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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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    2011 Pearson Education, Inc.

    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 0: Review of Algebra

    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

    Chapter 0: Review of Algebra

    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

    Chapter 0: Review of Algebra

    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

    Chapter 0: Review of Algebra

    0.2 Some Properties of Real Numbers

    cabbcacbacba and

    aaaa 1and0

    0 aa 11 aa

    cabaacbacabcba and

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    Chapter 0: Review of Algebra

    0.2 Some Properties of Real Numbers

    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

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    Chapter 0: Review of Algebra

    0.2 Some Properties of Real Numbers

    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

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    Properties:

    Chapter 0: Review of Algebra

    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

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    Chapter 0: Review of Algebra

    0.3 Exponents and Radicals

    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

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    Chapter 0: Review of Algebra

    0.3 Exponents and Radicals

    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

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    Chapter 0: Review of Algebra

    0.3 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

    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

    Chapter 0: Review of Algebra

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    Chapter 0: Review of Algebra

    0.3 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

    xxx

    xxx

    Chapter 0: Review of Algebra

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    Chapter 0: Review of Algebra

    0.3 Exponents and Radicals

    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

    Chapter 0: Review of Algebra

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    Chapter 0: Review of Algebra

    0.3 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

    Chapter 0: Review of Algebra

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    Chapter 0: Review of Algebra

    0.3 Exponents and Radicals

    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

    Chapter 0: Review of Algebra

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    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

    Chapter 0: Review of Algebra

    0.4 Operations with Algebraic Expressions

    01

    1

    1 cxcxcxc n

    n

    n

    n

    Chapter 0: Review of Algebra

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    Chapter 0: Review of Algebra

    0.4 Operations with Algebraic Expressions

    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

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    Chapter 0: Review of Algebra

    0.4 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

    Chapter 0: Review of Algebra

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    A list of products may be obtained from the

    distributive property:

    Chapter 0: Review of Algebra

    0.4 Operations with Algebraic Expressions

    Chapter 0: Review of Algebra

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    Chapter 0: Review of Algebra

    0.4 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

    Chapter 0: Review of Algebra

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    Chapter 0: Review of Algebra

    0.4 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

    Chapter 0: Review of Algebra

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    Chapter 0: Review of Algebra

    0.4 Operations with Algebraic Expressions

    Example 7 Dividing a Multinomial by a Monomial

    zzz

    z

    zzz

    xx

    xx

    3

    2

    342

    2

    6384b.

    33

    a.

    223

    23

    Chapter 0: Review of Algebra

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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

    Chapter 0: Review of Algebra

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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

    Chapter 0: Review of Algebra

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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

    Chapter 0: Review of Algebra

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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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    0.7 Equations, in Particular Linear Equations

    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.

    Chapter 0: Review of Algebra

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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.

    0.7 Equations, in Particular Linear Equations

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    0.7 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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    0.7 Equations, in Particular Linear Equations

    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

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    0.7 Equations, in Particular Linear Equations

    Example 3 Solving a Linear Equation

    SolveSolution:

    .365 xx

    3

    26

    22

    62

    60662

    062

    33365

    365

    x

    x

    x

    x

    x

    xxxx

    xx

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

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    0.7 Equations, in Particular Linear Equations

    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

    Chapter 0: Review of Algebra

    0 7 E ti i P ti l Li E ti

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    0.7 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.

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

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    0.7 Equations, in Particular Linear Equations

    Example 7 Solving a Literal Equation

    Solve for x.Solution:

    ac

    ax

    aacxaaxxxcxax

    axxxca

    2

    2

    222

    22

    2

    22

    axxxca

    Chapter 0: Review of Algebra

    0 7 Equations in Particular Linear Equations

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    0.7 Equations, in Particular Linear Equations

    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

    Chapter 0: Review of Algebra

    0 7 Equations in Particular Linear Equations

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    0.7 Equations, in Particular Linear Equations

    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

    Chapter 0: Review of Algebra

    0 8 Quadratic Equations

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    0.8 Quadratic Equations

    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

    Chapter 0: Review of Algebra

    0 8 Quadratic Equations

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    0.8 Quadratic Equations

    Example 1 Solving a Quadratic Equation by Factoring

    b. Solve

    Solution:

    .56 2 ww

    6

    5or0

    056

    56 2

    ww

    ww

    ww

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    0.8 Quadratic Equations

    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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    Chapter 0: Review of Algebra

    0 8 Quadratic Equations

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    Quadratic Formula

    The roots of the quadratic equation

    can be given as

    0.8 Quadratic Equations

    02 cbxax

    a

    acbbx

    2

    42

    Chapter 0: Review of Algebra0.8 Quadratic Equations

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    0.8 Quadratic Equations

    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

    Chapter 0: Review of Algebra

    0 8 Quadratic Equations

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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.

    0.8 Quadratic Equations

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