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

2011 Pearson Education, Inc.

Chapter 0 Chapter 0 Review of AlgebraReview of Algebra

2011 Pearson Education, Inc.

• 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

2011 Pearson Education, Inc.

• 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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

• 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

2011 Pearson Education, Inc.

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.

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

• 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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

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

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

2011 Pearson Education, Inc.

• 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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

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

2011 Pearson Education, Inc.

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.