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CHAPTER 1
Introduction to
Algebra
1-1 Variables
DEFINITION
Variable – is a symbol used
to represent one or more
numbers. The numbers are
called values of the
variable.
DEFINITION
Variable Expression – an
expression that contains a
variable.
Examples: 4x, 10y, -1/2z
DEFINITION
Numerical Expression – an
expression that names a
particular number
Examples: 4.50 x 4, 6 + 2, 10-3
DEFINITION
Value of the Expression – the
number named by an
expression
Examples: 4.50 x 4 = 18
6+2=8
10-3 = 7
SUBSTITUTION PRINCIPLE
An expression may be replaced
by another expression that
has the same value.
Example: (42 ÷ 6) + 8
7+8
15
Section 1-2
Grouping Symbols
Order of Operations
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
DEFINITION
Grouping symbol – is a device,
such as a pair of parentheses,
used to enclose an expression
that should be simplified first.
Examples: ( ), { }, [ ], _____
Example 1
Simplify:
a) 6(5 – 3)
12
b) 6(5) – 3
27
Example 2
Simplify:
12 + 4
15 – 7
2
Example 3
Simplify:
18 – [52 ÷ (7 + 6)]
14
Example 4
Simplify:
19 – 7 + 12 ∙ 2 ÷ 8
15
Example 5
Evaluate
4x + 5y
3x – y
when x = 3 and y = 8.
Section 1-3
Equations
DEFINITION
Equation – is formed by placing
an equals sign between two
numerical or variable
expressions, called the sides of
the equation.
Examples: 11-7 = 4, 5x -1 = 9
DEFINITION
Open sentences – an equation or
inequality containing a variable.
Examples: y + 1= 1 + y
5x -1 = 9
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
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}
Section 1-4
Translating Words into
Symbols
Addition - Phrases
• The sum of 8 and x
• A number increased by 7
• 5 more than a number
Subtraction - Phrases
• The difference between a
number and 4
• A number decreased by 8
• 5 less than a number
• 6 minus a number
Multiplication - Phrases
• The product of 4 and a
number
• Seven times a number
• One third of a number
Division - Phrases
• The quotient of a number
and 8
• A number divided by 10
Section 1-5
Translating Sentences
into Equations
EXAMPLES
Twice the sum of a number
and four is 10
2(n +4) = 10
EXAMPLES
When a number is multiplied
by four and the result
decreased by six, the final
result is 10.
4n - 6 = 10
EXAMPLES
Three less than a number is
12.
x – 3 = 12
EXAMPLES
The quotient of a number and
4 is 8.
b/4 = 8
EXAMPLES
Write an equation to represent
the given information.
The distance traveled in 3 hours
of driving was 240 km.
Section 1-6
Translating Problems
into Equations
PROCEDURE
• Read the problem carefully
• Choose a variable to
represent the unknowns
• Reread the problem and write
an equation.
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
Section 1-7
A Problem Solving Plan
SOLVING A WORD
PROBLEM
1. Read the problems carefully.
Decide what unknown numbers
are asked for and what facts are
known. Making a sketch may
help
SOLVING A WORD
PROBLEM
2. Choose a variable and use it
with the given facts to
represent the unknowns
described in the problem.
SOLVING A WORD
PROBLEM
2. Reread the problem and write
an equation that represents
relationships among the
numbers in the problem.
SOLVING A WORD
PROBLEM
4. Solve the equation and find the
unknowns asked for.
5. Check your results with the
words of the problem. Give the
answer.
EXAMPLES
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
• Jason has one and a half
times as many books as
Ramone. Together they
have 45 books. How many
books does each boy have?
Translation
Let r = number of Ramone’s
books
Then 1.5r = number of
Jason’s books
r + 1.5r = 45
Solution
r + 1.5r = 45
2.5r = 45
r = 18
Examples
Phillip has $23 more than
Kevin. Together they
have $187. How much
does each have?
Section 1-8
Number Lines
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
Inequality Symbols
Are used to show the order of
two real numbers
> means “is greater than”
< means “is less than”
Section 1-9
Opposites and Absolute
Values
OPPOSITES A pair of numbers
differing in sign only
{-4, 4} , {10, -10}, {½, -½}
RULES
1. If a is positive, then –a
is negative
2. If a is negative, then
–a is positive.
RULES
3. If a = 0, then –a = 0
4.The opposite of –a is
a; that is, -(-a) = a
ABSOLUTE VALUES
The absolute value of a
number a is denoted by |a|,
and it may be thought of as
the distance between the
graph of the number and the
origin on a number line.
EXAMPLES
|8| = 8
|-8| = 8
|0|= 0
|-4| + |7|= 11
The End