Transcript Inequalities and their Graphs

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