Chapter 1

Basic Concepts of Algebra

LANGUAGE OF ALGEBRA

SET– a collection or group of, things, objects,

numbers, etc.

INFINITE SET – a set whose members cannot

be counted.

If A= {1, 2, 3, 4, 5,…} then A is infinite

FINITE SET – a set whose members can be

counted.

If A= {e, f, g, h, i, j} then A is finite and

contains six elements

SUBSET – all members of a set are members of

another set

If A= {e, f, g, h, i, j} and B = {e, i} ,

then BA

EMPTY SET or NULL SET – a set having no

elements.

A= { } or B = { } are empty sets or null sets written as

1-1

Real Numbers and Their Graphs

Real Numbers

NATURAL NUMBERS - set of counting numbers

{1, 2, 3, 4, 5, 6, 7, 8…}

WHOLE NUMBERS - set of counting numbers plus

zero

{0, 1, 2, 3, 4, 5, 6, 7, 8…}

INTEGERS - set of the whole numbers plus their

opposites

{…, -3, -2, -1, 0, 1, 2, 3, …}

RATIONAL NUMBERS - numbers that can be

expressed as a ratio of two integers a and b and

includes fractions, repeating decimals, and terminating

decimals

EXAMPLES OF RATIONAL NUMBERS

½, ¾, ¼, - ½, -¾, -¼, .05

.76, .333…, .666…, etc.

IRRATIONAL NUMBERS -numbers that cannot be

expressed as a ratio of two integers a and b and can still be designated on a number

line

EXAMPLES OF IRRATIONAL NUMBERS

, 6, -29, 8.11211121114…,

etc.

1. Each point on a number line is paired with exactly one real number, called the coordinate of the point.

2. Each real number is paired with exactly one point on the line, called the graph of the number

1-2

Simplifying Expressions

Definitions

NUMERICAL EXPRESSION or NUMERAL

a symbol or group of symbols used to represent a number

3 x 4 5 + 5 +2 15 - 324 ÷2 12 2 x 6

VALUE of a Numerical Expression

The number represented by the expression

Twelve is the value of 3 x 4 5 + 5 +2 15 - 324 ÷2 12 2 x 6

EQUATION a sentence formed by placing an equals sign = between two

expressions, called the sides of the equation. The equation is a

true statement if both sides have the same value.

EXAMPLES OF EQUATIONS

-6 + 10 = 6 – 2 or 4x + 3 = 19

INEQUALITY SYMBOL One of the symbols

< - less than > greater than

≠ - does not equal≤ - less than or equal to

≥ - greater than or equal to

INEQUALITYa sentence formed by placing an inequality symbol between

two expressions, called the sides of the inequality

-3 > -5-3 < - 0.3

SUMthe result of adding numbers, called the terms of the sum

6 + 15 = 2110 + 2 = 12

terms sum

DIFFERENCEthe result of subtracting one

number from another8 – 6 = 210 - 2 = 8

difference

PRODUCTthe result of multiplying

numbers, called the factors of the product6 x 15 = 8010 · 2 = 20

factors product

QUOTIENTthe result of dividing one

number by another35 ÷ 7 = 510 ÷ 2 = 5

quotient

POWER, BASE, and EXPONENT

A power is a product of equal factors. The repeated factor is the base. A positive exponent tells the number of times the

base occurs as a factor.

EXAMPLES OF POWER, BASE, and EXPONENT

Let the base be 3.First power: 3 = 31

Second power: 3 x 3 = 32

Third power 3 x 3 x 3 = 33

Exponent is 1,2,3

GROUPING SYMBOLSPairs of parentheses ( ),

brackets [ ], braces { }, or a bar — used to enclose part

of an expression that represents a single number.

{ 3 + 4[(2 x 6) -22] ÷ 2}

VARIABLE – a symbol, usually a letter, used to

represent any member of a given set, called the domain or replacement set, of the

variable a, x, or y

EXAMPLES OF VARIABLES

If the domain of x is {0,1,2,3}, we write

x {0,1,2,3}

VALUE of a Variable - the members of the domain of the variable. If the domain of a is the set of positive integers, then a can have these values: 1,2,3,4,…

Algebraic Expression – a numerical expression; a

variable; or a sum, difference, product, or

quotient that contains one or more variables

EXAMPLES OF ALGEBRAIC

EXPRESSIONS 24 + 3 + x y2 – 2y + 6

a + b 2c2d – 4 c d

SUBSTITUTION PRINCIPLE

An expression may be replaced by another

expression that has the same value.

ORDER OF OPERATIONS1. Grouping symbols

2. Simplify powers

3. Perform multiplications and divisions in order from left to right. and

ORDER OF OPERATIONS4. Perform additions and

subtractions in order from left to right

5. Simplify the expression within each grouping symbol, working outward from the innermost grouping

DEFINITION of ABSOLUTE VALUE

For each real number a,

l a l = a if a >0

0 if a = 0

- a if a < 0

1-3

Basic Properties of Real Numbers

Properties of Equality

Reflexive Property - a = aSymmetric Property - If a =

b, then b = aTransitive Property - If a = b,

and b = c, then a = c

Addition Property - If a = b, then a + c = b + c and c + a = c + b

Multiplication Property -If a = b, then ac = bc and ca = cb

Properties of Real Numbers

CLOSURE PROPERTIES

a + b and ab are unique7 + 5 = 127 x 5 = 35

COMMUTATIVE PROPERTIES

a + b = b + a

ab = ba

2 + 6 = 6 + 22 x 6 = 6 x 2

ASSOCIATIVE PROPERTIES

(a + b) + c = a + (b

+c)(ab)c = a(bc)(5 + 15) + 20 = 5 + (15

+20)(5·15)20 = 5(15·20)

IDENTITY PROPERTIES

There are unique real numbers 0 and 1 (1≠0)

such that:a + 0 = 0 + a = a

a · 1 = 1 ·a-3 + 0 = 0 + -3 = -3

3 x 1 = 1 x 3 = 3

INVERSE PROPERTIES

PROPERTY OF OPPOSITES

For each a, there is a unique real number – a such that:

a + (-a) = 0 and (-a)+ a = 0 (-a is called the opposite or additive inverse of a

INVERSE PROPERTIESPROPERTY OF RECIPROCALS For each a except 0, there is a

unique real number 1/a such that:

a · (1/a) = 1 and (1/a)· a = 1 (1/a is called the reciprocal or multiplicative inverse of a

DISTRIBUTIVE PROPERTY

a(b + c) = ab + ac(b +c)a = ba + ca

5(12 + 3) = 5•12 + 5 •3 = 75

(12 + 3)5 = 12• 5 + 3 • 5 = 75

1-4

Sums and Differences

Rules for Addition

For real numbers a and b1. If a and b are negative

numbers, then a + b is negative and a + b = -(lal + lbl) -5 + (-9) = - (l-5l + l-9l) = -14

For real numbers a and b2. If a is a positive number, b

is a negative number, and lal is greater than lbl, then a + b is a positive number and a + b = lal – lbl 9 + (-5) = l9l – l-5l = 4

For real numbers a and b3. If a is a positive number, b

is a negative number, and lal is less than lbl, then a + b is a negative number and a + b = -lbl – lal 5 + (-9) = -l-9l – l5l = -4

DEFINITION of SUBTRACTION

For all real number a and b,

a – b = a + (-b)

To subtract any real number, add its opposite

DISTRIBUTIVE PROPERTY

For all real number a ,b, and c

a(b - c)= ab – ac

and

(b – c)a = ba - ca

1-5

Products

MULTIPLICATIVE PROPERTY OF 0

For every real number a,

a · 0 = 0 and 0 · a = 0

MULTIPLICATIVE PROPERTY OF -1

For every real number a,a(-1) = -a and (-1)a = -a

Rules for Multiplication

1. The product of two positive numbers or two negative numbers is a positive number.(5)(9) = 45 or (-5)(-9) = 45

2. The product of a positive number and a negative number is a negative number.(-5)(9) = -45 or (5)(-9) = -45

3. The absolute value of the product of two or more numbers is the product of their absolute valuesl(-5)(9)l = l-5l l9l = 45

PROPERTY of the OPPOSITE of a

PRODUCT For all real number a and

b,-ab = (-a)b

and-ab = a(-b)

PROPERTY of the OPPOSITE of a SUM

For all real number a and

b,-(a + b) = (-a) + (-b)

1-6

Quotients

DEFINITION OF DIVISION

The quotient a divided by b is written a/b or a÷b. For every real number a and nonzero real number b,

a/b = a·1/b, or a÷b = a·1/b

DEFINITION OF DIVISION

To divide by any nonzero number, multiply by its reciprocal. Since 0 has no reciprocal, division by 0 is not defined.

Rules for Division

1. The quotient of two positive numbers or two negative numbers is a positive number-24/-3 = 8 and 24/3 = 8

2. The quotient of two numbers when one is positive and the other negative is a negative number.

24/-3 = -8 and -24/3 = -8

PROPERTY

For all real numbers a and b and nonzero real number c,

(a + b)/c = a/c + b/c and

(a-b)/c = a/c – b/c

1-7 Solving Equations in One

Variable

DEFINITIONOpen sentences – an equation or inequality containing a variable.

Examples: y + 1= 1 + y

5x -1 = 9

DEFINITION

Solution – any value of the variable that makes an open sentence a true statement.

Examples: 2t – 1 = 53 is a solution or root because

2·3 -1= 5 is true

DEFINITIONSolution Set – the set of all

solutions of an open sentence. Finding the solution set is called solving the sentence.

Examples: y(4 - y) = 3when y{0,1,2,3}

y {1,3}

DEFINITION

Domain – the given set of numbers that a variable may represent

Example:

5x – 1 = 9

The domain of x is {1,2,3}

DEFINITION

Equivalent equations – equations having the same solution set over a given domain.

Examples: y(4 - y) = 3when y{0,1,2,3} and

y2 – 4y = -3 y {1,3}

DEFINITION

Empty set – the set with no members and is denoted by

DEFINITIONIdentity – the solution set is

the set of all real numbers.

DEFINITIONFormula – is an equation that

states a relationship between two or more variables usually representing physical or geometric quantities.

Examples: d = rtA = lw

Transformations that Produce Equivalent

Equations1. Simplifying either side of an

equation.

2. Adding to (or subtracting from) each side of an equation the same number or the same expression.

3. Multiplying (or dividing) each side of an equation by the same nonzero number.

1-8

Words into Symbols

CONSECUTIVE NUMBERSIntegers – {n-1, n, n+1}

{… -3, -2, -1, 0, 1, 2, 3,….}

Even Integers – {n-2, n, n+2}{…-4,-2, 0, 2, 4,….}

Odd Integers – {n-2, n, n+2}{…-5,-3, -1, 1, 3, 5,….}

Addition - Phrases

• The sum of 8 and x• A number increased by 7• 5 more than a number

Addition - Translation

• 8 + x• n + 7• n + 5

Subtraction - Phrases

• The difference between a number and 4

• A number decreased by 8• 5 less than a number• 6 minus a number

Subtraction - Translation

• x - 4• x- 8• n – 5• 6 - n

Multiplication - Phrases

• The product of 4 and a number

• Seven times a number• One third of a number

Multiplication - Translation

• 4n• 7n• 1/3x

Division - Phrases

• The quotient of a number and 8

• A number divided by 10

Division - Translation

• n/8• n/10

1-9 Problem Solving with

Equations

Plan for Solving Word Problems

1. Read the problem carefully. Decide what numbers are asked for and what information is given. Making a sketch may be helpful.

Plan for Solving Word Problems2. Choose a variable and use it

with the given facts to represent the number(s) described in the problem. Labeling your sketch or arranging the given information in a chart may help.

Plan for Solving Word Problems3. Reread the problem. Then

write an equation that represents relationships among the numbers in the problem.

Plan for Solving Word Problems4. Solve the equation and find the

required numbers.

5. Check your results with the original statement of the problem. Give the answer

EXAMPLES

Solve using the five-step plan.• Two numbers have a sum

of 44. The larger number is 8 more than the smaller. Find the numbers.

Solution

n + (n + 8) = 44

2n + 8 = 44

2n = 36

n = 18

EXAMPLES

Translate the problem into an equation.

(1) Marta has twice as much money as Heidi.

(2) Together they have $36.How much money does each have?

Translation

Let h = Heidi’s amount

Then 2h = Marta’s amount

h + 2h = 36

EXAMPLES

Translate the problem into an equation.

(1) A wooden rod 60 in. long is sawed into two pieces.

(2) One piece is 4 in. longer than the other.

What are the lengths of the pieces?

Translation

Let x = the shorter length

Then x + 4 = longer length

x + (x + 4) = 60

EXAMPLES

Translate the problem into an equation.

(1) The area of a rectangle is 102 cm2.

(2) The length of the rectangle is 6 cm.

Find the width of the rectangle?

Translation

Let w = width of rectangle

Then 6 = length of rectangle

6w = 102

EXAMPLES

Solve using the five-step plan.• Jason has one and a half

times as many books as Ramon. Together they have 45 books. How many books does each boy have?

Translation

Let b = number of Ramon’s books

Then 1.5b = number of Jason’s books

b + 1.5b = 45

Solution

b + 1.5b = 45

2.5b = 45

b = 18

The End