Intermediate Algebra Intermediate Algebra

Full text

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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

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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”

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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

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Individual numbers are dots

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-1 0 1 2 3 4 -2

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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

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Intervals including end points

[

[ ]

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-1 0 1 2 3 4 -2

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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

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Intervals not including end points

(

( )

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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)

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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

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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

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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

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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

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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

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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

nn

a 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

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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.

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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

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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!

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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.

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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.

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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

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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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