INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007...

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

2007 Pearson Education Asia

Chapter 0 Chapter 0 Review of AlgebraReview of Algebra

2007 Pearson Education Asia

INTRODUCTORY MATHEMATICAL ANALYSIS

0. Review of Algebra

1. Applications and More Algebra

2. Functions and Graphs

3. Lines, Parabolas, and Systems

4. Exponential and Logarithmic Functions

5. Mathematics of Finance

6. Matrix Algebra

7. Linear Programming

8. Introduction to Probability and Statistics

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9. Additional Topics in Probability

10. Limits and Continuity

11. Differentiation

12. Additional Differentiation Topics

13. Curve Sketching

14. Integration

15. Methods and Applications of Integration

16. Continuous Random Variables

17. Multivariable Calculus

INTRODUCTORY MATHEMATICAL ANALYSIS

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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 the denominator.

• 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 ObjectivesChapter 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 0: Review of Algebra

Chapter OutlineChapter 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 that set.

• Different type of integers:

• The real-number line is shown as

Chapter 0: Review of Algebra

0.1 Sets of Real Numbers0.1 Sets of Real Numbers

... ,3 ,2 ,1integers positive of Set

1 ,2 ,3 ..., integers negative of Set

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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

Chapter 0: Review of Algebra

0.2 Some Properties of Real Numbers0.2 Some Properties of Real Numbers

. then , and If cacbba

. and

numbers real unique are there numbers, real all For

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

Chapter 0: Review of Algebra

0.2 Some Properties of Real Numbers

cabbcacbacba and

aaaa 1 and 0

0 aa 11 aa

cabaacbacabcba and

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Chapter 0: Review of Algebra0.2 Some Properties of Real Numbers

Example 1 – Applying Properties of Real Numbers

Example 3 – Applying Properties of Real Numbers

354543 b.

2323 a.

xwzywzyxSolution:

a. Show that

Solution:

.0 for

c

c

ba

c

ab

c

ba

cba

cab

c

ab 11

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Chapter 0: Review of Algebra0.2 Some Properties of Real Numbers

Example 3 – Applying Properties of Real Numbers

b. Show that

Solution:

.0 for c

b

c

c

a

c

ba

c

bc

ac

bac

ba 111

c

b

c

a

c

bac

b

c

a

cb

ca

11

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• Properties:

Chapter 0: Review of Algebra

0.3 Exponents and Radicals0.3 Exponents and Radicals

1 4.

1 3.

0 for 11

2.

1.

0

x

xx

x xxxxx

x

xxxxx

nn

factorsn

nn

factorsn

n

nxexponent

base

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Chapter 0: Review of Algebra0.3 Exponents and Radicals

Example 1 – Exponents

xx

π

1

000

55-

55-

4

e.

1)5( ,1 ,12 d.

24333

1 c.

243

1

3

13 b.

16

1

2

1

2

1

2

1

2

1

2

1 a.

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Chapter 0: Review of Algebra0.3 Exponents and Radicals

• The symbol is called a radical.

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

n x

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Chapter 0: Review of Algebra0.3 Exponents and Radicals

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

2 a.

6 55

6 566 5

1

61

65

61

21

21

21

21

21

a. Eliminate negative exponents in and simplify.

Solution:

11 yx

xy

xy

yxyx

1111

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Chapter 0: Review of Algebra0.3 Exponents and Radicals

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

xxxxxx

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Chapter 0: Review of Algebra0.3 Exponents and Radicals

Example 5 – Exponents

d. Eliminate negative exponents in

Solution:

e. Apply the distributive law to

Solution:

.211 yx

2222

22211

11

xy

yx

xy

xy

xy

xy

yxyx

.2 56

21

52

xyx

58

21

52

56

21

52

22 xyxxyx

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Chapter 0: Review of Algebra0.3 Exponents and Radicals

Example 7 – Radicals

a. Simplify

Solution:

b. Simplify

Solution:

3233 33 323 46 )( yyxyyxyx

7

14

77

72

7

2

.3 46yx

.7

2

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Chapter 0: Review of Algebra0.3 Exponents and Radicals

Example 7 – Radicals

c. Simplify

Solution:

d. If x is any real number, simplify

Solution:

Thus, and

210105

2152510521550250

.21550250

.2x

0 if

0 if 2

xx

xxx

222 .33 2

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• If symbols are combined by any or all of the operations, the resulting expression is called an algebraic expression.

• A polynomial in x is an algebraic expression of the form:

where n = non-negative integer cn = constants

Chapter 0: Review of Algebra

0.4 Operations with Algebraic Expressions0.4 Operations with Algebraic Expressions

011

1 cxcxcxc nn

nn

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Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions

Example 1 – Algebraic Expressions

a. is an algebraic expression in the

variable x.

b. is an algebraic expression in the

variable y.

c. is an algebraic expression in the

variables x and y.

3

3

10

253

x

xx

2

3

y

xyyx

27

5310

yy

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Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions

Example 3 – Subtracting Algebraic Expressions

Simplify

Solution:

.364123 22 xyxxyx

48

316243

)364()123(

364123

2

2

22

22

xyx

xyx

xyxxyx

xyxxyx

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• A list of products may be obtained from the distributive property:

Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions

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Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions

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

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Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions

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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Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions

Example 7 – Dividing a Multinomial by a Monomial

zzz

z

zzz

xx

xx

3

2

342

2

6384 b.

33

a.

223

23

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• If two or more expressions are multiplied together, the expressions are called the factors of the product.

Chapter 0: Review of Algebra

0.5 Factoring0.5 Factoring

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Chapter 0: Review of Algebra0.5 Factoring

Example 1 – Common Factors

a. Factor completely.

Solution:

b. Factor completely.

Solution:

xkxk 322 93

kxxkxkxk 3393 2322

224432325 268 zxybayzbayxa

24232232

224432325

342

268

xyzbazbyxaya

zxybayzbayxa

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Chapter 0: Review of Algebra0.5 Factoring

Example 3 – Factoring

zzzz

xxx

yyyyyy

xxxx

xxx

1 e.

396 d.

23231836 c.

2313299 b.

4168 a.

4/14/54/1

22

23

2

42

2222

333366

23

2222

3/13/13/13/2

24

j.

2428 i.

bbaa h.

4145 g.

1111 f.

yxyxyxyxyxyx

yxyxyx

xxxx

bayxyxyxyx

xxxx

xxxx

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Simplifying Fractions

• Allows us to multiply/divide the numerator and denominator by the same nonzero quantity.

Multiplication and Division of Fractions

• The rule for multiplying and dividing is

Chapter 0: Review of Algebra

0.6 Fractions0.6 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 be rationalized by multiplying an expression that makes the denominator a difference of two squares.

Addition and Subtraction of Fractions

• If we add two fractions having the same denominator, we get a fraction whose denominator is the common denominator.

Chapter 0: Review of Algebra

0.6 Fractions

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Chapter 0: Review of Algebra0.6 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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Chapter 0: Review of Algebra0.6 Fractions

Example 3 – Dividing Fractions

41

2

82

1

1

4

1821

4

c.

32

5

2

1

3

5

235

b.

32

5

3

5

25

3

2 a.

222

2

xxxx

x

x

x

xxx

xx

xx

x

xx

x

xxx

xx

xx

x

x

x

x

x

x

x

x

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Chapter 0: Review of Algebra0.6 Fractions

Example 5 – Adding and Subtracting Fractions

2

33

2

235

2

23

2

5 a.

2

2

2

p

pp

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

xxxx

xx

xx

xx

17

7

7

425

149

84

7

2

7

5 c.

22

2

22

x

xx

xxx

xx

x

x

x

x

xx

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Chapter 0: Review of Algebra0.6 Fractions

Example 7 – Subtracting Fractions

332

615

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 two expressions are equal.

• The two expressions that make up an equation are called its sides.

• They are separated by the equality sign, =.

Chapter 0: Review of Algebra

0.7 Equations, in Particular Linear Equations0.7 Equations, in Particular Linear Equations

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

Example 1 – Examples of Equations

zw

y

y

xx

x

7 d.

64

c.

023 b.

32 a.2

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

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Equivalent Equations

• Two equations are said to be equivalent if they have exactly the same solutions.

• There are three operations that guarantee equivalence:

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.

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

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

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

Linear Equations

• A linear equation in the variable x can be written 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.

0 bax

0a

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

Example 3 – Solving a Linear Equation

Solve

Solution:

.365 xx

3 2

6

2

2

62

60662

062

33365

365

x

x

x

x

x

xxxx

xx

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

Example 5 – Solving a Linear Equations

Solve

Solution:

.64

89

2

37

xx

2

105

24145

2489372

644

89

2

374

x

x

x

xx

xx

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

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

Example 7 – Solving a Literal Equation

Solve for x.

Solution:

ac

ax

aacx

aaxxxcxax

axxxca

2

2

222

22

2

22 axxxca

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

Example 9 – Solving a Fractional Equation

Solve

Solution:

Fractional Equations

• A fractional equation is an equation in which an unknown is in a denominator.

.3

6

4

5

xx

x

xx

xxx

xxx

9

4635

3

634

4

534

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

Example 11 – Literal Equation

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

Solution:

,vau

us

sa

svu

svsau

usvsau

uvausvau

us

1

1

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

Radical Equations

• A radical equation is one in which an unknown occurs in a radicand.

Example 13 – Solving a Radical Equation

Solve

Solution:

.33 yy

4

2

126

963

33

y

y

y

yyy

yy

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• A quadratic equation in the variable x is an equation 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.

Chapter 0: Review of Algebra

0.8 Quadratic Equations0.8 Quadratic Equations

02 cbxax

.0a

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Chapter 0: Review of Algebra0.8 Quadratic Equations

Example 1 – Solving a Quadratic Equation by Factoring

a. Solve

Solution: Factor the left side factor:

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

.0122 xx

043 xx

4 3

04 or 03

x x

xx

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Chapter 0: Review of Algebra0.8 Quadratic Equations

Example 1 – Solving a Quadratic Equation by Factoring

b. Solve

Solution:

.56 2 ww

6

5 or 0

056

56 2

ww

ww

ww

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Chapter 0: Review of Algebra0.8 Quadratic Equations

Example 3 – Solving a Higher-Degree Equation by Factoring

a. Solve

Solution:

b. Solve

Solution:

.044 3 xx

1 or 1 or 0

0114

014

0442

3

xxx

xxx

xx

xx

.0252 32 xxxxx

7/2 or 2 or 0

0722

0252

0252

2

2

32

xxx

xxx

xxxx

xxxxx

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Chapter 0: Review of Algebra0.8 Quadratic Equations

Example 5 – Solution by Factoring

Solve

Solution:

.32 x

3 Thus,

3 or 3

033

03

32

2

x

xx

xx

x

x

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Quadratic Formula

• The roots of the quadratic equation

can be given as

Chapter 0: Review of Algebra

0.8 Quadratic Equations

02 cbxax

a

acbbx

2

42

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Chapter 0: Review of Algebra0.8 Quadratic Equations

Example 7 – A Quadratic Equation with One Real Root

Solve by the quadratic formula.

Solution: Here a = 9, b = 6√2, and c = 2. The roots are

09262 2 yy

3

2

18

026 or

3

2

18

026

92

026

yy

y

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Quadratic-Form Equation

• When a non-quadratic equation can be transformed into a quadratic equation by an appropriate substitution, the given equation is said to have quadratic-form.

Chapter 0: Review of Algebra

0.8 Quadratic Equations

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Chapter 0: Review of Algebra

0.8 Quadratic Equations

Example 9 –– Solving a Quadratic-Form Equation

Solve

Solution:

This equation can be written as Substituting w =1/x3, we have

Thus, the roots are

.0891

36

xx

081

91

3

2

3

xx

1 or 8

018

0892

ww

ww

ww

1 or 2

1

11

or 81

33

xx

xx