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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
BA
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.
2.
Each point on a number line is
paired with exactly one real
number, called the
coordinate of the point.
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
3x4
24 ÷2
5 + 5 +2 15 - 3
12
2x6
VALUE of a Numerical
Expression
The number represented by the
expression
Twelve is the value of
3x4
5 + 5 +2 15 - 3
24 ÷2
12
2x6
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
INEQUALITY
a sentence formed by placing
an inequality symbol between
two expressions, called the
sides of the inequality
-3 > -5
-3 < - 0.3
SUM
the result of adding numbers,
called the terms of the sum
6 + 15 = 21
10 + 2 = 12
terms
sum
DIFFERENCE
the result of subtracting one
number from another
8–6=2
10 - 2 = 8
difference
PRODUCT
the result of multiplying
numbers, called the factors of
the product
6 x 15 = 80
10 · 2 = 20
factors product
QUOTIENT
the result of dividing one
number by another
35 ÷ 7 = 5
10 ÷ 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.
1
First power: 3 = 3
Second power: 3 x 3 = 32
3
Third power 3 x 3 x 3 = 3
Exponent is 1,2,3
GROUPING SYMBOLS
Pairs 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 OPERATIONS
1.
2.
3.
Grouping symbols
Simplify powers
Perform multiplications and
divisions in order from left to
right. and
ORDER OF OPERATIONS
4.
5.
Perform additions and
subtractions in order from left to
right
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 = a
 Symmetric Property - If a =
b, then b = a
 Transitive 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 unique
7 + 5 = 12
7 x 5 = 35
COMMUTATIVE
PROPERTIES
a+b =b+a
ab = ba
2+6=6+2
2x6=6x2
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
3x1=1x3=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 PROPERTIES
 PROPERTY
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 b
1.
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 b
2.
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 b
3.
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 values
l(-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
DEFINITION
Open 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 = 5
3 is a solution or root because
2·3 -1= 5 is true
DEFINITION
Solution Set – the set of all
solutions of an open sentence.
Finding the solution set is called
solving the sentence.
Examples: y(4 - y) = 3
when 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) = 3
when y{0,1,2,3}
and
2
y – 4y = -3
y  {1,3}
DEFINITION
Empty set – the set with no
members and is denoted
by 
DEFINITION
Identity – the solution set is
the set of all real numbers.
DEFINITION
Formula – is an equation that
states a relationship between
two or more variables usually
representing physical or
geometric quantities.
Examples: d = rt
A = lw
Transformations that Produce
Equivalent Equations
1.
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 NUMBERS
Integers – {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 Problems
2.
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 Problems
3.
Reread the problem. Then
write an equation that
represents relationships
among the numbers in the
problem.
Plan for Solving Word Problems
4.
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