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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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
• 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
2007 Pearson Education Asia
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)
2007 Pearson Education Asia
• 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
2007 Pearson Education Asia
• 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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
• 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
2007 Pearson Education Asia
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.
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
• 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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
• A list of products may be obtained from the distributive property:
Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
• 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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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.
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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.
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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.
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
• 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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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
2007 Pearson Education Asia
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