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Elementary Algebra

Exam 1 Material

Familiar Sets of NumbersNatural numbersNumbers used in counting:1, 2, 3, (Does not include zero)Whole numbersIncludes zero and all natural numbers:0, 1, 2, 3, (Does not include negative numbers)FractionsRatios of whole numbers where bottom number can not be zero:

Prime NumbersNatural Numbers, not including 1, whose only factors are themselves and 12, 3, 5, 7, 11, 13, 17, 19, 23, etc.

What is the next biggest prime number?29

Composite NumbersNatural Numbers, bigger than 1, that are not prime4, 6, 8, 9, 10, 12, 14, 15, 16, etc.

Composite numbers can always be factored as a product (multiplication) of prime numbers

Factoring NumbersTo factor a number is to write it as a product of two or more other numbers, each of which is called a factor12 = (3)(4)3 & 4 are factors12 = (6)(2)6 & 2 are factors12 = (12)(1)12 and 1 are factors12 = (2)(2)(3)2, 2, and 3 are factorsIn the last case we say the 12 is completely factored because all the factors are prime numbers

Hints for Factoring NumbersTo factor a number we can get two factors by writing any multiplication problem that comes to mind that is equal to the given numberAny factor that is not prime can then be written as a product of two other factorsThis process continues until all factors are primeCompletely factor 2828 = (4)(7)4 & 7 are factors, but 4 is not prime28 = (2)(2)(7)4 is written as (2)(2), both prime

In the last case we say the 28 is completely factored because all the factors are prime numbers

Other Hints for FactoringSome people prefer to begin factoring by thinking of the smallest prime number that evenly divides the given numberIf the second factor is not prime, they again think of the smallest prime number that evenly divides itThis process continues until all factors are primeCompletely factor 120120 = (2)(60) 60 is not prime, and is divisible by 2120 = (2)(2)(30) 30 is not prime, and is divisible by 2120 = (2)(2)(2)(15) 30 is not prime, and is divisible by 3120 = (2)(2)(2)(3)(5) all factors are prime

In the last case we say the 120 is completely factored because all the factors are prime numbers

Fundamental Principle of FractionsIf the numerator and denominator of a fraction contain a common factor, that factor may be divided out to reduce the fraction to lowest terms:Reduce to lowest terms by factoring:

Summarizing the Process of Reducing FractionsCompletely factor both numerator and denominatorApply the fundamental principle of fractions: divide out common factors that are found in both the numerator and the denominator

When to Reduce Fractions to Lowest TermsUnless there is a specific reason not to reduce, fractions should always be reduced to lowest termsA little later we will see that, when adding or subtracting fractions, it may be more important to have fractions with a common denominator than to have fractions in lowest terms

Multiplying FractionsFactor each numerator and denominatorDivide out common factors Write answer Example:

Dividing FractionsInvert the divisor and change problem to multiplication

Example:

Adding Fractions Having a Common Denominator Add the numerators and keep the common denominator

Example:

Adding Fractions Having a Different Denominators Write equivalent fractions having a least common denominatorAdd the numerators and keep the common denominatorReduce the answer to lowest terms

Finding the Least Common Denominator, LCD, of Fractions Completely factor each denominatorConstruct the LCD by writing down each factor the maximum number of times it is found in any denominator

Example of Finding the LCDGiven two denominators, find the LCD:,Factor each denominator:Construct LCD by writing each factor the maximum number of times its found in any denominator:

Writing Equivalent FractionsGiven a fraction, an equivalent fraction is found by multiplying the numerator and denominator by a common factorGiven the following fraction, write an equivalent fraction having a denominator of 72:Multiply numerator and denominator by 4:

Adding FractionsFind a least common denominator, LCD, for the fractionsWrite each fraction as an equivalent fraction having the LCDWrite the answer by adding numerators as indicated, and keeping the LCDIf possible, reduce the answer to lowest terms

Example

Find a least common denominator, LCD, for the rational expressions:Write each fraction as an equivalent fraction having the LCD:

Write the answer by adding or subtracting numerators as indicated, and keeping the LCD:

If possible, reduce the answer to lowest terms

Subtracting FractionsFind a least common denominator, LCD, for the fractionsWrite each fraction as an equivalent fraction having the LCDWrite the answer by subtracting numerators as indicated, and keeping the LCDIf possible, reduce the answer to lowest terms

Example

Find a least common denominator, LCD, for the rational expressions:Write each fraction as an equivalent fraction having the LCD:

Write the answer by adding or subtracting numerators as indicated, and keeping the LCD:

If possible, reduce the answer to lowest terms

Improper Fractions& Mixed NumbersA fraction is called improper if the numerator is bigger than the denominator

There is nothing wrong with leaving an improper fraction as an answer, but they can be changed to mixed numbers by doing the indicated division to get a whole number plus a fraction remainder

Likewise, mixed numbers can be changed to improper fractions by multiplying denominator times whole number, plus the numerator, all over the denominator

Doing Math Involving Improper Fractions & Mixed NumbersConvert all numbers to improper fractions then proceed as previously discussed

Homework ProblemsSection: 1.1Page: 11Problems: Odd: 7 29, 33 51, 55 69

MyMathLab Homework 1.1 for practiceMyMathLab Homework Quiz 1.1 is due for a grade on the date of our next class meeting

Exponential Expressions

3 is called the base4 is called the exponentAn exponent that is a natural number tells how many times to multiply the base by itselfExample: What is the value of 34 ?(3)(3)(3)(3) = 81

An exponent applies only to the base (what it touches)

Meanings of exponents that are not natural numbers will be discussed later

Order of OperationsMany math problems involve more than one math operationOperations must be performed in the following order:Parentheses (and other grouping symbols)ExponentsMultiplication and Division (left to right)Addition and Subtraction (left to right)It might help to memorize:Please Excuse My Dear Aunt Sally

Order of OperationsExample:PEMDAS

Example of Order of OperationsEvaluate the following expression:

Inequality SymbolsAn inequality symbol is used to compare numbers:Symbols include:greater than:greater than or equal to:less than:less than or equal to:not equal to:Examples:

.

Expressions InvolvingInequality SymbolsExpressions involving inequality symbols may be either true or falseDetermine whether each of the following is true or false:

Translating to Expressions Involving Inequality SymbolsEnglish expressions may sometimes be translated to math expressions involving inequality symbols:

Seven plus three is less than or equal to twelve

Nine is greater than eleven minus four

Three is not equal to eight minus six

Equivalent Expressions Involving Inequality SymbolsA true expression involving a greater than symbol can be converted to an equivalent statement involving a less then symbolReverse the expressions and reverse the direction of the inequality symbol5 > 2 is equivalent to:2 < 5

Likewise, a true expression involving a less than symbol can be converted to an equivalent statement involving a greater than symbol by the same processReverse the expressions and reverse the direction of the inequality symbol3 < 7 is equivalent to:7 > 3

Homework ProblemsSection: 1.2Page: 21Problems: Odd: 5 19, 23 49, 53 79, 83 85

MyMathLab Homework 1.2 for practiceMyMathLab Homework Quiz 1.2 is due for a grade on the date of our next class meeting

Terminology of AlgebraConstant A specific numberExamples of constants:

Variable A letter or other symbol used to represent a number whose value varies or is unknownExamples of variables:

Terminology of AlgebraExpression constants and/or variables combined in a meaningful way with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and rootsExamples of expressions:

Only the first of these expressions can be simplified, because we dont know the numbers represented by the variables

Terminology of AlgebraIf we know the number value of each variable in an expression, we can evaluate the expressionGiven the value of each variable in an expression, evaluate the expression means:Replace each variable with empty parenthesesPut the given number inside the pair of parentheses that has replaced the variableDo the math problem and simplify the answer

ExampleEvaluate the expression for :

Consider the next similar, but slightly different, example

ExampleEvaluate the expression for :

Notice the difference between this example and the previous one it illustrates the importance of using a parenthesis in place of the variable

ExampleEvaluate the expression for :

ExampleEvaluate the expression for :

Translating English Phrases Into Algebraic ExpressionsMany English phrases can be translated into algebraic expressions:Use a variable to indicate an unspecified numberIdentify key words that imply:AddSubtractMultiplyDivide

Phrases that Translate to AdditionEnglish Phrase

A number plus 5The sum of 3 and a number4 more than a numberA number increased by 8Algebra Expression

Phrases that Translate to SubtractionEnglish Phrase

4 less than a numberA number subtracted from 76 subtracted from a numbera number decreased by 92 minus a numberAlgebra Expression

Phrases that Translate to MultiplicationEnglish Phrase

7 times a numberthe product of 4 and a numberdouble a numberthe square of a numberAlgebra Expression

Phrases that Translate to DivisionEnglish Phrase

the quotient of 2 and a number

a number divided by 8

6 divided by a number

Algebra Expression

Phrases Translating to Expressions Involving Multiple Math OperationsEnglish Phrase

4 less than 3 times a number

the quotient of 5 and twice a number

6 times the difference between a number and 5Algebra Expression

Phrases Translating to Expressions Involving Multiple Math OperationsEnglish Phrase

the difference between 4 and 7 times a number

the quotient of a number and 5, subtracted from the number

the product of 3, and a number increased by 4Algebra Expression

EquationsEquation a statement that two expressions are equalEquations always contain an equal sign, but an expression does not have an equal signLike a statement in English, an equation may be true or falseExamples:

.

EquationsMost equations contain one or more variables and the truthfulness of the equation depends on the numbers that replace the variablesExample:What value of x makes this true?A number that can replace a variable to make an equation true is called a solution

Distinguishing Between Expressions & EquationsExpressions contain constants, variables and math operations, but NO EQUAL SIGN

Equations always CONTAIN AN EQUAL SIGN that indicates that two expressions have the same value

Solutions to EquationsEarlier we said that any numbers that can replace variables in an equation to make a true statement are called solutions to the equationSoon we will learn procedures for finding solutions to an equationFor now, if we have a set of possible solutions, we can find solutions by replacing the variables with possible solutions to see if doing so makes a true statement

Finding Solutions to Equations from a Given Set of NumbersFrom the following set of numbers, find a solution for the equation:

Check x = 3

Check x = 4

Check x = 5

Writing Equationsfrom Word StatementsThe same procedure is used as in translating English expressions to algebraic expressions, except that any statement of equality in the English statement is replaced by an equal signChange the following English statement to an equation, then find a solution from the set of numbersFour more than twice a number is ten

Homework ProblemsSection: 1.3Page: 29Problems: Odd: 13 55, 59 81

MyMathLab Homework 1.3 for practiceMyMathLab Homework Quiz 1.3 is due for a grade on the date of our next class meeting

Sets of NumbersNatural numbersNumbers used in counting:1, 2, 3, (Does not include zero)Whole numbersIncludes zero and all natural numbers:0, 1, 2, 3, (Does not include negative numbers)IntegersIncludes all whole numbers and their opposites (negatives):, -3, -2, -1, 0, 1, 2, 3,

Number LineDraw a line, choose a point on the line, and label it as 0Choose some unit of length and place a series of points, spaced by that length, left and right of the 0 pointPoints to the right of zero are labeled in order 1, 2, 3, Points to the left of zero starting at the point closest to zero and moving left are labeled in order, -1, -2, -3,

Notice that for any integer on the number line, there is another integer the same distance on the other side of zero that is the opposite of the firstA number line is used for graphing integers and other numbers

Graphing Integerson a Number LineTo graph an integer on a number line we place a dot at the point that corresponds to the given number and we label the point with the numberThe number label is called the coordinate of the pointGraph -2:

Rational NumbersThe next set of numbers to be considered will fill in some of the gaps between the integers on a number line

Rational numbersNumbers that can be written as the ratio of two integersThis includes all integers since they can be written as themselves over 1This includes all fractions and their opposites (- , , etc.)It also includes all decimals that either terminate ( .57 ) or have a a sequence of digits that form an infinitely repeating pattern at the end (.666, written as .6, etc.)

Graphing Rational NumbersPositive rational numbers will correspond to a point right of zero and negative rational numbers will correspond to a point left of zeroTo find the location of the point, consider the mixed number equivalent of the given numberIf the number is positive:go to the right to the whole numberdivide the next interval into the number of divisions indicated by the denominator of the fractioncontinue to the right from the whole number to the division indicated by the numeratorPlace a dot at that point and label it with the coordinate If the number is negative:go to the left to the whole numberdivide the next interval into the number of divisions indicated by the denominator of the fractioncontinue to the left from the whole number to the division indicated by the numeratorPlace a dot at that point and label it with the coordinate

Examples of GraphingRational NumbersGraph

Graph

Irr...