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- Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers.

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Elementary AlgebraExam 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?29Composite 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 numbersHints 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 primeIn the last case we say the 28 is completely factored because all the factors are prime numbersOther 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 primeIn the last case we say the 120 is completely factored because all the factors are prime numbersFundamental 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 denominatorWhen 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 termsMultiplying FractionsFactor each numerator and denominatorDivide out common factors Write answer Example:Dividing FractionsInvert the divisor and change problem to multiplicationExample:Adding Fractions Having a Common Denominator Add the numerators and keep the common denominatorExample: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 termsFinding 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 denominatorExample 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 termsExampleFind 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 termsSubtracting 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 termsExampleFind 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 termsImproper Fractions& Mixed NumbersA fraction is called improper if the numerator is bigger than the denominatorThere 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 remainderLikewise, mixed numbers can be changed to improper fractions by multiplying denominator times whole number, plus the numerator, all over the denominatorDoing Math Involving Improper Fractions & Mixed NumbersConvert all numbers to improper fractions then proceed as previously discussedHomework 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 meetingExponential Expressions3 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) = 81An exponent applies only to the base (what it touches)Meanings of exponents that are not natural numbers will be discussed laterOrder 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 SallyOrder of OperationsExample:PEMDASExample 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 twelveNine is greater than eleven minus fourThree is not equal to eight minus sixEquivalent 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 < 5Likewise, 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 > 3Homework 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 meetingTerminology 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 variablesTerminology 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 answerExampleEvaluate 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 variableExampleEvaluate 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:AddSubtractMultiplyDividePhrases that Translate to AdditionEnglish PhraseA number plus 5The sum of 3 and a number4 more than a numberA number increased by 8Algebra ExpressionPhrases that Translate to SubtractionEnglish Phrase4 less than a numberA number subtracted from 76 subtracted from a numbera number decreased by 92 minus a numberAlgebra ExpressionPhrases that Translate to MultiplicationEnglish Phrase7 times a numberthe product of 4 and a numberdouble a numberthe square of a numberAlgebra ExpressionPhrases that Translate to DivisionEnglish Phrasethe quotient of 2 and a numbera number divided by 86 divided by a numberAlgebra ExpressionPhrases Translating to Expressions Involving Multiple Math OperationsEnglish Phrase4 less than 3 times a numberthe quotient of 5 and twice a number6 times the difference between a number and 5Algebra ExpressionPhrases Translating to Expressions Involving Multiple Math OperationsEnglish Phrasethe difference between 4 and 7 times a numberthe quotient of a number and 5, subtracted from the numberthe product of 3, and a number increased by 4Algebra ExpressionEquationsEquation 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 solutionDistinguishing Between Expressions & EquationsExpressions contain constants, variables and math operations, but NO EQUAL SIGNEquations always CONTAIN AN EQUAL SIGN that indicates that two expressions have the same valueSolutions 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 statementFinding Solutions to Equations from a Given Set of NumbersFrom the following set of numbers, find a solution for the equation:Check x = 3Check x = 4Check x = 5Writing 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 tenHomework 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 numbersGraphing 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 lineRational 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 NumbersGraphGraphIrrational NumbersIt may seem that rational numbers would fill up all the gaps between integers on a number line, but they dontThe next set of numbers to be considered will fill in the rest of the gaps between the integers and rational numbers on a number lineIrrational numbersNumbers that can not be written as the ratio of two integersThis includes all decimals that do not terminate and do not have a sequence of digits that form an infinitely repeating pattern at the endIncluded in this set of numbers are any square roots of positive numbers that will not simplify to get rid of square root signExamples:Notes on Square RootsThe square root of is written as and represents a number that multiplies by itself to giveWe know that the number that multiplies by itself to give is , so we write is a terminating decimal, so is a rational numberThe square root of is written as and represents a number that multiplies by itself to giveWe know of no number that multiplies by itself to give , but a calculator gives a decimal approximation that fills the screen without showing a repeating pattern at the end. is an irrational numberSquare roots may be rational, irrational, or neitherMore Notes on Square RootsThe square root of is written as , but it does not exist in the real number system (no real number multiplies by itself to give a negative is not rational or irrational. Its not real, but is a type of number called an imaginary number, that will be studied in college algebra Graphing Irrational NumbersPositive irrational numbers will correspond to a point right of zero and negative irrational numbers will correspond to a point left of zeroTo find the approximate location of the point, consider the decimal approximationIf the number is positive:go to the right to the whole numberdivide the next interval into the number of divisions of accuracy desired (tenths, hundredths, etc.)continue to the right from the whole number to the division indicated by the digits right of the decimal pointPlace 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 of accuracy desired (tenths, hundredths, etc.)continue to the left from the whole number to the division indicated by the digits right of the decimal pointPlace a dot at that point and label it with the coordinate Example of GraphingIrrational NumbersGraphReal NumbersThe set of rational numbers and the set of irrational numbers have no numbers in commonWhen the two sets of numbers are put together they make up a new set of numbers called real numbersEvery real number is either rational or irrationalThere is a one-to-one correspondence between points on a number line and the set of real numbersThere are some numbers that are not real numbers, an example is: . These type of numbers (complex numbers) will be discussed in college algebra.Ordering Real NumbersGiven two real numbers, represented by the variables a and b, one of the following order relationships is true:a = ba equals b if they graph at the same locationa < ba is less than b, if a is left of b on a number linea > ba is greater than b, if a is right of b on a number lineAdditive Inversesof Real NumbersEvery real number has an additive inverseThe additive inverse of a real number is the number located on a number line the same distance from zero, but in the opposite directionThe additive inverse of a number is the same as its oppositeThe additive inverse of 5 is:The additive inverse of -3 is:Placing in negative sign in front of a number is a way of indicating the additive inverse of the numberIf we want to indicate the additive inverse of -7, we can place a negative sign in front of -7:- (-7) is the same as:Absolute Valueof Real NumbersEvery real number has an absolute valueThe absolute value of a real number is its distance from zeroDistance is never negative, so absolute value is never negativeAbsolute value of a number is indicated by placing vertical bars around the numberThe absolute value of 5 is shown by :and is equal to:The absolute value of -3 is shown by:and is equal to:Homework ProblemsSection: 1.4Page: 39Problems: All: 9 20 Odd: 23 27, 35 63 MyMathLab Homework 1.4 for practiceMyMathLab Homework Quiz 1.4 is due for a grade on the date of our next class meetingAddition of Real NumbersAddition like a game between two teams, Positive and Negative, the answer to the problem is the answer to the question, Who won the game, and by how much?Example:Reasoning: Negatives scored:Positives scored:_________ won by ____, soSecond Example of AdditionExample:Reasoning: Negatives scored:Positives scored:_________ won by ____, so:Addition of Signed FractionsAddition rule is the same for all signed numbers, but you must first write each fraction as an equivalent fraction where all fractions have a common denominatorExample:Reasoning: Negatives scored:Positives scored:_________ won by ________, so:Addition of Signed DecimalsAddition rule is the same for all signed numbers, but be sure to line up decimal points before adding or subtractingExample:Reasoning: Negatives scored:Positives scored:_________ won by ____, so:Subtraction of Real NumbersSubtract means add the oppositeAll subtractions are changed to add the opposite and then the problem is done according to addition rules already discussedIn identifying a subtraction problem remember that the same symbol, - , is used between numbers to mean subtract and in front of a number to mean negative number .Problems Involving BothAddition and SubtractionExample:Identify subtraction:Add opposite:Reasoning: Negatives scored:Positives scored:_________ won by ____, so:Homework ProblemsSection: 1.5Page: 49Problems: Odd: 7 97 MyMathLab Homework 1.5 for practiceMyMathLab Homework Quiz 1.5 is due for a grade on the date of our next class meetingMultiplying and DividingReal NumbersMultiplication and Division of signed numbers follows the rule:Do problem as if both were positiveAnswer is positive if signs were the sameAnswer is negative if signs were oppositeExamples: .Multiplying Signed FractionsBasic rule has already been discussedOtherwise, remember to:Divide out factors common to top & bottomMultiply top factors to get topMultiply bottom factors to get bottomExample:Dividing Signed FractionsBasic rule has already been discussedOtherwise, remember to:Invert the second fraction and change problem to multiplicationComplete using rules for multiplicationExample:Division Involving ZeroPeople are often confused when division involves zero the rule must be memorized!Division by zero is always undefinedOtherwise, division into zero is always zeroExplanation comes from checking answer: .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 SallyHomework ProblemsSection: 1.6Page: 63Problems: Odd: 11 73, 77 113 MyMathLab Homework 1.6 for practiceMyMathLab Homework Quiz 1.6 is due for a grade on the date of our next class meetingAveraging Real NumbersTo average a set of real numbers we add all the numbers and then divide by the number of numbers in the setFind the average of the following set of numbers: .DivisibilityA real number is divisible by another if the division has no remainderOn the following slides are tests for divisibility by all the numbers between 2 and 9, except for 7 (there is no test for divisibility by 7)Memorize these testsTest for Divisibility by 2A real number is divisible by 2 only if its last digit is evenWhich of the following numbers are divisible by 2?31,976,1042571,34835,750Test for Divisibility by 3A real number is divisible by 3 only if the sum of its digits is divisible by 3Which of the following numbers are divisible by 3?51,976,1043571,34845,750Test for Divisibility by 4A real number is divisible by 4 only if the last two digits form a number that is divisible by 4Which of the following numbers are divisible by 4?51,976,1043571,34845,750Test for Divisibility by 5A real number is divisible by 5 only if the last digit is 5 or 0Which of the following numbers are divisible by 5?51,976,1043571,34845,750Test for Divisibility by 6A real number is divisible by 6 only if it passes both the test for divisibility by 2 and divisibility by 3Which of the following numbers are divisible by 6?51,976,1043571,34845,750Test for Divisibility by 8A real number is divisible by 8 only if its last three digits form a number divisible by 8Which of the following numbers are divisible by 8?51,976,1043571,34845,750Test for Divisibility by 9A real number is divisible by 9 only if the sum of its digits is divisible by 9Which of the following numbers are divisible by 9?51,976,1043571,34845,750Homework ProblemsSection: 1.6Page: 63Problems: All: 115 119, 121 127 MyMathLab Homework 1.6a for practiceMyMathLab Homework Quiz 1.6a is due for a grade on the date of our next class meetingProperties of Real NumbersCommutative Property the order in which real numbers are added or multiplied does not effect the result:Associative Property the way real numbers are grouped during addition or multiplication does not effect the result:Properties of Real NumbersCommutative Property Examples:Associative Property Examples:Properties of Real NumbersIdentity Property for Addition when zero is added to a number, the result is still the number:Identity Property for Multiplication when one is multiplied by a number, the result is still the number:Properties of Real NumbersIdentity Property for Addition Example:Identity Property for Multiplication Examples:Properties of Real NumbersInverse Property for Addition when the opposite (negative) of a number is added to the number, the result is zero:Inverse Property for Multiplication when the reciprocal of a number is multiplied by the number, the result is oneReciprocals of Real NumbersZero has no reciprocalReciprocals of other integers are formed by putting 1 over the numberReciprocals of fraction are formed by switching the numerator and denominatorProperties of Real NumbersInverse Property for Addition Examples:Inverse Property for Multiplication Examples:Properties of Real NumbersDistributive Property multiplication can be distributed over addition or subtraction without changing the resultIllustration of Distributive PropertyIllustration of Distributive PropertyDistributive Property works both directions:If two terms contain a common factor, that factor can be written outside parentheses with the remaining factors remaining as terms inside parenthesesIllustration of Distributive PropertyUse the Distributive Property backwards to write each of the following in a different way:Homework ProblemsSection: 1.7Page: 74Problems: All: 1 30, 35 50, 55 80 MyMathLab Homework 1.7 for practiceMyMathLab Homework Quiz 1.7 is due for a grade on the date of our next class meetingTerminology 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 with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and roots in a meaningful wayExamples of expressions:Only the first of these expressions can be simplified, because we dont know the numbers represented by the variablesTerminology of AlgebraTerm an expression that involves only a single constant, a single variable, or a product (multiplication) of a constant and variablesExamples of terms:Note: When constants and variables are multiplied, or when two variables are multiplied, it is common to omit the multiplication symbolPrevious example is commonly written:Terminology of AlgebraEvery term has a coefficientCoefficient the constant factor of a term(If no constant is seen, it is assumed to be 1)What is the coefficient of each of the following terms?Like TermsRecall that a term is a _________ , a ________, or a _______ of a ________ and _________Like Terms: terms are called like terms if they have exactly the same variables with exactly the same exponents, but may have different coefficientsExample of Like Terms:Determine Like TermsGiven the term:Which of the following are like terms?Adding Like TermsWhen like terms are added, the result is a like term and its coefficient is the sum of the coefficients of the other termsExample:The reason for this can be shown by the distributive property:Subtracting Like TermsWhen like terms are subtracted, the result is a like term with coefficient equal to the difference of the coefficients of the other termsExample:Reasoning:Simplifying Expressions by Combining Like TermsAny expression containing more than one term may contain like terms, if it does, all like terms can be combined into a single like term by adding or subtracting as indicated by the sign in front of each termExample: Simplify:Review of Distributive PropertyDistributive Property multiplication can be distributed over addition or subtractionSome people make the mistake of trying to distribute multiplication over multiplicationExample:Associative Property justifies answer! !!+ or in Front of ParenthesesWhen a + or is found in front of a parentheses, we assume that it means positive one or negative oneExamples:Multiplying TermsTerms can be combined into a single term by addition or subtraction only if they are like termsTerms can always be multiplied to form a single term by using commutative and associative properties of multiplicationExample: Simplifying an ExpressionGet rid of parentheses by multiplying or distributingCombine like termsExample:Homework ProblemsSection: 1.8Page: 80Problems: All: 5 30 Odd: 33 75 MyMathLab Homework 1.8 for practiceMyMathLab Homework Quiz 1.8 is due for a grade on the date of our next class meeting