Introduction to Algebra

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CHAPTER 1

• Introduction to Algebra

2

• 1-1 Variables

3
DEFINITION

• Variable is a symbol used to represent one or
  more numbers. The numbers are called values of
  the variable.

4
DEFINITION

• Variable Expression an expression that contains
  a variable.
• Examples 4x, 10y, -1/2z

5
DEFINITION

• Numerical Expression an expression that names a
  particular number
• Examples 4.50 x 4, 6 2, 10-3

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DEFINITION

• Value of the Expression the number named by an
  expression
• Examples 4.50 x 4 18
• 6 2 8
• 10-3 7

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SUBSTITUTION PRINCIPLE

• An expression may be replaced by another
  expression that has the same value.
• Example (42 6) 8
• 7 8
• 15

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Section 1-2

• Grouping Symbols

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Order of Operations

• Parenthesis
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

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DEFINITION

• Grouping symbol is a device, such as a pair of
  parentheses, used to enclose an expression that
  should be simplified first.
• Examples ( ), , , _____

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Example 1

• Simplify
• a) 6(5 3)
• 12
• b) 6(5) 3
• 27

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Example 2

• Simplify
• 12 4
• 15 7
• 2

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Example 3

• Simplify
• 18 52 (7 6)
• 14

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Example 4

• Simplify
• 19 7 12 2 8
• 15

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Example 5

• Evaluate 4x 5y
• 3x y
• when x 3 and y 8.

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Section 1-3

• Equations

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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

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DEFINITION

• Open sentences an equation or inequality
  containing a variable.
• Examples y 1 1 y
• 5x -1 9

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DEFINITION

• Domain the given set of numbers that a variable
  may represent
• Example
• 5x 1 9
• The domain of x is 1,2,3

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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

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Section 1-4

• Translating Words into Symbols

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Addition - Phrases

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

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Subtraction - Phrases

• The difference between a number and 4
• A number decreased by 8
• 5 less than a number
• 6 minus a number

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Multiplication - Phrases

• The product of 4 and a number
• Seven times a number
• One third of a number

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Division - Phrases

• The quotient of a number and 8
• A number divided by 10

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Section 1-5

• Translating Sentences into Equations

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EXAMPLES

• Twice the sum of a number and four is 10
• 2(n 4) 10

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EXAMPLES

• When a number is multiplied by four and the
  result decreased by six, the final result is 10.
• 4n - 6 10

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EXAMPLES

• Three less than a number is 12.
• x 3 12

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EXAMPLES

• The quotient of a number and 4 is 8.
• b/4 8

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EXAMPLES

• Write an equation to represent the given
  information.
• The distance traveled in 3 hours of driving was
  240 km.

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Section 1-6

• Translating Problems into Equations

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PROCEDURE

• Read the problem carefully
• Choose a variable to represent the unknowns
• Reread the problem and write an equation.

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EXAMPLES

• Translate the problem into an equation.
• Marta has twice as much money as Heidi.
• Together they have 36.
• How much money does each have?

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Translation

• Let h Heidis amount
• Then 2h Martas amount
• h 2h 36

36
EXAMPLES

• Translate the problem into an equation.
• A wooden rod 60 in. long is sawed into two
  pieces.
• One piece is 4 in. longer than the other.
• What are the lengths of the pieces?

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Translation

• Let x the shorter length
• Then x 4 longer length
• x (x 4) 60

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EXAMPLES

• Translate the problem into an equation.
• The area of a rectangle is 102 cm2.
• The length of the rectangle is 6 cm.
• Find the width of the rectangle?

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Translation

• Let w width of rectangle
• Then 6 length of rectangle
• 6w 102

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Section 1-7

• A Problem Solving Plan

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SOLVING A WORD PROBLEM

• Read the problems carefully. Decide what unknown
  numbers are asked for and what facts are known.
  Making a sketch may help

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SOLVING A WORD PROBLEM

1. Choose a variable and use it with the given facts
   to represent the unknowns described in the
   problem.

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SOLVING A WORD PROBLEM

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

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SOLVING A WORD PROBLEM

• Solve the equation and find the unknowns asked
  for.
• Check your results with the words of the problem.
  Give the answer.

45
EXAMPLES

• Two numbers have a sum of 44. The larger number
  is 8 more than the smaller. Find the numbers.

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Solution

• n (n 8) 44
• 2n 8 44
• 2n 36
• n 18

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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?

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Translation

• Let r number of Ramones books
• Then 1.5r number of Jasons books
• r 1.5r 45

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Solution

• r 1.5r 45
• 2.5r 45
• r 18

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Examples

• Phillip has 23 more than Kevin. Together they
  have 187. How much does each have?

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Section 1-8

• Number Lines

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NATURAL NUMBERS - set of counting numbers
1, 2, 3, 4, 5, 6, 7, 8
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WHOLE NUMBERS set of counting numbers plus
zero
0, 1, 2, 3, 4, 5, 6, 7, 8
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INTEGERS set of the whole numbers plus their
opposites
, -3, -2, -1, 0, 1, 2, 3,
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RATIONAL NUMBERS - numbers that can be
expressed as a ratio of two integers a and b and
includes fractions, repeating decimals, and
terminating decimals
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EXAMPLES OF RATIONAL NUMBERS
½, ¾, ¼, - ½, -¾, -¼, .05 .76, .333, .666, etc.
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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
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Inequality Symbols

• Are used to show the order of two real numbers
• gt means is greater than
• lt means is less than

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Section 1-9

• Opposites and Absolute Values

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OPPOSITES - A pair of numbers differing in sign
only
-4, 4 , 10, -10, ½, -½
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• RULES
• If a is positive, then a is negative
• If a is negative, then
• a is positive.

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• RULES
• If a 0, then a 0
• The opposite of a is a that is, -(-a) a

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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.

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EXAMPLES

• 8 8
• -8 8
• 0 0
• -4 7 11

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• The End