Algebra review

Intermediate AlgebraIntermediate Algebraby Gustafson and Friskby Gustafson and Frisk

Chapter 1Chapter 1

A Review of Basic AlgebraA Review of Basic Algebra

Section 1.1: The Real Number Section 1.1: The Real Number SystemSystem

SETS: SETS: collections of objects.collections of objects.

Natural NumbersNatural Numbers Whole NumbersWhole Numbers Rational NumbersRational Numbers Irrational NumbersIrrational Numbers Real NumbersReal Numbers

IntegersIntegers Positive NumbersPositive Numbers Negative NumbersNegative Numbers Even NumbersEven Numbers Odd NumbersOdd Numbers

Use { } {x | x > 5}{x | x > 5}is read “the set of all x such is read “the set of all x such that x is greater than 5”that x is greater than 5”


GRAPHS: GRAPHS: plot on the number line.plot on the number line.

-1 0 31 42 -2 -3

Individual numbers are dots

-1 0 31 42 -2 -3



-1 0 31 42 -2 -3

Intervals including end points

-1 0 31 42 -2 -3



-1 0 31 42 -2 -3

Intervals not including end points

Section 1.2: Arithmetic & Properties of Real Section 1.2: Arithmetic & Properties of Real NumbersNumbers

OPERATIONS: OPERATIONS:

AdditionAddition Subtraction (the same as adding Subtraction (the same as adding

a number with the opposite sign)a number with the opposite sign)

MultiplicationMultiplication Division (the same as multiplying Division (the same as multiplying

by the reciprocal)by the reciprocal)


ADDITION: ADDITION:

Addends that have opposite signsAddends that have opposite signs Subtract absolute valuesSubtract absolute values Keep the sign of the addend with Keep the sign of the addend with

the largest absolute valuethe largest absolute value

Addends that have the same signsAddends that have the same signs Add absolute valuesAdd absolute values Keep the sign of the addendsKeep the sign of the addends


MULTIPLICATION: MULTIPLICATION:

Multiply absolute valuesMultiply absolute values If the factors have the same If the factors have the same

signs, signs, the product is positivethe product is positive

If the factors have opposite signs, If the factors have opposite signs, the product is negativethe product is negative


STATISTICS: STATISTICS: measures of central measures of central tendencytendency

Mean Mean Median Median ModeMode


Properties: Properties:

Associative – addition, Associative – addition, multiplicationmultiplication

Commutative – addition, multiplicationCommutative – addition, multiplication

Distributive – multiplication is Distributive – multiplication is distributed over additiondistributed over additiona (b + c) = ab + aca (b + c) = ab + ac


Identities: Identities:

Addition – zeroAddition – zero Multiplication – oneMultiplication – one

Inverses: Inverses:

Addition – oppositesAddition – opposites Multiplication – reciprocalsMultiplication – reciprocals

Section 1.3: Definition of ExponentsSection 1.3: Definition of Exponents

EXPONENTS: EXPONENTS: repeated multiplicationrepeated multiplication

In the expression: aIn the expression: ann a is the base and n is the a is the base and n is the exponentexponent

Exponents are Exponents are NOTNOT factors factors Means to multiply “a” n timesMeans to multiply “a” n times

Section 1.3: Definition of ExponentsSection 1.3: Definition of Exponents

ORDER OF OPERATIONS: ORDER OF OPERATIONS:

If an algebraic expression has more than If an algebraic expression has more than one operation, the following order one operation, the following order applies: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 Do multiplication and division from left to right.to right.

4.4. Do addition and subtraction from left Do addition and subtraction from left to right.to right.

Section 1.5: Solving EquationsSection 1.5: Solving Equations

Algebraic Expression vs. EquationAlgebraic Expression vs. Equation

Expressions: a combination of Expressions: a combination of numbers and operationsnumbers and operations

Equation: a statement that two Equation: a statement that two expressions are equalexpressions are equal


EXPRESSIONS: EXPRESSIONS:

TermsTerms Like termsLike terms When multiplying, the terms do When multiplying, the terms do

not need to be alikenot need to be alike Can only add like terms!Can only add like terms!


TO SOLVE AN EQUATION IN ONE TO SOLVE AN EQUATION IN ONE VARIABLE:VARIABLE:

If you see fractions, If you see fractions, multiply both sides by the multiply both sides by the LCDLCD. This will eliminate the fractions.. This will eliminate the fractions.

SimplifySimplify the algebraic expressions on each side the algebraic expressions on each side of the equal sign (eliminate parentheses and of the equal sign (eliminate parentheses and combine like terms).combine like terms).

Use the addition property of equality to Use the addition property of equality to isolate isolate the variable terms from the constant termsthe variable terms from the constant terms on on opposite sides of the equal sign.opposite 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.


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 model a relationship between the variables.variables.

Section 1.6 & 1.7: Solving Problems, Section 1.6 & 1.7: Solving Problems, ApplicationsApplications

TYPES OF PROBLEMS:TYPES OF PROBLEMS: GeometryGeometry PercentPercent Physics (forces)Physics (forces) Uniform motionUniform motion MixturesMixtures Good ‘ole common sense analysisGood ‘ole common sense analysis

Chapter 1: Basic Algebra ReviewChapter 1: Basic Algebra Review

SUMMARY:SUMMARY:

KNOW YOUR VOCABULARY! KNOW YOUR VOCABULARY! You can’t You can’t follow directions if you don’t know follow directions if you don’t know what the words in the instructions what the words in the instructions mean.mean.

Memorize the processes and the Memorize the processes and the properties.properties.

I will provide formulas for your I will provide formulas for your reference.reference.

Ask questions if you are unsure.Ask questions if you are unsure. Always check your work to make sure Always check your work to make sure

that you answered the question, and that you answered the question, and that your answer is reasonable.that your answer is reasonable.

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Algebra review