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Chapter P
Prerequisites:
Fundamental
Concepts of
Algebra 1
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1
P.1 Algebraic Expressions,
Mathematical Models, and
Real Numbers

• Evaluate algebraic expressions.
• Use mathematical models.
• Find the intersection of two sets.
• Find the union of two sets.
• Recognize subsets of the real numbers.
• Use inequality symbols.
• Evaluate absolute value.
• Use absolute value to express distance.
• Identify properties of real numbers.
• Simplify algebraic expressions.
Objectives:

Algebraic Expressions
If a letter is used to represent various numbers, it is
called a variable. A combination of variables and
numbers using the operations of addition, subtraction,
multiplication, or division, as well as powers or roots, is
called an algebraic expression. Evaluating an
algebraic expression means to find the value of the
expression for a given value of the variable.
The expression bn
is called an exponential expression.

Exponential Notation

The Order of Operations Agreement

Example: Evaluating an Algebraic Expression
Evaluate 8 + 6(x – 3)2
for x = 13.
Solution:
8 + 6(x – 3)2
= 8 + 6(13 – 3)2
= 8 + 6 (10)2
= 8 + 6(100)
= 8 + 600
= 608

Formulas and Mathematical Models
An equation is formed when an equal sign is placed
between two algebraic expressions. A formula is an
equation that uses variables to express a relationship
between two or more quantities. The process of finding
formulas to describe real-world phenomena is called
mathematical modeling. Such formulas, together with
the meaning assigned to the variables, are called
mathematical models.

Example: Using a Mathematical Model
The formula models the
average cost of tuition and fees, T, for public U.S.
colleges for the school year ending x years after 2000.
Use the formula to project the average cost of tuition
and fees at public U.S. colleges for the school year
ending in 2015.
x = years after 2000 = 2015 – 2000 = 15
We substitute 15 for x in the formula.
2
4 341 3194T x x= + +

Example: Using a Mathematical Model
(continued)
2
4 341 3194T x x= + +
2
4(15) 314(15) 3194T = + +
4(225) 341(15) 3194T = + +
900 5115 3194T = + +
9209T =
The formula indicates that for the school year ending in
2015, the average cost of tuition and fees at public U.S.
colleges will be $9209.

Sets
A set is a collection of objects whose contents can be
clearly determined. The objects in a set are called the
elements of the set. We use braces, { }, to indicate that
we are representing a set.
{1, 2, 3, 4, 5, ...} is an example of the roster
method of representing a set. The three dots after the 5,
called an ellipsis, indicates that there is no final element
and that the listing goes on forever.
If a set has no elements, it is called the empty set, or
the null set, and is represented by the Greek letter phi, .∅

Set-Builder Notation
In set-builder notation, the elements of the set are
described but not listed.
is read, “The set of all x such that x is a counting
number less than 6”.
The same set written using the roster method is
{1, 2, 3, 4, 5}.
{ }is a counting number less than 6x x

Definition of the Intersection of Sets
The intersection of sets A and B, written is the
set of elements common to both set A and set B. This
definition can be expressed in set-builder notation as
follows:
,A BI
{ }is an element of and is an element of .=IA B x x A x B

Example: Finding the Intersection of Two Sets
Find the intersection:
The elements common to {3, 4, 5, 6, 7} and {3, 7, 8, 9}
are 3 and 7. Thus,
{ } { }3,4,5,6,7 3,7,8,9 .I
{ } { } { }3,4,5,6,7 3,7,8,9 3,7=I

Definition of the Union of Sets
The union of sets A and B, written , is the set of
elements that are members of set A or of set B or of
both sets. This definition can be expressed in
set-builder notation as follows:
We can find the union of set A and set B by listing the
elements of set A. Then we include any elements of
set B that have not already been listed. Enclose all
elements that are listed with braces. This shows that the
union of two sets is also a set.
A BU
{ }is an element of OR is an element of .=UA B x x A x B

Example: Finding the Union of Two Sets
Find the union:
To find the union of both sets, start by listing all
elements from the first set, namely, 3, 4, 5, 6, and 7.
Now list all elements from the second set that are not in
the first set, namely, 8 and 9. The union is the set
consisting of all these elements.
Thus,
{ } { }3,4,5,6,7 3,7,8,9 .U
{ } { } { }3,4,5,6,7 3,7,8,9 3,4,5,6,7,8,9 .=U

The Set of Real Numbers
The sets that make up the Real Numbers, , are:
the set of Natural Numbers,
the set of Whole Numbers,
the set of Integers,
the set of Rational Numbers,
and the set of Irrational Numbers. Irrational numbers
cannot be expressed as a quotient of integers.
{1,2,3,4,5,...}
{0,1,2,3,4,5,...}
{..., 5, 4, 3, 2, 1,0,1,2,3,4,5,...}− − − − −
{ }and are integers and 0
a
a b b
b
= ≠¤
¡

The Set of Real Numbers (continued)
The set of real numbers is the set of numbers that are
either rational or irrational:
{ }is rational or is irrational .x x x

Example: Recognizing Subsets of the Real Numbers
Consider the following set of numbers:
List the numbers in the set that are natural numbers.
, is the only natural number in the set.
List the numbers in the set that are whole numbers.
The whole numbers in the set are 0 and .
{ }9, 1.3,0,0.3, , 9, 10 .
2
π
− −
9 3= 9
9

(continued)
List the numbers in the set that are integers.
The numbers in the set that are integers are
–9, 0, and
{ }9, 1.3,0,0.3, , 9, 10 .
2
π
− −
9 3.=

List the numbers in the set that are rational numbers.
The numbers in the set that are rational numbers are
and
List the numbers in the set that are irrational numbers.
The numbers in the set that are irrational numbers are
and .
{ }9, 1.3,0,0.3, , 9, 10 .
2
π
− −
9 3.=
2
π 10
9, 1.3,0,0.3,− −

The Real Number Line
The real number line is a graph used to represent the set of
real numbers. An arbitrary point, called the origin, is labeled
0. The distance from 0 to 1 is called the unit distance.
Numbers to the right of the origin are positive and numbers
to the left of the origin are negative. On the real number
line, the real numbers increase from left to right.

The Real Number Line (continued)
We say that there is a one-to-one correspondence
between all the real numbers and all points on a real
number line: every real number corresponds to a point
on the number line and every point on the number line
corresponds to a real number.

Inequality Symbols
The following symbols are called inequality symbols.
These symbols always point to the lesser of the two real
numbers when the inequality statement is true.
means that a is less than b
means that a is less than or equal to b
means that a is greater than b
means that a is greater than or equal to b
When we compare the size of real numbers we say that
we are ordering the real numbers.
a b<
a b≤
a b>
a b≥

Absolute Value
The absolute value of a real number a, denoted , is
the distance from 0 to a on the number line. The
distance is always taken to be nonnegative.
Definition of absolute value
if 0
if 0
x x
x
x x
≥
= 
− <
x

Example: Evaluating Absolute Value
Rewrite the expression without absolute value bars:
answer:
Rewrite the expression without absolute value bars:
answer:
2 1− 2 1 2 1− = −
3π − 3 3π π− = −

Properties of Absolute Value

Distance between Points on a Real Number Line
If a and b are any two points on a real number line, then
the distance between a and b is given by
ora b− .−b a

Example: Distance between Two Points on a Number Line
Find the distance between –4 and 5 on the real number line.
Because the distance between –4 and 5 on the real number
line is given by , the distance between –4 and 5 isa b−
4 5 9 9− − = − =

Properties of the Real Numbers
The Commutative Property of Addition
a + b = b + a
The Commutative Property of Multiplication
ab = ba
The Associative Property of Addition
(a + b) + c = a + (b + c)
The Associative Property of Multiplication
(ab)c = a(bc)

Properties of the Real Numbers (continued)
The Distributive Property of Multiplication over
Addition
a(b + c) = ab + ac
The Identity Property of Addition
a + 0 = a
The Identity Property of Multiplication
1a a=g

Properties of the Real Numbers (continued)
The Inverse Property of Addition
The Inverse Property of Multiplication
( ) 0a a+ − =
1
1a
a
=g

Definitions of Subtraction and Division

Simplifying Algebraic Expressions
To simplify an algebraic expression we combine like
terms. Like terms are terms that have exactly the same
variable factors. An algebraic expression is simplified
when parentheses have been removed and like terms
have been combined.

Example: Simplifying an Algebraic Expression
Simplify:
2 2
7(4 3 ) 2(5 ).+ + +x x x x
2 2
7(4 3 ) 2(5 )x x x x+ + +
2 2
24 37 27 5x x x x= + + +g g g g
2 2
28 21 10 2x x x x= + + +
2 2
(28 10 ) (21 2 )x x x x= + + +
2
38 23x x= +

Properties of Negatives

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