Developmental Math II MAT0028C

Transcription

1 Developmental Math II Professor Sikora MAT008C Power Point Notes for Valencia College

2 FOUNDATIONS OF ALGEBRA

3 1.1 Variables VARIABLES = letters = symbols = represent possible number What word do the VARIABLES spell? m 4at -17h CONSTANT = Symbol that does not vary in value. ex: 5, -, 10, 00,

4 1.1 Expressions ALGEBRAIC EXPRESSIONS = 1 OR MORE: numbers variables arithmetic operators Exs: [ +, -,., ] 3x 1,, 4a + 7, bc (factors of a product) You evaluate an expression [when know values of the variables] w z 5

5 1.1 Equations have an = sign [the verb] To Solve an eq. Find values of the variable [solutions] to make it true Is 3 a solution of x + 1 = 19? Y N From n = {1,, 3, 4}, find the solution for n + 9 = 1

6 1.1 Inequalities [the verb] Math Relationship with the verb symbol (, <, >,, or ). Symbolic form Translation Eight is not equal to three. Five is less than seven. Seven is greater than five. x is less than or equal to three. y is greater than or equal to two.

7 1.1 Set = Collection of objects Objects are called elements or members. Write the set containing the first four days of the week. Answer: {Sunday, Monday, Tuesday, Wednesday} Write the set containing multiples of 3 to 15, inclusive. Answer: {3, 6, 9, 1, 15}

8 1.1 Rational # Rational number: Any real number that can be expressed in the form a, where a and b are integers and b 0. b Ex: Rules for Rationals: Fractions Terminating Decimals Non-terminating, repeating decimals

9 1.1 Irrational # = not rational Any real number that cannot be expressed in the form, b where a and b are integers and b 0. a Ex: 3 5 Rules for Irrationals: Non-terminating, non-repeating decimals

10 1.1 Real # s categorized Natural ==>{1,, 3, 4,...} ie. Counting # s Whole #s ==> {0, 1,, 3,...} Integers = {...-, -1, 0, 1,, 3,...} Irrat. Rat. I W N Rational #s = {x x is the quotient of integers} Irrational #s = {x x is not the quotient of integers} Ex., REAL #S = {x x is rational or irrational #}

11 1.1 Real # s have positions on # line Graph on a number line: Natural #s Whole #s sm lrg Integers Rational #s fractions [proper, improper, mixed, terminating decimals, repeating decimals] Irrational #s non-terminating, non-repeating decimals {see Venn Diagram on inside cover of book & page 5} Be able to categorize any given real #

12 1.1 Absolute Value Abs. Value of a # is it s distance [pos.] from 0 5 = 5 and -5 = 5 x = x if x > 0 -x if x < 0 Distances [and thus ABSOLUTE VALUES] are always positive! Simplify: -3 = - = 3 = -4.8 = - -3 = 3

13 1.1 Compare #s on number line a > b if a to right of b on number line b < a if b to left of a on number line Use =, <, or > in each box below:

14 1. Fractions numerator deno min ator Mult. fractions: # reciprocal of that # = 1 [KEEP CHANGE FLIP] Divide fractions: fraction _ bar 0 and if so, fraction undefined a c ac b & d 0 b d bd a b c d a b reciprocals d c

15 1. Build/Simplify Fractions fractions Equivalent if they represent the same # Build a fraction Mult. by a form of 1 Mult. Prop of 1: 1 a = a & a 1 = a a Reals Exs. Write 5 8 as an equivalent fraction Simplest form of a fraction [lowest terms] Remove a factor of 1 Ex: Simplify

16 1. + or Fractions often need an LCD Least common multiple (LCM): The smallest number that is a multiple of each number in a given set of numbers. Ex: LCM of and 3 is Ex: LCM of 3 and 4 is Least common denominator (LCD): The least common multiple of the denominators of a given set of fractions.

17 1. + or - Fractions A) Same Denominator a b a b d d d and Exs: B) Unlike Denominators Find LCD = smallest # each denom s into evenly 1)Factor each denom )Take each factor that appears to its highest pwr 3) Mult these factors for LCD a b a b d d d Ex: 3 4 5

18 1. Fractions use factors & factorization In multiplying: given factors find the product [ex: 5 7 = 70] In factoring: given product find the factors [ex: 70 = 5 7] A # w/ exactly factors [the # itself & 1] = PRIME number (whole #s > 1) Composite number (whole #s > 1) = not prime #s Prime Factorization = when whole # expressed as product of prime factors

19 1. Fractions use factors & factorization Find the PRIME FACTORIZATION of:

20 1.3 Properties of Addition[a,b,c R] Property Addition Commutative a + b = b + a Associative (a + b) + c = a + (b + c) Additive Identity a + 0 = a [0=Identity Elem for +] Additive Inverse a + (-a) = 0 Be able to identify these properties when used!

21 1.3 Add Real #s Methods: Using arrows on the Number Line Applying RULES: Add Integers [ SAME SIGN] * Add their Absolute Values * Ans. has SAME sign Ex: + 3 = Ex: -5 + (-4) = 0

22 1.3 Add Real #s Add Real #s [ SAME SIGN] Review * Add their Absolute Values * Ans. has SAME sign Ex: - + (-5) = -7 Add Real #s [ DIFFERENT SIGNS] * Subtract their Absolute Values [big - sm.] * Ans. has sign of larger Absolute Value Ex: 4 + (-5) = -1 Ex: =

23 1.3 Add Several Real #s (+9) + (+3) + (-7) = (-0) = (-5) + 10 =

24 1.3 Subtract Real #s To SUBTRACT a number, add its Additive Inverse: a - b = a + (-b) Ex: -8 (-1) = Use all those addition rules! Ex: 6 + [(-1 4) ] = Ex: - (- 14) =

25 MQ What does an algebraic equation have that an algebra expression doesn t? Which one can be solved?. Translate: w is less than or equal to 7 3. Give the Prime Factorization of 70 using exponents 4. What do we call a number that is the quotient of integers? 5. Write the set of natural #s less than = 7. Build an equivalent fraction to 9 w/ denom. of 64 8, Insert symbol:

26 1.4 Properties of Multiplication[a,b,c R] Property Multiplication Commutative ab = ba Associative (ab)c = a(bc) Mult. Prop of zero a 0 = 0 Mult. Identity a 1 = a [1=Identity Elem for ] Distributive a(b + c) = ab + ac Be able to identify these properties when used!

27 1.4 Multiplying Real #s Rules for products of signed #s: +. + = + and -. - = + & +. - = - and -. + = - Signs same Signs different Ex: (-3)(-5) = Ex: (-0.4)() = Ex:

28 1.4 Multiplying & Dividing Real #s * Multiplication & Division are Inverse Operations. Thus: If +. + = + and -. - = + & +. - = - and -. + = - Then + + = + and - - = + Signs different & + - = - and - + = Ex: cuz 5(-3) = -15 Signs same

29 1.4 Divide Real #s The Difference of numbersadd its Additive Inverse: a - b = a + (-b) The Quotient of numbersmult. by its Reciprocal or Multiplicative Inverse: x y 1 y Product = = x y 0 Ex. Ex. Dividing by 0 is undefined

30 1.5 Exponents x n --> x to the n th power base POWER exponent Repeated Multiples x. x. x. x... x --> n of these Ex: 4 3 = = (-4) 3 = = = = 4

31 1.5 Square Roots Squares: the square of 5 is 5 cuz 5 = 5 and the square of -5 is 5 cuz (-5) = 5 Square Roots: 5 is the square root of 5 cuz 5 = 5 and -5 is the square root of 5 cuz (-5) = 5 b is square root of a if b = a All positive #s have sq. roots. It s pos. sq. root = principal square root

32 1.5 Square Roots Square Root of a => [a = positive real #] Ex: Note: a 9 16 = represents the POSITIVE sq. root of a a

33 Order of Operations PLEASE ( ), [ ], { },, EXCUSE exponents or roots... MY multiplication DEAR division AUNT add SALLY subtraction LEFT RIGHT

34 1.5 Order of Operations Ex (- 5) = Ex (- 3) 3 = P E M D A S

35 1.5 Order of Operations Ex: 88 [7 (1 + 8) 4] = Ex:

36 1.5 Grouping Symbols Nested Parentheses: [5+3(4-1)] simplify inside outside * Fraction Bar: simplify numerator * & denominator * then Absolute Value Bars: work inside 1st * Using Order of Operations

37 1.5 Arithmetic Mean [Average] Mean of a set of values their sum by the # of values average Ex: Bruce has the following test scores in his biology class: 9, 96, 81, 89, 95, 93. Find the average of his test scores. Ans:

38 1.6 Translating Basic Phrases Fill in chart Addition Translatn Subtraction Translation The sum of x and 3 h plus k The difference of x and 3 h minus k 7 added to t 7 subtracted from t 3 more than a number y increased by 3 less than a number y decreased by

39 1.6 Translating Basic Phrases Fill in chart Multiplicatn Translation Division Translation The product of x and 3 h times k Twice a number n Triple the number n Two-thirds of a number n The quotient of x and 3 h divided by k h divided into k The ratio of a to b

40 1.6 Translating Basic Phrases Fill in chart Exponents Translatn Roots Translation c squared The square of b k cubed The cube of b n to the fourth power y raised to the fifth power The square root of x

41 MQ State the property: 1) -0 + (4 + 5) = -0 + (5 + 4) ) -0 (4 5)=(-0 4) 5 3) a 1 = a 4) 4(x+)=4x+8 5-7) Add or Subtract 5) 6) (-9) + (-6) 7) ( -3) ) Mult. or Divide 8) 9) ) Simplify

42 1.7 Evaluating Algebraic Expressions Find the value of p 3 if p = 3 Find the value of 4x y if x = 6 & y = 9 x 1 Find the value of 1) 3m ) (3m) if m = Evaluate m 3 6n when m = - and d = -5

43 1.7 Values causing undefined expressions Dividing by 0 is undefined Ex: 8 x 6 Ex: m m 3m 4

44 1.7 Distributive Prop. for exp. rewrite Distributive a(b + c) = ab + ac [also subtraction] Ex: 3(x + 1) = Ex: -3(x - 10) = y 10 Ex: =

45 1.7 Expressions Terms separated by + or - sign Product or quotient of #s and/or variables The coefficient is the numerical factor of a term ~ Identify it in these exs: 8y 3 1y +3y x 5 y 5

46 1.7 Combine Like Terms for exp. rewrite Use properties Ex: - (7 6k) + 9 = Be sure to arrow 1 thru parentheses Terms of an Alg. Expression: separated by + or - sign Combine like terms [same variables to same pwr] Ex: 5(a 6) 3(4a 9) = Ex: 3 3 y x y 5x = 8 4

47 MQ 1.7 & Review Evaluate: 1) 50 (5) 7 ) (3 4) - 4 3) ) ( ) + 1 5) ) Translate to algebraic expression: The difference between times a number (x) and 4 x 7 7 7) Evaluate: 3x - for x = - 8)When undefined? 9) Translate to algebraic expression: The absolute value of the quotient of a and two 10) Combine Like Terms: 1 1 a 4b 3 a b x x 1

48 SOLVING EQUATIONS & INEQUALITIES

49 .1 Solving Equations [ie math sentences] Solutions or roots = #s that when substituted for the variable(s), satisfies the equation makes it true Ex: Is 5 a solution of (46 - x) = 41? Y N Ex: Is - a solution of 8x + 18 =? Y N To solve an eq. find all variable values that make eq. true

50 .1 Identity Eq. = every real # is solution To determine if an equation is an identity: 1) Simplify expressions on each side of = sign. ) If, after simplifying, the expressions are identical, then the equation is an identity. Ex: (3x 4) 10x = 15-4(x + ) Identity? Y or N Ex: 0.5(3y 8) = y y Identity? Y or N

51 .1 Formulas from Geometry Perimeter = distance around Area = surface enclosed P sq = 4s P rect = L + W s A sq = s A rect = LW P = a+b+c C circle = r a c A = ½bh A circle = r h L W r b Memorize these! Other figures will have formulas given to you.

52 .1 Formulas=a relationship betw variables Sale price* = orig price discount s = p - d [x = orig. $. If discounted 5%, subtract.5x for sale $] Interest* = principal rate time I = Prt Distance* = rate time d = rt [units same] * Must memorize as they are required for the Final Exam

53 .1 Problem Solving Technique for word problems Steps: 1) Chose a variable for what you are to find [Write: Let x = ] Write out facts [w/ your var.], pictures,... ) Translate prob. to an equation [is =, etc.] 3) Solve equation [legally w/ properties] 4) Answer question(s) asked [may be more than what x equals] 5) Check [Sub. solution into original and work down til both sides of = sign the same. Then ]

54 .1 Problems using Formulas A 13 x 0 room needs crown molding where the walls & ceiling meet. a) Find the total molding length needed b) It comes in 1 strips. How many strips needed? c) Strips cost $9 ea. Find total $. a) b) c)

55 . Addition Principle of Equality Linear Eq. variables raised to power of 1 only ADDITION Principle of equality: a,b,creals Whatever you do to one side, you must do to the other! If Then a = b a + c = b + c Exs: Solve & check [by substituting ans. back into original and work down both sides until same] w 15 = = b - 38 EQUIVALENT EQS. = same solution

56 . Addition Principle of Equality Used to SUBTRACT quantity from both sides: a,b,cr Whatever you do to one side, you must do to the other! If Then a = b a - c = b c EQUIVALENT EQS. = same solution Exs: Solve & check [by substituting ans. back into original and work down both sides until same] x + 1 = y = n =9

57 . Addition Principle of Equality Ex: Solve and Check: n 4n 8 6 n 3 8n Ans: 10 n

58 . SOLVING EQS. with INFINITE # of Solutions Ex: 3y y + 10 = y Eqs. true for all values of variables = IDENTITY

59 . SOLVING EQS. with NO Solutions Ex: 3w + 8 = 6(w 1) 3w Final Eq. x=a # ex:x= True ex: 0=0 False ex:5=8 # of Solutns. Solutn Set called solved identity contradiction

60 . Addition Principle of Equality Ex: Raul wants to buy a membership to a gym which costs $395 for the year. He currently has saved $149. How much more does he need? Let x = the amount Raul needs. x = $46

61 .3 Multiplication Principle of Equality MULT. Principle of equality: a,b,creals Whatever you do to one side, you must do to the other! If a = b Then ca = cb (c 0) Exs: Solve & check x s 3 = 5 = y = How can I get the variable lonely?

62 .3 Multiplication Principle of Equality Used to DIVIDE each side by same quantity : a,b,creals Whatever you do to one side, you must do to the other! If a = b a c Then (c 0) Exs: Solve & check b c w = 40 1k = 48 How can I get the variable lonely?

63 .3 Steps for Solving Equations 4 Steps to solve a Linear Equation: ax + b = c; a,b,cr, a0 1) Simplify each side separately [clear ( ) with Distrib, combine Like terms] ) Isolate variable term on 1 side [use addition or subtraction prop. so var. term on 1 side & # term on other] Tip: Clear the variable term that has the lesser coefficient to avoid negative coefficients. 3) Isolate variable [use mult. or division prop. so variable term = #] Box ans. 4) Check [substitute solution into original to see if a true statement results ]

64 .3 Multiplication Principle of Equality Ex: Solve and check. 5 3 x y5 3 y 1 Ex: Solve and check. 3 x = 10 y = -

65 Ex:.3 Multiplication Principle of Equality The perimeter of the figure shown is 7 inches. Find the width and the length. x x +11 x = 1.5

66 Ex:.3 Multiplication Principle of Equality The Smith family is planning a 510-mile trip. If their average speed is 68 MPH, how long will it take them to complete the trip? x = 7.5 hrs

67 Mini-Quiz.1.3 ~ SHOW ALL WORK on bottom or back of strip 1) Solve: (x - 4) + x = 7 ) Ck #1) 40 8w 3) Solve: w 4) Ck #3) 5 5) Solve: 4(y - 3) - y = 3(y - 4) 6) Solve: -3(v + ) = -v 8 - v 3 7) Check if is a solution of: 6 (3x + ) = 6x 4 8) In Solving: x 8 mult. each side by 3 9) Solve & ck: x ) If an equation has fractions, multiply thru by their

68 .4 Solving Formulas for a variable Solve A = ½bh for h Solve p = r - c for r Solve P = L + W for W Solve jm + c = n for m

69 .5 Translating Basic Phrases Fill in chart Addition Translatn Subtraction Translation The sum of x and 3 h plus k The difference of x and 3 h minus k 7 added to t 7 subtracted from t 3 more than a number y increased by 3 less than a number y decreased by

70 .5 Translating Basic Phrases Fill in chart Multiplicatn Translation Division Translation The product of x and 3 h times k The quotient of x and 3 h divided by k Twice a number n Triple the number n Two-thirds of a number n h divided into k The ratio of a to b

71 .5 Translating Basic Phrases The sum of thirty-five and a number is equal to n = 18 Answer: n = -17

72 .5 Translating Basic Phrases Eight less than five times a number is equal to thirty-seven. 8 5n = 37 Answer: n = 9

73 .5 Translating Basic Phrases Ex: Nine times the sum of a number and seven subtracted from three times the number results in negative twenty-seven. Answer x = 6

74 Mini-Quiz.4.5 ~ SHOW ALL WORK on bottom or back of strip 1) Solve A = P + Prt, for t ) Solve for x: y = mx + b 3) Solve for h: V = r h 4) Translate to an eq.: Four less than three times a # is 5 5) Solve eq. in 4) 6) Translate to an eq.: Three times the difference of a # and 5 equals twice the # and 3 7) Solve eq. in 6) 8) Solve: (3x + 4) - (10-3x) = 5x 9) Ck 8) 10) Solve: w w

75 .6 Linear Inequality ~ Set-builder notation Linear inequality: An inequality containing expressions in which each variable term contains a single variable with an exponent of 1. Examples of linear inequalities: x > 5 n + < 6 (y 3) 5y 9 Set builder notation { x x 5} The set of all x such that x is greater than or equal to 5.

76 .6 Linear Inequality ~ Interval Notation [#, or,#]: Graph Includes the associated point (#, or,#): Do not Include the associated point (-, Begin with lowest unbounded neg. #, ) End with highest unbounded pos. # Graph these solutions & use Interval Notation to describe: x > 4 x < - -1 x < 3

77 .6 Addition Property of Inequality x < Graph: ] 0 1 x > - Graph: ( < x < : ( ] = Interval -1 ADDITION PROP. OF INEQUALITY: A,B,C A < B EQUIVALENT EQS. A + C < B + C [also >, <, and > ] Solve & Graph: 1 + 8r < 7r + 4 < x + < 10 Reals: Also for subtraction

78 .6 Mult. Property of Inequality MULTIP. PROP. OF INEQUALITY: A,B,C Reals, C 0: If C is positive A < B AC < B C If C is negative A < B AC > B C [reverse inequality] Solve & Graph: -r > -1 Solve & Graph: 36 < -9y

79 .6 Solving Linear Inequalities 1. Simplify both sides of the inequality as needed. a. Distribute to clear parentheses. b. Clear fractions or decimals by multiplying through by the LCD just as we did for equations. c. Combine like terms.. Legally get all variable terms are on one side of the inequality and all constants are on the other 3. Use the multiplication principle to clear any remaining coefficient. If you multiply (or divide) both sides by a negative number, reverse the direction of the inequality symbol!

80 .6 Solving Linear Inequalities Solve 8x + 13 > 3x 1. ( Set builder notation: {x x > 5} Interval notation: (5, )

81 .6 Translations for Linear Inequalities Less Than: Greater Than: A number is less than 7. A number must be smaller than 5. Less Than or Equal to: A number is at most 9. The maximum is 14. n < 7 n < 5 n 9 n 14 A number is greater than. A number must be greater than 3. A number must be more than 6. Greater Than or Equal to: A number is at least. The minimum is 18. n > n > 3 n > 6 n n 18

82 .6 Translations for Linear Inequalities Ex:Seven-eighths of a number is at least twenty-one. 7 8 n 1 n 4

83 Mini-Quiz.6 & Rev ~ SHOW ALL WORK on bottom or back of strip 1) Solve: 6x + 1 > -1 ) Write solution for 1) in setbuilder notation 3) Graph solution in 1) 4) Solve: 6(y + 1) < 4 8y + 3(5y 1) 5) Write solution for 4) in set-builder notation 6) Graph solution in 4) 7) Solve for C: 8) Solve: 1 9 C c 3 9) Solve: 5(b - ) > - (b 3) + b F 10) Will your Study Sheet for the Ch Test include all of the important facts is RED, explanations in BLUE & examples in PENCIL? 3 6c 1 & write solution in Interval Notation & write solution in Interval Notation

84 POLYNOMIALS

85 6.1 Positive Integer Exponents x n = x x x x x [n of these x factors] base exponent Numerical: Ex: -3 4 = where as Ex: (-3) 4 = Ex: -3 3 = and Ex: (-3) 3 = Rule: Neg # to even exponent = #, Neg # to odd exponent = #, Ex: = 3 3

86 6.1 Power Rules for Exponents POWER OF A QUOTIENT b 0 a b m = a b m m Ex: Simplify x 0 Ex: x 3 y x 3

87 6.1 Zero Exponents ZERO EXPONENT [x 0] x o = 1 cuz Ex: (-7) 0 = & same Ex: -7 0 = Ex: -(-7) 0 =

88 6.1 Neg. Integer Exponents NEGATIVE EXPONENT[a 0, n integer] 1 a -n = cuz & a n same Ex: Simplify 4 - Ex: Simplify -4 - Ex: Simplify (-4) - Ex: Simplify a 1 n = a n

89 6.1 Neg. Integer Exponents NEGATIVE to POSITIVE EXPONENTS [a, b 0; m,n integers] x y m n = y x n m Ex: Simplify Ex: h m 5 k Ex: x 6 8y 7

90 6.1 Scientific Notation:application of exponents Scientific Notation form: a x 10 n 1 < a < 10 n Integers Move Decimal pt. to right of 1 st nonzero digit Count # of places moved [no move 10 0 ] Large #s have positive power of 10 Small #s have negative power of 10 Ex: Write in Sci. notation: ,000 Ex: Write w/o exponents: 8.7 x x 10-4

91 6. Defining Polynomials Polynomial Ex: 3x 3 + 5x + x + 8x + 1 Terms separated by + or - signs Coefficient number (w/ sign) in front of var. Like Terms same variable to same power Unlike Terms diff. variable or diff. power Polynomial = a term or sum of terms where all variables have whole # exponents {0,1,,3, }

92 6. Classifying Polynomials Polynomial = Monomial or sum of Monomials Monomial: a number or Product a variable of these Exponents must be positive Names for Special Polynomials: Monomial(1 term) Ex: 3y or abc 3 or -5 Binomial( terms) Ex: 3y + abc 3 or -5+x Trinomial (3 terms) Ex: 3y + abc 3-5

93 6. Classifying Polynomials State if each of the following is a polynomial. If it is, state if it is a Monomial, Binomial, or Trinomial 3a - 7bc 3x + 7x - 4 7y 3-4y + 10 x 3 y z 8 r r

94 6. Classifying Polynomials Degree of a Monomial = sum of variables of exponents Degree of a Polynomial = greatest degree of any monomial term Monomial Degree Polynomial Degree 3y 3y + abc 3 5 abc 3 y 7 + y 6 + 3x 4 m 4-14 p 5 + p 3 m 3 +4m 9xyz x + xy +4abc

95 6. Evaluating Polynomial Functions Evaluate: -x 3 + x x + 3 for x = 0 for x = 1 for x = -3 Evaluate: c + 4c + 7 for c = -6 19

96 6. Arranging Polynomials Polynomials are arranged in powers of one variable: ascending order or descending order ascending order descending order 4 + 5a - 6a + a 3 a 3-6a + 5a x +4x 4x -x -5 When several variables are in the terms, write in order of only one variable.

97 6. Combine Like Terms a a b b a b a b Strike through like terms in the given polynomial as they are combined a a b b a b a b a 5 3ab a +3a b5

98 6.3 Adding Polynomials To ADD Polynomials: Group LIKE terms together [LIKE terms have same variable to same power ==> can be combined!] OR Place in COLUMN form [In DESCENDING order with LIKE terms aligned!]

99 6.3 Adding Polynomials ADD these Polynomials: Ex: (9y - 7x + 15a) + (-3y + 8x - 8a) Ex: (3a + 3ab - b ) + (4ab + 6b )

100 6.3 Subtracting Polynomials To SUBTRACT Polynomials: Find the Additive Inverse (opposite) of polynomial after (-) sign[ & place opposite sign above] Group LIKE terms together & Add OR Place in COLUMN form [ original signs original signs & place opposite sign above]& Add

101 6.3 Subtracting Polynomials SUBTRACT these Polynomials: Ex:(7a - 10b) - (3a + 4b) Ex:(3y + 7y + 8) - (y - 4y + 3) Ex:(-8t 3 + 3t 7t - 9) - (-10t 3 5t + 3t - 11)

102 Mini-Quiz ) 3xy 3 3) x 3 + x + 1 4) Eval: -x + 3x 1 for x = - 5-7) Simplify: 5) 4(3y y) y(y + 4) 6) (3x + x) + (5x 8x) 7) 3(9x + 3x + 7) (11x 5x + 9) 8) Subtract: 0 x 3 + 1x 1x 3 + 7x 7x 1) Simplify: ) 3) Classify the Polynomial & state degree 9) Copy & Complete the table: x a) Write in scientific notation: 78 & b) Write in standard notation: 7.53 x x y 3 3 z z 4

103 6.4 Product Rule for Same Base Exponents PROD. OF POWERS a m. a n = a m + n Bases must be the same! Simplify each [Multiply coefficents 1 st ]: (1c 6 )(c 7 ) (8x 4 )(3x) (a 4 )(a 3 b )(-3ab 3 )

104 6.4 Product Rule for Exponents We multiply numbers in scientific notation using the same procedure we used to multiply monomials. Monomials: 4a a 4 ga a 9 Scientific notation: Ex: (4. x 10 5 )(.8 x 10 8 ) Ans: x 10 14

105 6.4 Power of a power Rule (a m ) n = a mn Ex: (x 3 ) 4 = x 3 x 3 x 3 x 3 =x 1 Simplify: Ex: ( ) 3 = Ex: (a b 4 ) 5 = Ex: (-5a b 6 ) 3 = Ex: Simplify (-5mn 4 )(-mp ) 3 (1.5m n) = Ans: 60m 6 n 5 p 6

106 6.4 Power Rules for Exponents POWER OF A PRODUCT (ab) m = a m b m Simplify means: No powers of powers, each base only once, & fractions reduced Ex: (3x y) 3 = Ex: (a 3 ) 3 (a 4 ) =

107 6.4 Using more than 1 rule Simplify: (6b 4 y) [(-y) ] 4 Simplify: (x 4 y 3 z 6 ) [(x y ) ] 4

108 6.5 Multiply Polynomial by Monomial Remove parentheses & Simplify: Ex: (a 3a) + 5(a + a) Use distributive Prop w/ ARROWS Ex: 5x(x + x + 1) - x(x - 3) Ex: a b 3a b ab 5a 4bc

109 6.5 Multiplying Binomial by Binomial F L If Binomials (x - 3)(x + 4) = I + O (3a + 11)(5a - ) = (5y z)(y + 3z) = (x + 3)(x - 1) + x(x 1) = use FOIL - add outer & inner terms under = sign

110 6.5 Multiplying Polynomial by Binomial Use 1 of these 3 Methods: 1. BOX. LONG MULTIPLICATION 3. MULTIPLE DISTRIBUTIVE and

111 6.5 Multiplying Polynomial by Binomial Use 1 st of these 3 Methods: (x + 3)(x + 3x + 8) = x x 3x 8 3

112 6.5 Multiplying Polynomial by Binomial Use nd of these 3 Methods:. LONG MULTIPLICATION (x + 3)(x + 3x + 8) =

113 6.5 Multiplying Polynomial by Binomial Use 3 rd of these 3 Methods: 3. MULTIPLE DISTRIBUTIVE and (x + 3)(x + 3x + 8) =

114 6.5 Multiplying Polynomial by Binomial Solve using your favorite Method: 1. BOX. LONG MULT. 3. MULTIPLE DISTRIB. (x - 1)(x - 4x + 3) =

115 FOIL 6.5 Multiplying Conjugates (a + b)(a - b) = a - b (x - 3)(x + 3) = EASY! Same binomials with different middle signs (x - y)(x + y) = (6x - 0y)(6x + 0y) =

116 6.5 Special Binomial Products Exs. using SQUARE OF SUM: (a + b) = a + ab + b Sq. 1 st term; twice prod. of terms; sq. last term (x + 5) = (3a + ) = (5b + 7c) =

117 6.5 Special Binomial Products SQUARE OF DIFFERENCE: FOIL (a - b) = (a - b)(a - b) = a - ab + b Sq. 1 st term; twice prod. of terms; sq. last term (x - 3) = (4a - ) = (6x - 10y) =

118 6.6 Quotient Rule for Exponents Develop a pattern: = 1 QUOTIENT OF POWERS [a 0] = a a m n = a m - n

119 6.6 Dividing By Monomials Exs. of QUOTIENT OF POWERS: 18x y z 4 3 4xy 3 6 = 3 yz 4x = x x x 30 3

120 Exponents Summary Assume that no denominators are 0, that a and b are real numbers, and that m and n are integers. Zero as an exponent: a 0 = 1, where a is indeterminate. Negative exponents: n n Product rule for exponents: Quotient rule for exponents: Raising a power to a power: Raising a product to a power: Raising a quotient to a power: a, a, 1 1 n a a n a ga a m n m n a a a a a ab a b m n m n m n mn n n n a a n n b n b a b n b a n

121 6.6 Division of Polynomials Polynomial by Monomial: each term of poly. by mono. = + Ex: Divide 1m m m 4 Note: Omit Objective 5 pages =>Dividing by a Binomial by 6m a b c c 0 a c b c Ans: m 4 + 3m 3 + 5m

122 6.6 Division of Polynomials Ex: xy x y x y x Ex: Divide: (x + y) (x y) by Simplify first! xy 1x y 3x xy 4

123 Mini-Quiz Review 1-5 Multiply 1) (3x 3 y 4 )(-4x 5 y) ) -5x 3 (4x + 3x 6) 3) (3a 4)(5a + ) 4) (5y 3) 5) (x 1)(5x + 4x 3) 6-9 Simplify: 6) p 1 p 4 7) (w) 4 (3w ) 8) 9) 3 a 6 5 a 3z a 3 5 y z 1 10) 4 5 w 1w 4w

124 FACTORING and QUADRATIC EQUATIONS

125 7.1 Factors & G. C. F s In multiplying: given factors find the product [ex: 5 7 = 70] In factoring: given product find the factors [ex: 70 = 5 7] A # w/ exactly factors [the # itself & 1] = PRIME number (whole #s > 1) Prime Factorization = when whole # expressed as product of prime factors

126 7.1 Factors & G. C. F s Find the PRIME FACTORIZATION of:

127 7.1 Factors & G. C. F s The Greatest Common Factor (G.C.F.) of or more integers is the greatest # that is a factor of all the integers: 1) Find the Prime Factorization of all the # s ) Multiply the common PRIME factors Find the G.C.F. of 45, 60, & 54

128 7.1 Factors & G. C. F s The Greatest Common Factor (G.C.F.) of or more monomials is the product of their common factors. Find the G.C.F. 0a b 3, 1ab 4, 40a 3 b 45x 3 y & 30x

129 7.1 Factoring Using Distributive Prop. A polynomial is in Factored form when expressed as a product of monomial & polynomial. Multiplying Polynomials (Ch. 6): x(x + 75) = x + 75x 5x(3x - 4y) = 15x - 0xy reverse Factoring Polynomials (Ch. 7): x + 75x = x(x + 75) 15x - 0xy = 5x(3x - 4y) In the above exs., we are factoring out the GCF

130 7.1 Factoring out the G.C.F. Use the Distributive Prop. to Factor each polynomial: 1) Find G.C.F. of each term ) Use Distrib. to write product of G.C.F. & remaining factor [Ck by multiplying back!] Ex: 1mn - 18m n Ex: 0abc + 15a c - 5ac

131 7.1 Factoring out the G.C.F. Exs: 8p 5 q + 16p 6 q 3-1p 4 q x yz 15x y 18x y Ex: y(y 1) 7(y 1) [Common binomial factor] Ck [in your head] w/ the Distributive Prop.

132 7.1 Factoring by Grouping Ex. w/ + between groups: 49x + 1y + 35x 3 y + 15x y 3 Ex. w/ - between groups: 49c - 7cd d

133 7.1 Factoring by Grouping Rules: 1.) Group terms in groups each w/ common factor.) Factor w/in groups [If can t, rearrange middle terms & try again] 3.) Factor out common binomial from entire polynomial Ex. Need to rearrange terms [rarely necessary]: 6y 0w +15yw 8yw

134 7.1 Factoring by Grouping If ALL terms have a GCF, factor it out first: Ex. -4t 4s 4tz 4sz

135 7. Factoring Trinomials: x +bx+c [lead coeff. = 1] Review: Factor. Factor = Product Binomial. Binomial = Trinomial [FOIL] FOIL (x + )(x + 3) = + Reverse FOIL: Product = Factor. Factor Given: x + 5x + 6 Find #s that mult. to 6 & add to 5 ADD MULTIPLY [ans. & 3] So x + 5x + 6 = (x + )(x + 3)

136 7. Factoring Trinomials [lead coeff. = 1] Ex: Factor: x + 1x + 0 Prod. = 0 Sum = 1 #s? Ans: (x + )(x + ) + Ex: Factor: x - 13x + 36 Ck by FOILing Ex: Factor: x - 9x Ex: Factor: x - x + 5 It s Prime!

137 7. Factoring Trinomials [lead coeff. = 1] Facts in Factoring x + bx + c Find integers whose prod. = c and sum = b Both integers positive if b & c both positive, Both ints. negative if c positive and b neg. 1 integer positive & 1 neg. if c neg. Ex: Factor: q q - 4 Ex Factoring out common monomial 1st: Factor: 3x 4-15x x

138 7. Factoring Trinomials [lead coeff. = 1] Trinomials in variables factor best if both variables are placed in ( )( ) 1st Ex: Factor: r 6rs + 8s = ( )( ) Ex: Factor: s + 6st 7t = ( )( )

139 Mini-Quiz ) 1a 3 b + 15ab 4 1) GCF = ) Factor: ( ) 3-4) a 3 x 3 - a x + ax 3) GCF = 4) Factor: ( ) 5-6) Factor by Grouping:5) 3a 6a + ab - b 6) w 3 - w + w ) Factor: 7) m + 10m + 1 8) 5b + 0b - 60 GCF? 9) n - 11n ) x - x - 30

140 7.3 Factoring Trinomials: ax +bx+c [lead coeff. 1] Trial & Check Factoring Method: Factor: 3x + 7x + = ( )( ) FOIL CK! Factor: 6h 7hk + k = ( )( ) + FOIL CK!

141 7.3Factoring Trinomials [lead coeff. 1] Group Method Review: Factor. Factor = Product Binomial. Binomial = Trinomial [FOIL] FOIL (3x + )(5x + 3) = + PRODUCT of 1st Last term of trinomial = Find all double factors of this 90 & say yes to the pair that SUM up to middle term:

142 7.3 Group Factoring Trinomials [lead coeff. 1] To Factor Trinomial: 1) Find factors of (1st term. Last term product) ) Which factor pair sums up to middle term [yes] 3) Rewrite the trinomial w/ yes factors added as equivalent middle term 4) Factor by grouping 5) FOIL check Ex: x + 9x + 10 Factors of 0 ==>0. 1 N 10. N 5. 4 Y Prod. = 0 Sum

143 7.3 Group Factoring Trinomials [lead coeff. 1] Ex continued: x + 9x + 10 Prod. = Y Sum = 9 3)Rewrite the trinomial w/ yes factors added as equivalent middle term 4)Factor by grouping x + 5x + 4x +10 x(x +5) + (x + 5) 5)FOIL check (x + 5)(x + ) +

144 7.3 Group Factoring Trinomials [lead coeff. 1] Follow 5 steps for Factoring Trinomials Ex: 3a + 17a + 10 Ex: 6x + 5x + 14

145 7.3 Group Factoring Trinomials [lead coeff. 1] Follow 5 steps for Factoring Trinomials Ex: 3p 4p + 1 Ex: 8x + 9x -1

146 7.3 Group Factoring Trinomials [lead coeff. 1] Follow 5 steps for Factoring Trinomials Ex: 5x + 13x + 3 Ex: 1a 13a +

147 7.3 Group Factoring Trinomials [lead coeff. 1] Follow 5 steps for Factoring Trinomials Ex: variables Ex: Common Factor 6m + 13mn 5n 1a a - 0

148 7.4 Factoring - Perfect Squares Ch 6 (x + y) = x + xy + y Ch 6 (x - y) = x xy + y PERFECT SQUARE TRINOMIAL Ch 7 Factor x + xy + y to (x + y) Ch 7 Factor x - xy + y to (x - y)

149 7.4 Factoring - Perfect Squares Determine if each is a PERFECT TRINOMIAL, if so Factor:[Caution: 1 st arrange descending] x - 1x + 36 Y N ( ) x + 14x - 49 Y N ( ) 4y + 36yz + 81z Y N ( ) 9n n Y N ( )

150 7.4 Factoring Diff. of Squares [to congugates] FOIL Chapter 5 product of (x + 8)(x - 8) = x + 8x - 8x - 64 = x - 64 [Diff. of sqs: x & 64] Rule: x - y = (x + y)(x - y) Ex: m - 81 Ex: 100x - 5y m - 9 GCF = ( )( ) ( ) Can binomial can be written as x - y? [( ) - ( ) ] ( )( )

151 7.4 Factoring Differences of Squares Rule: x - y = (x + y)(x - y) Look for GCF! Ex: 16a - 5b Ex: 0cd - 15c 5

152 7.4 Factoring Differences of Squares Rule: x - y = (x + y)(x - y) Keep going! Ex: 3k 4-48 Note: Omit Objective 3 & 4 pages 504 & 505 => Difference & Sum of cubes

153 Mini-Quiz ) Factor: 1) 5n + 7n + ) 5w + 9w + 4 3) x - 5x 3 4) 6c + 13c - 5 GCF? 5) 8x y 0xy 7y 6) 49w - 14w + 1 7) 5m + 0m + 4 8) b 49 GCF? 9) 50a 3 18a 10) y 4 16

154 7.5 Start Factor Flow Chart Factor it out Y GCF? N Diff of squares (x + y)(x - y) Factor it out Y # of terms 3 GCF? N Perfect Trinomial? Y (x + y) or (x - y) 4 GCF? N N Y Grouping ( )( ) The Hoo k Leading N Coeff. 1? Y Factor it out Prod/ Sum/ Group ( )( ) Mult/Add( )( )

155 7.5 Factor via the Flow Chart Use the flow chart to decide how to factor each then do: 4k 100 = 1y y 3 = 1y 3 (y + 7) t - 18t + 36 = 150x 3 y 10x y + 4xy 3 = x 5 - x 3-8x + 8 = 6xy(5x 0xy + 4y ) 6xy(5x y)

156 7.6 Solve Quadratic Eqs. by Factoring 1 st Degree polynomials in Linear Equations Ex: 6x 1 = 0 Solve: [1 solution] nd Degree polynomials in Quadratic Equations Ex: 6x 3x = 0 [ solutions] Quadratic Equation in form ax + bx + c = 0 (a 0) [a, b, c Reals] Quadratic Form If equation isn t in Quadratic Form, use + & - properties to get 0 alone on rt. side

157 7.6 Solve Quadratic Eqs. by Factoring ZERO PRODUCT PROPERTY: For all #s a & b, IF a. b = 0 THEN a = 0, b = 0, or both a and b = 0 Ex: (x + 3)(x - 5) = 0 Solve & Check: Ex: (a + 4)(a + 7) = 0

158 7.6 Solve Quadratic Eqs. by Factoring ZERO PRODUCT PROPERTY: IF a. b = 0 THEN a = 0, b = 0, or both a and b = 0 Solve & check: Ex: x = 9x[Don t both sides by a variable cuz it could = 0] Ex: x - 36 = 5x Ex: a - 4a = -144

159 7.6 Solve Quadratic Eqs. by Factoring Ex: 4m + 5 = 0m Solve & check: Ex: x 3 + x = 15x [Look for common factor] Ex: a 3-13a + 4a = 0 [Look for common factor]

160 7.6 Applications of Quadratic Equations Ex: The length of a hall is 5 times the width. The area of the floor is 45 sq. meters. Find the hall s length & width. [Mark diagram with the facts]

161 7.6 Applications of Quadratic Equations Pythagorean Formula: a + b = c a Ex: The hypotenuse of a rt. is 3 longer than the longer leg. The shorter leg is 3 shorter than the longer leg. Find the lengths of all sides. [Mark diagram with the facts] b c

162 Mini-Quiz ) Perf. Sq. Trinomials? If Y, FACTOR; If N, why? 1) x + 10x + 5 Y N ( ) ) w 6w - 9 Y N ( ) 3) 5y 30yz + 9z Y N ( ) 4-6) Factor 4) 5a x 3 y 0b xy 5) n 4 p 3p 6) 81p 4 16q 4 7 & 9) Solve by Factoring: 7) x + x - 15 = 0 8) Check #7 [All Solutions!] Extra Credit 9) k 3 8k = - 1k 10) w The length of the rectangle is 1 cm more than twice the width. The Area = 36 sq. cm. Find the Perimeter. [Solve by factoring!]

163 RATIONAL ¾ POLOYNOMIAL EXPRESSIONS

164 8.1 Simplifying Rational Expressions Rational Expression = Quotient of polynomials [divisor 0] P Form: Both Polynomials Q 0 Q Ex: Ex: Ex: 3a 3 b 4ab 4xy z 7xy 3x x 4 4 8x 9 Simplify using 3 ac bc a b [b & c 0]

165 8.1 Evaluate Rational Expressions Evaluate the expression 7x 9, x 1 when a. x = b. x = 1 To Find Value(s) That Make a Rational Expression Undefined: 1. Write an equation that has the denominator = 0. Solve the equation. Ex: When is this undefined? 3r r 6r 8

166 8.1 Simplifying Rational Expressions Write in Lowest Terms: Exs: 1 a a a b ab a b a FACTOR! k k x x x x 5 5 x x x x x x x x

167 8.1 Simplifying Rational Expressions Write in Lowest Terms [Factor & Slash]: Exs: 1 a a a b ab a b a 1 x x x x 5 5 x x x x x x x x

168 8. Mult. [] & of Rational Expressions Mult. [] of Rational Expressions: Ex: 8p 3 q 9 pq P Q R S PR QS Ans. in lowest terms! Ex: 3( p p q) ( q p q) Ex: x 7x 3x 10 6 x 6x x 6 15

169 8. Mult. [] & of Rational Expressions Rational Expressions: Ex: P Q 0 R S P Q S R PS QR Keep Change Flip Ans. in lowest terms! Ex: 9 p 3p 4 6 p 3p 3 4 Ex: 5a b 10ab 8

170 8. Mult. [] & of Rational Expressions Rational Expressions: Ex: Ex: Ans. in lowest terms! 1 4 3) ( 1 3) ( 4 x x x x x x 1 1 a a b a a a ab

171 8. Combining Operations Ex: Ex: Ans. in lowest terms! b b a ab a b a b ab ab a x x x x x x x x x

172 8.3 + & - of Rational Expressions~Like Denom. Adding Rational Expressions: Ex: Same Denominator x 4x P Q R Q P R Q Ex: x x y x 1 y

173 8.3 + & - of Rational Expressions~Like Denom. Subtracting Rational Expressions: Same Denominator Ex: Ex: Q R P Q R Q P n n n x x x x x

174 8.4 LCD ~ Least Common Denominator LCD = Least expression all denominators into w/out remainder To find LCD: 1) FACTOR each denom. ) LIST factors using greatest # of times anywhere 3) MULTIPLY factors in ) for LCD Ex: Find LCD for 9 8m 4 11 and 1m 6 Ex: for x 6 4x and 3x x 1 16

175 8.4 LCD ~ Least Common Denominator Write equivalent fractions over an LCD Ex: Rewrite over new denominator: Ex: k 5k FACTOR! 1 k k( k )( k 1) 7k 5 30

176 8.4 + & - of Rational Expressions Adding Rational Expressions: Different Denominators: 1) Find LCD ) Rewrite Fractions over this LCD 3) Add numerators & put over this LCD 4) Ans. in Lowest Terms Ex: m 3n 7n Ex: p 1 p 4 p 1

177 8.4 + & - of Rational Expressions Adding Rational Expressions: Different Denominators: Ex: k k 5k 4 k 3 1 Ex: m n m 3n 3n m

178 8.4 + & - of Rational Expressions Subtracting Rational Expressions: Review: Ex: Same Denominator Different Denominators Ex: 5x 5 x 1 x 1 6 a a 1 3 LCD P Q R Q P Q R Ex: 4x 3x x x LCD

179 8.4 + & - of Rational Expressions Different Denominators Ex: r 3r 5r r 4 10r 5 LCD Ex: a a a 1 1 6a 6

180 Ex: x x & - of Rational Expressions x x 4 1 Ex: 7y y y

181 Mini-Quiz Perform operation & simplify: 9 0xz 4xy 15y 1) ) 3) 6x xy xy 6x 4)&5) Find the LCD 3 z 9 6 x y 4) 5) 3, x y 5 z x 9y 3 y 6 y 3y 6y x 10x, 5 x 5 5 3k 3k 5 6k 3k ) 7) 8) b b 5 b 4 x x 3 3 c 4c c 9) 10) c Were Mini-Quizzes helpful? Y or N

182 PROBLEM SOLVING

183 3.1 Ratios & Proportions Ratio = Quotient of #s or quantities [a way to compare numerical quantities] 7 9 Ex: Ex: 1 to 7 Ex: 35:50 Are any of these ratios equal? Ex: Express the ratio 3. to 16 in lowest terms Ex: Express the ratio 1 foot to yds in lowest terms [make units same]

184 3.1 Ratios, Rates, & Proportions Unit Ratio = Ratio w/ denominator = 1 [ex: cost per pound or cost per oz.] Ex: Which is a better buy? 1 oz Coke for 79 or a 16 oz Coke for 99 Rate = Compare quantities of different units Ex: miles per hour [MPH] Ex: What is the hourly rate for $640 earned for 40 hrs work?

185 3.1 Ratios, Rates, & Proportions Ratio = Comparison by division a to b, a : b or a b [units the same] Proportion = equal ratios: [b,d 0] a b c d extremes = iff [if & only if] ad = bc [Read a is to b as c is to d] means Ex: Is = a proportion? Y or N

186 3.1 Solving Proportions Solve equations involving Proportions x a 6 a Ex Ex = 1 5 Set up ---- for word problems: Ex: If 7 shirts cost $87.50, find the cost of 11 shirts.

187 3.1 Similar s have corresponding sides pptnl Similar s = same shape; different size A b C ABC~ DEF means a c Ex: A tree casts a shadow of 18 when a 5 person casts a shadow of 1.5 How tall is the tree? B D e F a d d f b e E c f Note: Similar figures [trapezoids, pentagons, ] also have pptnl sides.

188 3. ~ % Problems Rewriting Percent: Ratio representing some part of 100 Ex: Write as reduced fraction & as a decimal: 6% = [ by 100] Ex: Write a fraction or decimal as a %: = [mult. By 100%] = 3 8

189 3. ~ % Problems Use translation & algebra Translation: What is 8% of 70? What => n of => times is => equals 14 is what % of 5? % =>. 80 is 0% of what number?

190 3.3 Problems Translate word situation equation Steps: 1) Chose a variable for what you are to find [Write: Let x = ] Write out facts [w/ your var.], pictures,... ) Translate prob. to an equation [is =, etc.] 3) Solve equation [legally w/ properties] 4) Answer question(s) asked [may be more than what x equals] 5) Check [Sub. solution into original and work down til both sides of = sign the same. Then ]

191 3.3 ~ Problems Translate word situation equation Ex: One positive # is one-third another positive #. The larger Eq: # minus the smaller # is equal to 15. Find both #s. Let x = then = Solve:.5 & 7.5 Ex: The perimeter of a rectangular frame is 36 cm. The length is 3 less than twice the width. Find length & width. 11cm & 7cm

192 3.3 ~ Problems w/ or more unknowns Note: Supplementary /s sum to180 o & Complementary /s sum to 90 o Ex: One / is 6 o less than 3 times it s complement. Find both /s Ex: The sum of 3 consecutive integers is 96. Find them. 66 o & 4 o 31, 3, 33 Ex: Find the meas. of all 3 /s of a if the nd / is twice the 1 st / and the 3 rd is 0 less than the nd / Note: Sum of / measures of a = 180 o 40 o, 80 o, 60 o

193 Mini-Quiz ~ SHOW ALL WORK 1) Translate to eq.:what is 60% of 00? ) Solve 1) 3) Translate to eq.: 75 is 0% of what #? 4) Solve ) 5-8) Show set up: [---] & use proportions to solve: 5) If a car travels 91 mi. in hrs., how far will it travel in 7 hrs? 6) shirts cost $5, how much will 5 shirts cost? 7) A chef needs 4 large bottles of ketchup to make gal. of 3 sauce. How many bottles for 10 gal. sauce? 8) If a 5 ft. person casts a 6 ft. shadow, how tall is a building that casts a 30 ft. shadow in the same sun? 9) Translate to eq.:the sum of Sam & Jan s age is 51. Sam is yrs older than 6 times Jan s age. Their ages?10) Solve 9) 1

194 3.4 ~ % Decrease or % Increase Probs Decrease: $Orig - %Decr. of $Orig = $Sale Ex: Find the Original price (x) of a shirt that is now 30% off resulting in a sale price of $8.00 Increase: $Orig + %Incr. of $Orig = $Final

195 Distance = Rate Time [d = rt] Diagram Motion Problems ~ Ex: cars are traveling in opposite directions, 40mph & the 50mph. When will they be 180 miles apart? 40mph 3.4 ~ Rate Problems ~ DISTANCE 50mph Looking for t Eq. 180 mi.

196 3.4 ~ Rate Problems Note: d = r t Objects traveling in opposite directions: Ex: trains leave the station at the same time in opposite directions. 40 MPH & 55 MPH. In how many hrs will they be 190 miles apart? hrs

197 3.4 ~ Rate Problems Note: d = r t Objects traveling in same direction: Ex: Two trains are delivering to the same site. One leaves at 8:00 a.m. and the other leaves at 8:15 a.m. If the first train is traveling at 55 mph and the second at 60 mph, at what time will the second catch up to the first? Train 1 st : Train nd : 8:00 am 8:15 am Time = t Time = 11:00 am

198 3.5 ~ Investment Problems (I)nterest=(P)rincipal (r)ate [% as decimal] (t)ime[in yrs] Ex [easy]: Plug & Chug: How much interest on $000 invested at 5% for 8 years? Problem below = Extra Credit I = Prt Ex:You invest a total of $4000 in two different accounts. The first account earns 6% while the second account earns 4%. If the total interest earned is $10 after one year, what principle was invested in each account? Accounts Principal Rate Interest First 4000 P 0.06 Second P (4000 p) p = 10 What can you mult. each term by? Answer The principal invested in the second account is $1500. In the first account is $4000 $1500 = $500.

199 3.5 ~ Investment Problems I = Prt Problem below = Extra Credit (I)nterest=(P)rincipal (r)ate [% as decimal] (t)ime[in yrs] Ex: You invest some 8% and $3000 more than twice as 10%. Total annual income from this is $540. How much 8%? Principal Rate Interest P.08.08P P (P+3000) Eq:.08P +.10(P ) = 540 What can you mult. each term by? Ans: $8000

200 Mini-Quiz ~ SHOW ALL WORK 1-5) Distance Prob: Two trains are traveling toward each other from a distance of 08 miles. One train is traveling at 18 miles per hour and the other at 46 miles per hour. How long will it take for them to pass each other? Problem below = Extra Credit 6-10) Distance Prob: [see page 14 ~ Your Turn ] Juan & Angela are bicycling along the same trail. Juan passes a 9:00 am, and Angela passes the same 9:05 am. Juan is traveling 8mph while Angela is traveling 10 mph. What time will Angela catch up to Juan? J: A: 9:00 am 9:05 am Time = t Time = 3.5 hours 9:5 am

201 GRAPHING LINEAR EQUATIONS

202 4.1 The Rectangular Coordinate System II I Quadrant. B x y. A Pts. named by Ordered Pairs: (x,y). C III. D Origin (0,0) IV Find coord. of A, B, C, D

203 4.1 The Coordinate Plane y II I Quadrant (-, +). (+, +) < >x Pts. named by Ordered Pairs: (x,y) III (-, -) (+, -) IV Origin (0,0)

204 4.1 Pts linear or non-linear? (-5,7), (-1,3), (,0), (4,-), (6,-4) y < >x

205 4. Linear Eqs. ~ Variables Linear Equation in 1 variable: A,B Reals Ax + B = 0 A 0 Linear Equation in variables: A,B,C Ax + By = C A & B both 0 We are going to graph these now in the rectangular coordinate system Ch graphs = pt. on a line Reals

206 4. Linear Eqs. ~ Variables Solution to a Linear Equation in variables: Ax + By = C is (x, y) an ORDERED PAIR [Note: the x is always 1 st & y always nd ] Ex: Is (, -5) a solution of 5x + y = 0? Y N Ex: Complete the ordered pair (, 7) for y = x - 9

207 4. Linear Eqs. ~ Variables - Table Ex: Complete the table of values for x 3y = 1 then write results as ordered prs: x y

208 4. Graphing:Linear Eqs. ~ Variables Graph the Linear Eq: 5x + y = -10 Use 3 entries in a table of values for your 3 ordered pairs: x y Use x-intercept [when y=0] Use y-intercept [when x=0] Use 3 rd pair as a check

209 4. Identifying Linear Eqs. Linear Equation in variables: A,B,C Reals Ax + By = C GENERAL FORM [A & B both 0] Write with A, B & C as Integers & A > 0 Exponents of x & y must be 1 for eq. To be Linear Ex: Solve for y, generate a (x, y) table of Integers & Graph: x y 3y = 3 + x

210 4.3 Graphing:Eqs. ~ using Intercepts Find intercepts of: 5x + y = 10 (, );(, ) Find intercepts of: x - 3y = 1 (, );(, ) Find intercepts of: 4x - y = 0 Graph of equation: Ax + By = 0 passes through

211 4.3 Graphing:Linear Eqs. ~ Variables Graph of equation: y = k is Ex: y + 5 = 0 Graph of equation: x = k is Ex: x =

212 Mini-Quiz Draw 4 xy-axes on ans sheet back 1-3 [top axis] Plot & label A(-3, 1) B(4, -) C(-,0) [on front] Specify quadrant or axis for the location of your 3 pts:1) A ) B 3) C 4) Which of these equations are LINEAR? A) y = x B) y = x 3 C) y = x D) x + 3y = 7 5) Identify & write as ordered pairs the x & y intercepts for: - 4x + y = 8 6) Is (-3, 5) a solution of 4x + 3y = 3? Y or N 7) Is (-3, 5) a solution of y = + x? Y or N 8-10) Put table of 3 integer values sideways of front & graph on the 3 remaining axes on the back: 8) 3x + y = 6 9) y = 4 10) x = -

213 4.4 Slope of a Line Slope - Slant uphillor Slant downhill - Steepness of h very steep > least steep h h h h Black Diamond ==============> Easy Circle

214 4.4 Slope of a Line Slope of a line ==> Ratio of the rise [Vert. Change] to the run [Horiz. Change] Change in y Slope = rise Change in x run Rise 3 Run 5 Slope = 3 5

215 4.4 SLOPE = Rise Positive Rise up & Run forward Run Negative Rise up & Run bckwrds

216 4.4 Slope of a Line Every line has a SLOPE Pos. Slope 4 3 Neg. Slope 5 3 Horiz. = 0 slope Vert. = undefined slope

217 4.4 Slope of a Line Line positions that DO have slope:: Slope of line thru pts: (x 1, y 1 ) & (x, y ) is m = y x Pos. slope --> Incre. --> slant up 0 slope --> no change --> horizontal Neg. slope--> Decre. --> slant down y x 1 1 Vertical change (rise) Horizontal change (run)

218 4.4 Slope of a Line Slope = m = y y1 x 1 x x x 1 Ex: Find slope of line thru (-5, 3) & (, 1) Ex: Find slope of line thru (4, 3) & (1, -1) Always: Label one pt. x 1, y 1 & other x, y

219 4.4 Slope of a Line Slope = m = y y1 x 1 x x x 1 Ex: Find slope of a line thru (-1, -) and (1, -7) Ex. Graph a line thru (3, -5) with slope of 5

220 4.4 Slope of a Line y y1 Slope: m = x 1 x x x 1 Find slope of the Horizontal line thru (-5, 3) & (, 3) eq: y = k slope 0 Find slope of the Vertical line thru (-5, 3) & (-5, 1) eq: x = k slope undefined

221 4.4 Find Slope from the eq.of a Line Steps to Find Slope from the eq.of a Line: (1)Solve the equation for y ()The slope [m] is the coefficient of x (3) The y-intercept is the (0, b) in y = mx + b Ex: Find the y-int & slope of : 3x + y = 6 & graph it SLOPE-INTERCEPT FORM

222 4.4 Using y = mx + b to graph equations Ex: Find the y-int & slope of : 4x - 3y = 1 & graph it Ex: Find the y-int & slope of : x + y = 4 & graph it

223 4.4 Find Slope from the eq.of a Line Graph each of the following on the same grid. y = x y = 3x y = 4x Solution Complete a table of values. If x is y = x y = 3x y = 4x y = 4x y = x y = x y = 3x 3x

224 4.4 Slope of Parallel Lines Parallel Lines in coord. Plane: Never Intersect If non-vertical, then slopes are equal

225 4.4 Slope of Perpendicular Lines Perpendicular Lines in coord. Plane: If product of slopes = -1 ==> lines [Negative reciprocals] If vertical & horizontal ==> lines Ex: Decide if these lines are,, or neither: 3x y = 4 x + 3y = 9

226 Mini-Quiz 4.4 & Review 1) Which ordered pair is a solution of equation 3x y = 6? a) (4, 3) b) (1, 5) c) (, 0) d) (0, 3) ) & 3) For the equation 7x y = 14, find the ) x- and 3) y-intercepts. Write these as ordered pairs 4) & 5) Find the slope of line thru 4) (-5,-) & (7,-5) and 5) (4,-1) & (-3,-1) 6) For the equation x + 5y = 0, determine the slope and the y-intercept. Then draw axes & graph the equation. on back for 7) 8) Write the slope of a line parallel to: x - y + 7 = 0 9) Write the slope of a line perpendicular to that in 8) 10) Are 5x 3y = 11 & 3x + 5y = 8,, or neither?

227 RADICAL EXPRESSIONS

228 9.1 Square Roots Squares: the square of 5 is 5 cuz 5 = 5 and the square of -5 is 5 cuz (-5) = 5 Square Roots: 5 is the square root of 5 cuz 5 = 5 and -5 is the square root of 5 cuz (-5) = 5 b is square root of a if b = a All positive #s have sq. roots. It s pos. sq. root = principal square root

229 9.1 Square Roots Square Root of a => [a = positive real #] Also: = ( ) = a and Note: Ex: a a a a = represents the POSITIVE sq. root of a Ex: a 9 16 Ex: 8 36x Ex: 6 m 49u 4

230 9.1 Square Roots Note: If a = POSITIVE # that is NOT a perfect sq., then a is IRRATIONAL and If a = NEGATIVE #, then a is NOT REAL or Imaginary Identify as Rational, Irrational, or Imaginary: Ex: Ex: Ex:

231 9. Mult.& Simplifying Radical Expressions Product Rule for Radicals: [a & b nonneg. Reals] Find: a and visa versa Simplify Radicals no PERFECT SQUARE under the radical sign Simplify: b a b Find Product & Simplify:

232 9. Simplifying Radical Expressions Simplify: x y c 5q 63p q Mult. & Simplify: 3 3 w 4 8w 7 x 3 3x 4 3 5yz 45yz

233 9.3 & Simplify Radical Expressions Quotient Rule for Radicals: [a & b nonneg. Reals] a a y 0 and visa versa b b Simplify: y 9 160y b 16a x b

234 9.4 + and - of Radical Expressions Like Radicals: multiples of same root of the same # Ex: and Add or subtract LIKE radicals: Add: Simplify 1 st! Subtract: Simplify: 3r 6 8r