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Chapter PPrerequisites: 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.

24 341 3194T x x= + +


Example: Using a Mathematical Model (continued)

24 341 3194T x x= + +24(15) 314(15) 3194T = + +

4(225) 341(15) 3194T = + +

900 5115 3194T = + +9209T =

The formula indicates that for the school year ending in2015, 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 0aa 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 .2π− −

9 3= 9

9


Example: Recognizing Subsets of the Real Numbers (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 .2π− −

9 3.=


Example: Recognizing Subsets of the Real Numbers


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 .2π− −

9 3.=

2π 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 xx

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

or a 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 is a 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+ − =

11a

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 27(4 3 ) 2(5 ).+ + +x x x x

2 27(4 3 ) 2(5 )x x x x+ + +2 224 37 27 5x x x x= + + +g g g g

2 228 21 10 2x x x x= + + +2 2(28 10 ) (21 2 )x x x x= + + +

238 23x x= +


Properties of Negatives

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