real-number line is shown as


0.1 Sets of Real Numbers

...,3,2,1integerspositiveofSet

1,2,3..., integersnegativeofSet


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Important properties of real numbers

1. The Transitive Property of Equality

2. The Closure Properties of Addition and

Multiplication

3. The Commutative Properties of Addition

and Multiplication


0.2 Some Properties of Real Numbers

.then,andIf cacbba

.and

numbersrealuniquearetherenumbers,realallFor

abba

baababba

and


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4. The Commutative Properties of Addition

and Multiplication

5. The Identity Properties

6. The Inverse Properties

7. The Distributive Properties



cabbcacbacba and

aaaa 1and0

0 aa 11 aa

cabaacbacabcba and


7/532011 Pearson Education, Inc.



Example 1 Applying Properties of Real Numbers

Example 3 Applying Properties of Real

Numbers

354543b.

2323a.

xwzywzyxSolution:

a. Show that

Solution:

.0for

c

c

ba

c

ab

c

ba

c

ba

c

ab

c

ab 11





Example 3 Applying Properties of Real Numbers

b. Show thatSolution:

.0forc

b

cc

a

c

ba

c

bc

ac

bac

ba 111

c

b

c

a

c

bac

b

c

a

cb

ca

11

Ch t 0 R i f Al b



Properties:


0.3 Exponents and Radicals

14.

13.

0for11

2.

1.

0

x

xx

xxxxxx

x

xxxxx

n

n

factorsn

n

n

factorsn

n

nx exponentbase

Ch t 0 R i f Al b





Example 1 Exponents

xx

1

000

5

5-

5

5-

4

e.

1)5(,1,12d.

243331c.

243

1

3

13b.

161

21

21

21

21

21a.

Ch t 0 R i f Al b





The symbol is called a radical.

n is the index, x is the radicand, and is theradical sign.

nx

Ch t 0 R i f Al b





Example 3 Rationalizing Denominators

Solution:

Example 5 Exponents

x

x

x

x

xxx 3

32

3

32

3

2

3

2

3

2

b.

5

52

5

52

55

52

5

2

5

2a.

6 55

6 566 5

1

61

65

61

21

21

21

21

21

a. Eliminate negative exponents in andsimplify.

Solution:

11 yx

xy

xy

yx

yx

1111






Example 5 Exponents

b. Simplify by using the distributive law.

Solution:

c. Eliminate negative exponents in

Solution:

12/12/12/3 xxxx

2/12/3 xx

.77 22 xx

2222

22

49

17

7

1777

xxx

xxx






Example 5 Exponents

d. Eliminate negative exponents in

Solution:

e.Apply the distributive law to

Solution:

.211 yx

222

2

22211

11

xyyx

xyxy

xy

xy

yxyx

.2 56

21

52

xyx

58

21

52

56

21

52

22 xyxxyx






Example 7 Radicals

a.Simplify

Solution:

b. Simplify

Solution:

3233 33 323 46 )( yyxyyxyx

7

14

77

72

7

2

.3 46yx

.7

2






Example 7 Radicals

c. Simplify

Solution:

d. If x is any real number, simplify

Solution:

Thus, and

210105

2152510521550250

.21550250

.2

x

0if

0if2

xx

xxx

222 .332




If symbols are combined by any or all of theoperations, the resulting expression is called

an algebraic expression.

A polynomial in xis an algebraic expression

of the form:

where n = non-negative integer

cn= constants


0.4 Operations with Algebraic Expressions

01

1

1 cxcxcxc n

n

n

n






Example 1 Algebraic Expressions

a. is an algebraic expression in the

variablex.

b. is an algebraic expression in the

variable y.

c. is an algebraic expression in the

variablesx and y.

3

3

10

253

x

xx

2

3

y

xyyx

2

7

5310

y

y






Example 3 Subtracting Algebraic Expressions

Simplify

Solution:

.364123 22 xyxxyx

48

316243

)364()123(

364123

2

2

22

22

xyx

xyx

xyxxyx

xyxxyx




A list of products may be obtained from the

distributive property:








Example 5 Special Products

a. By Rule 2,

b. By Rule 3,

103

5252

52

2

2

xx

xx

xx

204721

45754373

4753

2

2

zz

zz

zz






Example 5 Special Products

c. By Rule 5,

d. By Rule 6,

e. By Rule 7,

168

442

4

2

22

2

xx

xx

x

8

31

3131

2

22

2

22

y

y

yy

8365427

23233233

23

23

3223

3

xxx

xxx

x



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Example 7 Dividing a Multinomial by a Monomial

zzz

z

zzz

xx

xx

3

2

342

2

6384b.

33

a.

223

23



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If two or more expressions are multipliedtogether, the expressions are called thefactorsof the product.

p g

0.5 Factoring



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p g

0.5 Factoring

Example 1 Common Factors

a.Factor completely.

Solution:

b.Factor completely.

Solution:

xkxk 322 93

kxxkxkxk 3393 2322

224432325 268 zxybayzbayxa

24232232224432325

342

268

xyzbazbyxaya

zxybayzbayxa



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p g

0.5 Factoring

Example 3 Factoring

zzzz

xxx

yyyyyy

xxxx

xxx

1e.

396d.

23231836c.

2313299b.

4168a.

4/14/54/1

22

23

2

42

2222

333366

23

2222

3/13/13/13/2

24

j.

2428i.

bbaah.

4145g.

1111f.

yxyxyxyxyxyx

yxyxyx

xxxx

bayxyxyxyx

xxxx

xxxx



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Simplifying Fractions

Allows us to multiply/divide the numerator anddenominator by the same nonzero quantity.

Multiplication and Division of Fractions

The rule for multiplying and dividing is

p g

0.6 Fractions

bd

ac

d

c

b

a

bc

ad

d

c

b

a



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Rationalizing the Denominator

For a denominator with square roots, it may berationalized by multiplying an expression thatmakes the denominator a difference of two

squares.

Addition and Subtraction of Fractions

If we add two fractions having the samedenominator, we get a fraction whosedenominator is the common denominator.

g

0.6 Fractions



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0.6 Fractions

Example 1 Simplifying Fractions

a.Simplify

Solution:

b.Simplify

Solution:

.

127

62

2

xx

xx

4

2

43

23

127

62

2

x

x

xx

xx

xx

xx

.448

8622

2

xx

xx

22

4

214

412

448

8622

2

x

x

xx

xx

xx

xx



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0.6 Fractions

Example 3 Dividing Fractions

412

82

1

1

4

1

82

1

4

c.

32

5

2

1

3

5

2

3

5

b.

325

35

253

2a.

222

2

xxxx

x

x

x

x

xx

x

xxx

x

xx

x

x

x

x

xxxx

xx

xx

xx

xx



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0.6 Fractions

Example 5 Adding and Subtracting Fractions

233

2

235

2

23

2

5

a.

2

2

2

ppp

p

pp

p

p

p

p

3

4

32

2

31

41

65

2

32

45

b. 2

2

2

2

x

xx

xx

xx

xx

xx

xx

xx

xx

17

7

7

425

149

84

7

2

7

5c.

22

2

22

x

x

x

xxx

xx

x

x

x

x

xx



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0.6 Fractions

Example 7 Subtracting Fractions

332615

332

6512102

332

32322

92

2

96

2

2

2

2

22

222

xx

xx

xx

xxxx

xx

xxxx

x

x

xx

x



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Equations

An equation is a statement that twoexpressions are equal.

The two expressions that make up an equationare called its sides.

They are separated by the equality sign, =.

0.7 Equations, in Particular Linear Equations

Chapter 0: Review of Algebra0 7 E ti i P ti l Li E ti


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Example 1 Examples of Equations

zw

y

y

xx

x

7d.

64

c.

023b.

32a.

2

A variable (e.g.x, y) is a symbol that can bereplaced by any one of a set of different

numbers.



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Equivalent Equations

Two equations are said to be equivalentif theyhave exactly the same solutions.

There are three operations that guaranteeequivalence:

1. Adding/subtracting the same polynomial

to/from both sides of an equation.

2. Multiplying/dividing both sides of an

equation by the same nonzero constant.3. Replacing either side of an equation by an

equal expression.




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Operations That May Not Produce Equivalent

Equations

4. Multiplying both sides of an equation by an

expression involving the variable.

5. Dividing both sides of an equation by an

expression involving the variable.

6. Raising both sides of an equation to equal

powers.



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Linear Equations

A linear equation

in the variablex can bewritten in the form

where a and b are constants and . A linear equation is also called a first-degree

equation or an equation of degree one.

0bax

0

a

Chapter 0: Review of Algebra0 7 Equations in Particular Linear Equations


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Example 3 Solving a Linear Equation

SolveSolution:

.365 xx

3

26

22

62

60662

062

33365

365

x

x

x

x

x

xxxx

xx



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Example 5 Solving a Linear Equations

SolveSolution:

.64

89

2

37

xx

2

105

24145

2489372

644

89

2

374

x

x

x

xx

xx


0 7 E ti i P ti l Li E ti


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Literal Equations

Equations where constants are not specified,but are represented as a, b, c, d, etc. are

called literal equations.

The letters are called literal constants.



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Example 7 Solving a Literal Equation

Solve for x.Solution:

ac

ax

aacxaaxxxcxax

axxxca

2

2

222

22

2

22

axxxca


0 7 Equations in Particular Linear Equations


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Example 9

Solving a Fractional Equation

Solve

Solution:

Fractional Equations

A fractional equation is an equation in whichan unknown is in a denominator.

.3

6

4

5

xx

x

xx

xxx

xxx

9

4635

3634

4534


0 7 Equations in Particular Linear Equations


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Example 11 Literal Equation

If express u in terms of the remainingletters; that is, solve for u.

Solution:

,vau

u

s

sa

svu

svsau

usvsau

uvausvau

us

1

1


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A quadrat ic equation in the variablex is anequation that can be written in the form

where a, b, and c are constants and

A quadratic equation is also called a second-degree equation or an equation of degree two.

0.8 Quadratic Equations

02 cbxax

.0a


0 8 Quadratic Equations


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Example 1Solving a Quadratic Equation by Factoring

a. SolveSolution:

Factor the left side factor:

Whenever the product of two or more quantitiesis zero, at least one of the quantities must be

zero.

.012

2

xx

043 xx

43

04or03

xx

xx




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Example 1 Solving a Quadratic Equation by Factoring

b. Solve

Solution:

.56 2 ww

6

5or0

056

56 2

ww

ww

ww

Chapter 0: Review of Algebra0 8 Quadratic Equations


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Example 3 Solving a Higher-Degree Equation by

Factoring

a. Solve

Solution:

b. Solve

Solution:

.044 3 xx

1or1or0

0114

014

044

2

3

xxx

xxx

xx

xx

.0252 32 xxxxx

7/2or2or0

0722

0252

0252

2

2

32

xxx

xxx

xxxx

xxxxx


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Quadratic Formula

The roots of the quadratic equation

can be given as


02 cbxax

a

acbbx

2

42

Chapter 0: Review of Algebra0.8 Quadratic Equations


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Example 7 A Quadratic Equation with One Real Root

Solve by the quadratic formula.

Solution:

Here a = 9, b = 62, and c = 2. The roots are

09262 2 yy

3

2

18

026or

3

2

18

026

92026

yy

y




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Quadratic-Form Equation

When a non-quadratic equation can betransformed into a quadratic equation by anappropriate substitution, the given equation is

said to have quadratic-form.



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