Introductory Maths Analysis

8/9/2019 Introductory Maths Analysis

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2007 Pearson Education Asia

MATHEMATICS FOR

BUSINESS

Handouts for University PreparatoryStudents

By

SURESH KUMAR, S.T.,M.SI


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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 NumbersSome Properties of Real Numbers

Exponents and Radicals

Operations with Algebraic Expressions

Factoring

Fractions

Equations, in Particular Linear Equations

Quadratic Equations


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 thatset.

Different type of integers:

The real-number line is shown as


0.1 Sets of Real Numbers0.1 Sets of Real Numbers

_ a...,3,2,1!integerspositiveofSet

_ a1,2,3..., !i t rsatift


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The set ofrational numbers consists of

numbers that can be written as aquotient of two integers.

Exe.

Numbers represented by nonterminatingnonrepeating decimals are called

irrational numbers. Exe. (pi) and .

Together rational and irrational numbersform the set of real numbers

2

%60,5.0,2

6,

7

2,

20

19


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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 Numbers0.2 Some Properties of Real Numbers

.th,a dIf cacbba !!!

.a d

umb rsr alu iquarth rumb rs,r alallF r

abba

baababba !! a d


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4. The Commutative Properties of Additionand Multiplication

5. The Identity Properties

6. The Inverse Properties

7. The Distributive Properties


0.2 Some Properties of Real Numbers

cabbcacbacba !! a d

aaaa !! 1n0

0! aa 11 ! aa

cabaacbacabcba !! a d


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Example 1 Applying Properties of Real Numbers

a. The commutative property of multiplication

b. The associative property of multiplication

Example 3 Applying Properties of RealNumbers

354543.

2323.!

! xwzywzyxSol tion:

a. Show that

Sol tion:

.0for {

! c

ca

c

ab

!

!!

c

ba

cba

cab

c

ab 11

By t e ssoci tive property

By t e efinition of ivision


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Example 3 Applying Properties of Real Numbers

b. Show that

Sol tion:

T e efinition of ivision n t e istri tive property.

.0forc

{! cca

cba

c

bc

ac

bac

ba 111!!

c

b

c

a

c

ba

c

b

c

a

cbca

!

!

11


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Properties:


0.3 Exponents and Radicals0.3 Exponents and Radicals

14.

13.

0for11

2.

1.

0

!

!

{

!!

!

x

xx

xxxxxx

x

xxxxx

n

n

factorsn

n

n

factorsn

n

.

.

nxexponent

se


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0.3 Exponents and Radicals

Example 1 Exponents

xx

!

!!!

!!

!!

!

!

1

000

55-

55-

4

e.

1)5(,1,12.

243331

c.

243

1

3

13.

161

21

21

21

21

21.


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T e symbol is c lle radical.

n is t e index, xis t e radicand, n is t e

radical sign.

n x


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Example 3 Rationalizing Denominators

Sol tion:

Example 5 Exponents

xxxxxxx 3323323 23 232b.

5

52

5

52

55

52

5

2

5

2.

6 55

6 566 5

1

61

65

61

21

21

21

21

21

!!!

!

!

!

!!

a. Eliminate negative exponents in andsimplify.

Sol tion:

11 yx

xy

xy

yx

yx

!! 1111


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Example 5 Exponents

b. Simplify by using the distributive law.Sol tion:

c. Eliminate negative exponents in

Sol tion:

12/12/12/3 ! xxxx

2/12/3 xx

. xx

2222

22

49

17

7

1777

xxxx

xx !!


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Example 5 Exponents

d. Eliminate negative exponents in

Sol tion:

e. Apply the distributive law toSol tion:

.211

yx

2222

22

211

11

xy

yx

xy

xy

xy

xy

yxyx

!

!

!

!

.2 5

62

15

2

xyx

582152562152 22 xyxxyx !


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Example 7 Radicals

a. Simplify

Sol tion:

b. Simplify

Sol tion:

3233 33 323 46 )( yyxyyxyx !!

7

14

77

72

7

2!

!

.3 46

yx

.2


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Example 7 Radicals

c. Simplify

Sol tion:

d. If x is any real number, simplify

Sol tion:

T s, n

210105

2152510521550250

!

!

.21550250

.2x

u!

0if

0if2

xx

xxx

222! .33

2!


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Simplify and express all answers in terms ofpositive exponents

1.

2.

3.



Problems 1 Exponents

23

22

59

53

yy

xx

43

2332

x

xx


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Evaluate the expressions.

1.

2.

3.




3

27

8

32

27

64

54

32

1


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Simplify the expressions

1.




3 12827582

43

12

32

3

256.3

8

27.2

x

t


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Write the expressions in terms ofpositive exponents only. Avoid all

radicals in the final form. For example,




y

xxy

211

!

4 322

5 1032

2

35

.3

.2.1

zxyx

zyxc

ba


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Rationalize the denominators




4 2

5

3 2

3

2.3

3

2.2

3

1.1

ba

y

x


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Express all answers in terms of positiveexponents. Rationalize the denominator

where necessary to avoid fractional

exponents in the denominator.




432233 23 32

2

3

2

23.4.3

16

2

1.2

3

243.1

z

zyyxxyyzx

x

x


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

an algebraic expression.

A polynomia

l in x is an algebraic expressionof the form:

where n=

non-negative integercn = constants


0.4 Operations with Algebraic Expressions0.4 Operations with Algebraic Expressions

011

1 cxcxcxcn

n

n

n

-


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0.4 Operations with Algebraic Expressions

Example 1 Algebraic Expressions

a. is an algebraic expression in t e

variablex.

b. is an algebraic expression in t e

variable y.

c. is an algebraic expression in t e

variablesxan y.

3

3

10253

xxx

2

3

y

xyyx

27

5310

yy


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Example 3 Subtracting Algebraic Expressions

Simplify

Sol tion: .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 thedistributive property:




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Example 5 Special Products

a. By R le 2,

b. By R le 3,

103

5252

52

2

2

!

!

xx

xx

xx

204721

45754373

4753

2

2

!

!

zz

zz

zz


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Example 5 Special Products

c. By R le 5,

. By R le 6,

e. By R le 7,

168

4424

2

22

2

!

!

xx

xx

x

8

31

3131

2

22

2

22

!

!

y

y

yy

8

365

427

23233233

23

23

3223

3

!

!

xxx

xxx

x


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Example 7 Dividing a Multinomial by a Monomial

z

zz

z

zzz

xx

xx

3

2

342

2

6384b.

33

a.

2

23

23

!

!


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Perform the indicated operations andsimplify



Example 1 Algebraic Expressions

? A_ a

2

35

23

22

22

22

2

2

146.7

2323.61234.5

54.4

522332.372

435

.2

422106.1

x

xx

xxxxxxx

xx

xxyxyxyxyx

xyzxyx

z


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


0.5 Factoring0.5 Factoring


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0.5 Factoring

Example 1 Common Factors

a. Factor

completely.

Sol tion:

b. Factor completely.

Sol tion:

xkxk 322 93

kxxkxkxk 3393 2322 !

224432325268 zxybayzbayxa

24232232224432325

342

268

xyzbazbyxayazxy

bayz

bayx

a

!


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0.5 Factoring

Example 3 Factoring

zzzz

xxx

yyyyyy

xxxxxxx

!

!

!

!!

1e.

396.

23231836c.

2313299b. 4168

a

.

4/14/54/1

22

23

2

42

2222

333366

23

2222

3/13/13/13/2

24

j.

2428i.bbaah.

4145.

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


0.6 Fractions0.6 Fractions

bd

ac

d

c

b

a

!

bc

ad

d

c

b

a

!z


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Factor the following expressions completely.


0.5 Factoring

Problems Factoring

yyxyx

xxxx

xx

pp

dcbcdabbca

24

223

2

2

22433

2.51313

.4

30255.3

34.2

4128.1


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Rationalizing the Denominator

For a denominator with square roots, it maybe rationalized by multiplying an expressionthat makes the denominator a difference oftwo squares.

Addition and Subtraction of Fractions

If we add two fractions having the same

denominator, we get a fraction whosedenominator is the common denominator.


0.6 Fractions

Ch t 0 R i f Al b


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0.6 Fractions

Example 1 Simplifying Fractions

a. SimplifySol tion:

b. Simplify

Sol tion:

.127

62

2

xx

xx

4

2

43

23

127

6

2

2

!

!

x

x

xx

xx

xx

xx

.448

862

2

2

xx

xx

22

4214

412448

8622

2

!

!

x

x

xx

xx

xx

xx

Ch t 0 R i f Al b


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0.6 Fractions

Example 3 Dividing Fractions

412

82

1

1

4

1

821

4

c.

32

5

2

1

3

5

2

35

b.

325

35

253

2a.

222

2

!

!

!

!

!

!

z

xxxx

x

x

x

x

xxx

x

xx

x

xx

x

x

x

x

xx

xx

x

x

x

x

x

x

x

x

Ch t 0 R i f Al b


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0.6 Fractions

Example 5 Adding and Subtracting Fractions

233

2

235

22325a.

2

2

2

!

!

p

pp

p

pp

pp

pp

34

32

2

31

41

65

2

324

5b. 2

2

2

2

!

!

x

xx

xx

xx

xx

xxxx

xxxx

17

7

7

425

149

84

7

2

7

5c.

22

2

22

!

!

!

x

x

x

xxx

xx

x

x

x

x

xx

Chapter 0 Re ie of Algebra


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0.6 Fractions

Example 7 Subtracting Fractions

332615

332

6512102

3323

2

322

92

2

96

2

2

2

2

22

222

!

!

!

xxxx

xx

xxxx

xxxxxx

xx

xxx



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Perform the operations and simplify as muchas possible


0.6 Fractions

Problems Fractions

63

2.8

2

2

1

3.7

45

33

1

4.6

3

65

3.5

45

1

82

22.4

.31

1.22

209.1

2

2

2

2

2

2

3

3

2

2

z

y

x

xx

x

xx

x

x

x

x

x

x

xx

x

xx

x

bax

xc

cx

bax

x

x

xx

xx



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Equations

An equation is a statement that twoexpressions are equal.

The two expressions that make up anequation are called its sides.

They are separated by the equality sign, =.


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



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0.7 Equations, in Particular Linear Equations

Example 1 Examples of Equations

zw

y

y

xx

x

!

!

!

!

7.

64

c.

023b.

32a.2

Avariable (e.g.x, y) is a symbol t at can bereplace byany one ofa set of ifferent

n mbers.

Ch t 0 R i f Al b


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Equivalent Equations

Two equations are said to be equivalent ifthey 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.





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Operations That May Not Produce Equivalent

Equations4. Multiplying both sides of an equation by an

expression involving the variable.

5. Dividing both sides of an equation by anexpression involving the variable.

6. Raising both sides of an equation to equal

powers.



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Linear Equations

A linear equation in the variable xcan bewritten in the form

where a and b are constants and .

A linear equation is also called a first-degreeequation or an equation of degree one.

0! bax

0{a



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Example 3 Solving a Linear Equation

Solve

Sol tion:

.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

Solve

Sol tion:

.64

89

2

37!

xx

2

105

24145

2489372

64

4

89

2

374

!

!

!

!

!

x

x

x

xx

xx



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Literal Equations

Equations where constants are notspecified, 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.

Sol tion:

ac

ax

aacxaaxxxcxax

axxxca

!

!!

!

2

2

222

22

2

2

2 axxxca !



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Example 9 Solving a Fractional Equation

Solve

Sol tion:

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



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Example 11 Literal Equation

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

Sol tion:

,vau

u

s

!

sa

svu

svsau

usvsau

uvausvau

us

!

!

!

!

!

1

1



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Radical Equations

Aradical equation is one in w ic an nknownocc rs in a radicand.

Example 13 Solving a Radical Equation

Solve

Sol tion:

.33 ! yy

4

2

126

963

33

!

!

!

!

!

y

y

y

yyy

yy



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Solve the equations

g


Problems Equation

025

21.803

7.4

32.729

47.3

32.6343275.2

2

2

1

1.5935.1

2

!

!

!!

!!

!

!

wwx

yyxx

zzzppp

ppx



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A quadratic equation in the variable xis anequation that can be written in the form

where a, b, and c are constants and

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

two.

0.8 Quadratic Equations0.8 Quadratic Equations

02 ! cbxax

.0{a



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p g

0.8 Quadratic Equations

Example 1 Solving a Quadratic Equation by Factoring

a. Solve

Sol tion:

actor t e left side factor:

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

zero.

.0122 !xx

043 ! xx

43

04or03

!!

!!

xx

xx



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p g


Example 1 Solving a Quadratic Equation by Factoring

b. Solve

Sol tion:

.56 2 ww !

6

5r0

056

562

!!

!

!

ww

ww

ww



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p g


Example 3 Solving a Higher-Degree Equation by

Factoring

a. Solve

Sol tion:

b. Solve

Sol tion:

.044 3 ! xx

1or1or0

0114

014

0442

3

!!!

!

!

!

xxx

xxx

xx

xx

.0252 32 ! xxxxx

? A

7/2r2r0

0722

0252

0252

2

2

32

!!!

!

!

!

xxx

xxx

xxxx

xxxxx



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Example 5 Solution by Factoring

Solve

Sol tion:

.32 !x

3T s,

3or3033

03

32

2

s!

!!

!

!

!

x

xx

xx

x

x


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Example 7 A Quadratic Equation with One Real Root

Solve by the quadratic formula.Sol tion:

Here a = 9, b = 62, andc= 2. T e roots are

092622

! yy

3

2

18

026r

3

2

18

026

92026

!

!!

!

s!

yy

y



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Quadratic-Form Equation

When a non-quadratic equation can betransformed into a quadratic equation by an

appropriate substitution, the given equation

is said to have qua

dra

tic-for

m.




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Example 9 Solving a Quadratic-Form Equation

Solve

Sol tion:

T is eation c

an be written

asS bstit ting w=1/x3, we ave

T s, t e roots are

.08

9136 !xx

081

91

3

2

3

!

xx

1r8

018

0892

!!

!

!

ww

ww

ww

1or2

1

11

or81

33

!!

!!

xx

xx



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Solve by factoring

Find all roots by using the quadratic

Solve the given quadratic-form equation.


Problems Quadratic Equation

049.404.33613.20158.13222 !!!! ttxuutt

22224.309124.20242.1 nnxxxx !!

035

2

12

2

1.2

0209.1

2

24

!

!

xx

xx

Documents

Introductory Maths Analysis