Date post: | 30-Jan-2016 |
Category: |
Documents |
Upload: | viega-theresa-maria |
View: | 214 times |
Download: | 0 times |
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.