Some Basics of AlgebraSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Part 2 of 1.1Slide 14Slide 15Converting Fractions to DecimalsSlide 17Slide 18Rationals include the following.Slide 20Identify as natural, whole, integers, rational, or irrational.AnswersSlide 23Write with Roster NotationWrite with Set-Builder NotationSlide 26Slide 27True or False?Slide 29Some Basics of Algebra•Algebraic Expressions and Their Use•Translating to Algebraic Expressions•Evaluating Algebraic Expressions•Sets of Numbers1.1TerminologyA letter that can be any one of various numbers is called a variable. If a letter always represents a particular number that never changes, it is called a constant.Algebraic ExpressionsAn algebraic expression consists of variables, numbers, and operation signs.Examples:, 2 2 , .4yt l w m x b- �+ � �+When an equal sign is placed between two expressions, an equation is formed.Translating to Algebraic Expressions per of less than more than ratio twicedecreased byincreased byquotient of times minus plusdivided byproduct ofdifference of sum of divide multiply subtract addDivisionMultiplicationSubtractionAdditionKey WordsExampleTranslate to an algebraic expression:Eight more than twice the product of 5 and a number. Solution( )8 2 5 n+ � �Eight more than twice the product of 5 and a number.Evaluating Algebraic ExpressionsWhen we replace a variable with a number, we are substituting for the variable. The calculation that follows is called evaluating the expression.ExampleEvaluate the expression 8 for 2, 7, and 3.xz y x y z- = = =Solution8xz – y = 8·2·3 – 7= 41= 48 – 7SubstitutingMultiplyingSubtractingExampleThe base of a triangle is 10 feet and the height is 3.1 feet. Find the area of the triangle. 1 12 2b h��= �Solution10·3.1= 15.5 square feethbExponential Notation The expression an, in which n is a counting number means n factors In an, a is called the base and n is called the exponent, or power. When no exponent appears, it is assumed to be 1. Thus a1 = a.a a a a a��������Rules for Order of Operations1. Simplify within any grouping symbols.2. Simplify all exponential expressions.3. Perform all multiplication and division working from left to right.4. Perform all addition and subtraction working from left to right.ExampleEvaluate the expression Solution2(x + 3)2 – 12 x2SubstitutingSimplifying 52 and 22Multiplying and DividingSubtracting( )222 3 12 for 2.x x x+ - � =�= 2(2 + 3)2 – 12 22�( )222 5 12 2 = - �2 25 12 4 = � - �Working within parentheses50 3 = -47 =ExampleEvaluate the expression 24 2 for 3, 2, and 8.x xy z x y z+ - = = =Solution4x2 + 2xy – z = 4·32 + 2·3·2 – 8= 36 + 12 – 8= 40= 4·9 + 2·3·2 – 8SubstitutingSimplifying 32MultiplyingAdding and SubtractingPart 2 of 1.1Sets of NumbersSets of NumbersNatural Numbers (Counting Numbers) Numbers used for counting: {1, 2, 3,…}Whole Numbers The set of natural numbers with 0 included: {0, 1, 2, 3,…}Integers The set of all whole numbers and their opposites: {…,-3, -2, -1, 0, 1, 2, 3,…}Rational Numbers Numbers that can be expressed as an integer divided by a nonzero integer are called rational numbers:Sets of Numbers is an integer, is an integer, and 0 .pp q qq� ��� ��Converting Fractions to DecimalsDivide the numerator by the denominator65 .8 3 3 36 5.0 0 0 4 8 2 0 1 8 2 0 38.0...8333.036.01146.032375.083Any fraction can be converted to a repeating decimal or a terminating decimal..All integers can be written as fractions. Insert a denominator of 1.11414133Look at the following conclusion.Rationals include the following.•All integers (-2, 5, 17, 0)•All fractions (proper, improper, or mixed)•All terminating decimals•All repeating decimals211,87,1191 -2.34, 0.0456784.3 ,2.1Sets of NumbersReal Numbers Numbers that are either rational or irrational are called real numbers:{ } is rational or irrational .x xNumbers like are said to be irrational. Decimal notation for irrational numbers neither terminates nor repeats. 5 and pIdentify as natural,whole, integers, rational, or irrational.5-3 10540.457160 6.23252342.09400Answers•Natural: 10•Whole: 10 0 •Integers: 10 0 -3•Rational: •Irrational:1616169400 ,342.0 ,325 ,2.6 ,0 ,16 ,457.0 ,54 ,10, 32 ,5Set NotationRoster notation: {2, 4, 6, 8} Set-builder notation: {x | x is an even number between 1 and 9}“The set of all x such that x is an even number between 1 and 9”Write with Roster Notation12} and 5between number even an is |{ xx}10,8,6{7}least at number natural a is |{ xx}7,6,5,4,3,2,1{Write with Set-Builder Notation1} and 6-between integer an is |{ xx}17,15,13,11,9{ )218} and 8between number oddan is |{ xx1) The set of all integers between -6 and 15)33} and 21between 4 of multiple a is |{ xx}32,28,24{Elements and SubsetsIf B = { 1, 3, 5, 7}, we can write 3 B to indicate that 3 is an element or member of set B. We can also write 4 B to indicate that 4 is not an element of set B.��When all the members of one set are members of a second set, the first is a subset of the second. If A = {1, 3} and B = { 1, 3, 5, 7}, we write A B to indicate that A is a subset of B. �True or False?Use the following sets: N= Naturals, W = Wholes,Z = integers, Q = Rationals, H = Irrationals,and R =
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