Intermediate Algebra Intermediate Algebra
by Gustafson and Frisk by Gustafson and Frisk
Chapter 1 Chapter 1
A Review of Basic Algebra
A Review of Basic Algebra
Section 1.1: The Real Number System Section 1.1: The Real Number System
SETS:
SETS: collections of objects. collections of objects.
Natural Numbers Natural Numbers
Whole Numbers Whole Numbers
Rational Numbers Rational Numbers
Irrational Numbers Irrational Numbers
Real Numbers Real Numbers
Integers Integers
Positive Numbers Positive Numbers
Negative Numbers Negative Numbers
Even Numbers Even Numbers
Odd Numbers Odd Numbers
Use { } {x | x > 5} {x | x > 5}
is read “the set of all x such that is read “the set of all x such that
x is greater than 5”
x is greater than 5”
Section 1.1: The Real Number System Section 1.1: The Real Number System
GRAPHS:
GRAPHS: plot on the number line. plot on the number line.
-1 0 1 2 3 4 -2
-3
Individual numbers are dots
-1 0 1 2 3 4 -2
-3
Section 1.1: The Real Number System Section 1.1: The Real Number System
GRAPHS:
GRAPHS: plot on the number line. plot on the number line.
-1 0 1 2 3 4 -2
-3
Intervals including end points
[
[ ]
-1 0 1 2 3 4 -2
-3
Section 1.1: The Real Number System Section 1.1: The Real Number System
GRAPHS:
GRAPHS: plot on the number line. plot on the number line.
-1 0 1 2 3 4 -2
-3
Intervals not including end points
(
( )
Section 1.2: Arithmetic & Properties of Real Numbers Section 1.2: Arithmetic & Properties of Real Numbers
OPERATIONS:
OPERATIONS:
Addition Addition
Subtraction (the same as adding a Subtraction (the same as adding a number with the opposite sign)
number with the opposite sign)
Multiplication Multiplication
Division (the same as multiplying by Division (the same as multiplying by the reciprocal)
the reciprocal)
Section 1.2: Arithmetic & Properties of Real Numbers Section 1.2: Arithmetic & Properties of Real Numbers
ADDITION:
ADDITION:
Addends that have opposite signs Addends that have opposite signs
Subtract absolute values Subtract absolute values
Keep the sign of the addend with the Keep the sign of the addend with the largest absolute value
largest absolute value
Addends that have the same signs Addends that have the same signs
Add absolute values Add absolute values
Keep the sign of the addends Keep the sign of the addends
Section 1.2: Arithmetic & Properties of Real Numbers Section 1.2: Arithmetic & Properties of Real Numbers
MULTIPLICATION:
MULTIPLICATION:
Multiply absolute values Multiply absolute values
If the factors have the same signs, If the factors have the same signs, the product is positive
the product is positive
If the factors have opposite signs, If the factors have opposite signs, the product is negative
the product is negative
Section 1.2: Arithmetic & Properties of Real Numbers Section 1.2: Arithmetic & Properties of Real Numbers
STATISTICS:
STATISTICS: measures of central tendency measures of central tendency
Mean Mean
Median Median
Mode Mode
Section 1.2: Arithmetic & Properties of Real Numbers Section 1.2: Arithmetic & Properties of Real Numbers
Properties:
Properties:
Associative – addition, multiplication Associative – addition, multiplication
Commutative – addition, multiplication Commutative – addition, multiplication
Distributive – multiplication is Distributive – multiplication is distributed over addition
distributed over addition a (b + c) = ab + ac
a (b + c) = ab + ac
Section 1.2: Arithmetic & Properties of Real Numbers Section 1.2: Arithmetic & Properties of Real Numbers
Identities:
Identities:
Addition – zero Addition – zero
Multiplication – one Multiplication – one
Inverses:
Inverses:
Addition – opposites Addition – opposites
Multiplication – reciprocals Multiplication – reciprocals
Section 1.3: Definition of Exponents Section 1.3: Definition of Exponents
EXPONENTS:
EXPONENTS: repeated multiplication repeated multiplication In the expression: a
In the expression: a
nna is the base and n is the exponent a is the base and n is the exponent
Exponents are Exponents are NOT NOT factors factors
Means to multiply “a” n times Means to multiply “a” n times
Section 1.3: Definition of Exponents Section 1.3: Definition of Exponents
ORDER OF OPERATIONS:
ORDER OF OPERATIONS:
If an algebraic expression has more than one If an algebraic expression has more than one
operation, the following order applies:
operation, the following order applies:
1.1.
Clear up any grouping. Clear up any grouping.
2.2.
Evaluate exponents. Evaluate exponents.
3.3.
Do multiplication and division from left to Do multiplication and division from left to right.
right.
4.4.
Do addition and subtraction from left to Do addition and subtraction from left to right.
right.
Section 1.5: Solving Equations Section 1.5: Solving Equations
Algebraic Expression vs. Equation Algebraic Expression vs. Equation
Expressions: a combination of Expressions: a combination of numbers and operations
numbers and operations
Equation: a statement that two Equation: a statement that two expressions are equal
expressions are equal
Section 1.5: Solving Equations Section 1.5: Solving Equations
EXPRESSIONS:
EXPRESSIONS:
Terms Terms
Like terms Like terms
When multiplying, the terms do not When multiplying, the terms do not need to be alike
need to be alike
Can only add like terms! Can only add like terms!
Section 1.5: Solving Equations Section 1.5: Solving Equations
TO SOLVE AN EQUATION IN ONE VARIABLE:
TO SOLVE AN EQUATION IN ONE VARIABLE:
If you see fractions, If you see fractions, multiply both sides by the LCD multiply both sides by the LCD . . This will eliminate the fractions.
This will eliminate the fractions.
Simplify Simplify the algebraic expressions on each side of the the algebraic expressions on each side of the equal sign (eliminate parentheses and combine like equal sign (eliminate parentheses and combine like
terms).
terms).
Use the addition property of equality to Use the addition property of equality to isolate isolate the the variable terms from the constant terms
variable terms from the constant terms on opposite on opposite sides of the equal sign.
sides of the equal sign.
Use the multiplication property to make the Use the multiplication property to make the coefficient of the variable equal to one.
coefficient of the variable equal to one.
Check your results by evaluating. Check your results by evaluating.
Section 1.5: Solving Equations Section 1.5: Solving Equations
TYPES OF EQUATIONS:
TYPES OF EQUATIONS:
CONDITIONAL: if x equals this, then y CONDITIONAL: if x equals this, then y equals that.
equals that.
IDENTITY: always true no matter what IDENTITY: always true no matter what numbers you use.
numbers you use.
CONTRADICTION: never true no matter CONTRADICTION: never true no matter what numbers you use.
what numbers you use.
FORMULAS: conditional equations that FORMULAS: conditional equations that
model a relationship between the variables.
model a relationship between the variables.
Section 1.6 & 1.7: Solving Problems, Applications Section 1.6 & 1.7: Solving Problems, Applications
TYPES OF PROBLEMS:
TYPES OF PROBLEMS:
Geometry Geometry
Percent Percent
Physics (forces) Physics (forces)
Uniform motion Uniform motion
Mixtures Mixtures
Good ‘ole common sense analysis Good ‘ole common sense analysis
Chapter 1: Basic Algebra Review Chapter 1: Basic Algebra Review
SUMMARY:
SUMMARY:
KNOW YOUR VOCABULARY! KNOW YOUR VOCABULARY! You can’t follow You can’t follow directions if you don’t know what the words directions if you don’t know what the words
in the instructions mean.
in the instructions mean.
Memorize the processes and the properties. Memorize the processes and the properties.
I will provide formulas for your reference. I will provide formulas for your reference.
Ask questions if you are unsure. Ask questions if you are unsure.
Always check your work to make sure that Always check your work to make sure that you answered the question, and that your you answered the question, and that your
answer is reasonable.
answer is reasonable.
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