Schaum's Easy Outlines: Intermediate Algebra

127 Pages·2003·1.59 MB·English

by  Ray Steege

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SCHAUM’S Easy OUTLINES
I
NTERMEDIATE
A
LGEBRA
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Other Books in Schaum’s
Easy Outlines Series Include:
Schaum’s Easy Outline: Calculus
Schaum’s Easy Outline: College Algebra
Schaum’s Easy Outline: College Mathematics
Schaum’s Easy Outline: Differential Equations
Schaum’s Easy Outline: Discrete Mathematics
Schaum’s Easy Outline: Elementary Algebra
Schaum’s Easy Outline: Geometry
Schaum’s Easy Outline: Linear Algebra
Schaum’s Easy Outline: Mathematical Handbook of
Formulas and Tables
Schaum’s Easy Outline: Precalculus
Schaum’s Easy Outline: Probability and Statistics
Schaum’s Easy Outline: Statistics
Schaum’s Easy Outline: Trigonometry
Schaum’s Easy Outline: Bookkeeping and Accounting
Schaum’s Easy Outline: Business Statistics
Schaum’s Easy Outline: Economics
Schaum’s Easy Outline: Principles of Accounting
Schaum’s Easy Outline: Beginning Chemistry
Schaum’s Easy Outline: Biology
Schaum’s Easy Outline: Biochemistry
Schaum’s Easy Outline: College Chemistry
Schaum’s Easy Outline: Genetics
Schaum’s Easy Outline: Human Anatomy and Physiology
Schaum’s Easy Outline: Molecular and Cell Biology
Schaum’s Easy Outline: Organic Chemistry
Schaum’s Easy Outline: Applied Physics
Schaum’s Easy Outline: Physics
Schaum’s Easy Outline: HTML
Schaum’s Easy Outline: Programming with C++
Schaum’s Easy Outline: Programming with Java
Schaum’s Easy Outline: Basic Electricity
Schaum’s Easy Outline: Electromagnetics
Schaum’s Easy Outline: Introduction to Psychology
Schaum’s Easy Outline: French
Schaum’s Easy Outline: German
Schaum’s Easy Outline: Spanish
Schaum’s Easy Outline: Writing and Grammar
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SCHAUM’S Easy OUTLINES
I
NTERMEDIATE
A
LGEBRA
Based on Schaum’s
Outline of Theory and Problems of
Intermediate Algebra
by
Ray Steege, M.A.
and
Kerry Bailey, M.A.
Abridgement Editor
George J. Hademenos, Ph.D.
SCHAUM’S OUTLINE SERIES
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DOI: 10.1036/007143108X
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Contents
Chapter 1 Fundamental Concepts 1
Chapter 2 Polynomials 13
Chapter 3 Rational Expressions 23
Chapter 4 First-Degree Equations
and Inequalities 30
Chapter 5 Exponents, Roots, and Radicals 46
Chapter 6 Second-Degree Equations
and Inequalities 54
Chapter 7 Systems of Equations and
Inequalities 67
Chapter 8 Relations and Functions 83
Chapter 9 Exponential and Logarithmic
Functions 95
Chapter 10 Sequences, Series, and the
Binomial Theorem 105
Index 115
v
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Chapter 1
Fundamental
Concepts
In This Chapter:
✔
Definitions
✔
Axioms of Equality and Order
✔
Properties of Real Numbers
✔
Operations with Real Numbers
✔
Order of Operations
Definitions
Aset is a collection of objects. The collection should be well defined.
That is, it must be clear that an object is either in the set or is not in the
set. The objects in the set are called elementsor membersof the set. The
members of a set may be listed or a description of the members may be
given.
We list the members or describe the members within braces {}. Cap-
ital letters such as A, B, C, S, T, and Uare employed to name sets. For ex-
ample,
A={2, 4, 6, 8, 12} B={3, 6, 9, 12}
U={people enrolled in algebra this semester}
The symbol used to represent the phrase “is an element of” or “is a
member of” is “∈.” Thus, we write 4 ∈Ato state that 4 is a member of
1
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2 INTERMEDIATE ALGEBRA
set A. The symbol used to represent the phrase “is not an element of” is
“∉.” Hence, 4 ∉B is written to indicate that 4 is not an element of setB.
Sets are said to be equal if they contain the same elements. Hence
{1, 5, 9} ={5, 9, 1}.
Note!
Order is disregarded when the members of a set
are listed.
Sometimes a set contains infinitely many elements. In that case, it is
impossible to list all of the elements. We simply list a sufficient number
of elements to establish a pattern followed by a series of dots “…”. For
example, the set of numbers employed in counting is called the set of nat-
ural numbers or the set of counting numbers. We write N = {1, 2, 3, 4,
…} to represent that infinite set. If zero is included with the set of natu-
ral numbers, the set of whole numbers is obtained. In this case, the sym-
bol used is W={0, 1, 2, …}.
Set B above could be described as the set of multiples of three be-
tween0 and 15. Note that the term “between” does not include the num-
bers 0 and 15. Set-builder notation is sometimes used to define sets. We
write, for example,
{ }
B= x x is a multiple of three between 0 and 15
There are occasions when a set contains no elements. This set is
called the emptyor nullset. The symbol used to represent the empty set
is “∅” or “{}.” Note that no braces are used when we represent the emp-
ty set by ∅. {negative natural numbers} is an example of an empty set.
Definition 1. Set A is a subset of B if all elements of A are elements of
B. We write A⊆B.
Hence, if A={2, 4, 6} and B={1, 2, 3, 4, 5, 6, 7}, A⊂B. Ais called
a proper subset of B. If C = {4, 2, 6}, A ⊆ C since the sets are the same
set. Ais called an improper subsetof C.
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CHAPTER 1: Fundamental Concepts 3
New sets may be formed by performing operations on existing sets.
The operations used are union and intersection.
Definition 2. The union of two sets A and B, written A ∪ B, is the set
containing all of the elements in set A or B, or in both A
and B.
{ }
If set-builder notation is used, we write A∪B= x x∈A or x∈B . If
A={1, 5, x, z} and B={3, 5, 7, z}, then A∪B={1, 5, x, z, 3, 7}. Recall
that the members of a set may be listed without regard to order.
Definition 3. The intersectionof two sets Aand B, written A∩B, is the
set containing the elements common to both sets.
{ }
If set-builder notation is used, we write A∩B= x x∈A and x∈B .
Hence, if set A={1, 5, x, z} and B={3, 5, 7, z}, then A∩B={5, z}.
Venn diagrams are sometimes used to illustrate relationships be-
tween sets. Figure 1-1 (a) and (b) shown below illustrate the concepts dis-
cussed above. The shaded regions represent the specified set.
Figure 1-1(a) Figure 1-1(b)
Urepresents the universal set. It is a set that contains all of the ele-
ments under discussion in a given situation. The universal set is typical-
ly represented by a rectangular region.
Figure 1-2 shows a Venn diagram that illustrates A⊂B.
Note that all of set Ais completely contained in set B.
If set S is not a subset of set T, we write S ⊄ T. This occurs when S
contains at least one element that is not in T.
There are several additional sets of numbers that will be referred to
often. Their definitions follow.