1. Int roduc t ory Mat hem at ic al Analysis forB u s i n e s s , Ec o n o m i c s a n d t h e L i f e a n d So c i a l Sc i e n c e s

2. Review of algebraMore Algebra 3. 25 POINTS: 5 QUIZES25 POINTS: FINAL50 POINTS: CALULAS (after the mid-term) 4. Review of the Subsets of theReal Number System 5. N- counting numbers- 6. WHOLE NUMBERS (W (W) 7. INTEGERS (Z) , 8. RATIONAL NUMBERS (Q (Q)numbers that can be expressed as aquotient a/b, where a and b are integers.terminating or repeating decimals 9. cAny number that cannot be written as a 3 1.732050808 ratio (fraction) 32 1.492106248 Any number whose e1 2.718281828 decimal 3.141592654 representation neither repeats or stops. 10. Set of all rational and irrational numbers. c A Real Number is any number that can be graphed in the number line 11. . Graph the numbers 0.2, 7 , 1, 2 , and4 on a number line. 10 7 410.22 104321 0 1 2 34 12. I M c Q A G Z I N W A R Y N 13. Properties of Addition & MultiplicationProperties of Equivalence or EqualityProperties of Order or Inequality 14. CLOSURE PROPERTYGiven real numbers a and b, Then, or 15. Is the set of integers CLOSED with respect to: 1. Addition 2. Subtraction 3. Multiplication 4. Division 16. Given real numbers a and b, Addition: Multiplication: 17. Given real numbers a, b and c, Addition: Multiplication: 18. Given real numbers a, b and c, 19. Given a real number a, Addition: Multiplication: 20. Given a real number a, Addition: Multiplication:-1 a 0 21. For any real numbers a, b, and cAdditionMultiplication Commutative a b b aa bb a Associative (a b) c a (b c) (a *b)* c a *(b * c) Identity a+0=a=0+aa aaIf a in not zero, then Inverse a ( a) 0 ( a) a 11a *a 1a *a 22. Pr o p e r t i e s o f Eq u i v a l e n c e o r Eq u a l i t y Addition and For any reals a, b, and c, if Subtraction a=b then a+c=b+c anda-c=b-c Multiplication For any reals a, b, and c, if and Division a=b then a*c=b*c, and, if cis not zero, a/c=b/c 23. Pr o p e r t i e s o f Or d e r o r Inequalit y Result: Between any two real numbers there is a rational number and an irrational number.1- For any real numbers a and b either a < b , b < a or a=b2- If a < b and b < c then a < c 24. 3- If a < b then a+c < b+c c R4- If a < b and c > 0 ,then ac < bc and a b c c If a < b and c < 0, then ac > bcand ab c c 25. 5- If a < b and a = c , then c < b6- If 0 < a < b or a < b < 0, then 11 ab 26. 7- If 0 < a < b and n > 0 , then an < bnand n a < n b 27. - is an element of ~ - not or negation of - Union - Intersection - is a subset of - and (conjunctive/ intersection of two sets) - or (disjunctive/ union of two sets)> - Greater than & Greater than or equal to < - Less than & Less than or equal to = - equal and not equal to 28. Tell w hic h of t he propert ies of real num bers just ifies eac h of t he follow ing st at em ent s. 1. (2 )(3 ) + (2 )(5 ) = 2 (3 + 5 ) 2. (1 0 + 5 ) + 3 = 1 0 + (5 + 3 ) 3. (2 )(1 0 ) + (3 )(1 0 ) = (2 + 3 )(1 0 ) 4. (1 0 )(4 )(1 0 ) = (4 )(1 0 )(1 0 ) 5. 1 0 + (4 + 1 0 ) = 1 0 + (1 0 + 4 ) 6. 1 0 [ (4 )(1 0 )] = [ (4 )(1 0 )] 1 0 7. [ (4 )(1 0 )] 1 0 = 4 [ (1 0 )(1 0 )] 8. 3 + 0 .3 3 i s a r e a l n u m b e r 29. n 1 n n 30. 4 31. n one +ve rootno real one -ve root rootsone real one +ve root no +ve rootsroot, 0one -ve root no -ve roots 32. 1 3 3 31 4 4 4 33. 2 2 2 4 4 2 22 34. 3 2 3 2 3 6 3 3 3 3 3 35. Review basic laws of exponents and radicals on page 10 36. Simplify and express all answers in terms of positive exponents Answer: 37. Simplify the expressions Answer: 38. Write the expression in terms of positive exponents only. Avoid all radicals in the final form.Answer: 39. Write the exponential forms involving radicalsAnswer: 40. Simplify the expressions. Express all answers in terms of positive exponents. Rationalize the denominator where necessary to avoid fractional exponents of denominator.Answer: 41. Simplify the expressions. Express all answers in terms of positive exponents. Rationalize the denominator where necessary to avoid fractional exponents of denominator. Answer: 42. Algebraic expressions are numbers represented by symbols which are combined by any or all of the arithmetic operations such as addition, subtraction, multiplication and division as well as exponentiation and extraction of roots. 43. Algebraic expressions with exactly one term : monomials with exactly two terms: binomials with exactly three terms: trinomials with more than one term: multinomials 44. Adding Algebraic Expression Prob.7 Prob.7 (Sec. 0.4) Perform the indicated operations and simplify (6x2 + 10xy + 2) (2z xy + 4)10xyAnswer: (6x2 + 10xy + 2) (2z xy + 4) (6 10xy= 6x2 + 10xy + 2 2z + xy - 410xy= 6x2 + 10xy + xy - 2z + 2 410xy= 6x2 + 11xy - 2z + 2 411xy 45. Subtracting Algebraic Expression Prob.13 Prob.13 (Sec. 0.4) Perform the indicated operations and simplify 3(x2 + y2) x(y + 2x) + 2y(x + 3y)Answer: 3x2 + 3y2 xy - 2x2 + 2xy + 6y2= 3x2 - 2x2 + 3y2 + 6y2 xy + 2xy= x2 + 9y2 + xy 46. Removing Grouping Symbols Prob.15 Prob.15 (Sec. 0.4) Perform the indicated operations and simplify 2[3[3(x2 + 2) 2(x2 5)]] )]]Answer: 2[3[3x2 + 6 2x2 + 10]]10] = 2[3[3x2 - 2x2 + 6 + 10]] (rearranging) 10] = 2[3[x2 + 16]] = 2[3x2 + 48] = 6x2 + 9616]48] 47. Special Products Refer to page 18 of textbook for list of rules for special products Prob.19 (Sec. 0.4) Perform the indicated Prob.19 operations and simplify (x + 4)(x + 5)Answer: (x +4)(x + 5) = x2 + 5x + 4x + 20= x2 + 9x + 20 48. Multiplying Multinomials Prob.35 Prob.35 (Sec. 0.4) Perform the indicated operations and simplify (x2 - 4)(3x2 + 2x - 1))(3Answer: x2(3x2 + 2x 1) 4(3x2 + 2x1)= 3x4 + 2x3 - x2 - 12x2 - 8x + 4 12x= 3x4 + 2x3 - 13x2 - 8x + 413x 49. Dividing a Multinomial by a Multinomial Prob.47 Prob.47 (Sec. 0.4) Perform the indicated operations and simplify Answer: 50. Long Division Prob.51 Prob.51 (Sec. 0.4) Perform the indicated operations and simplify(3x3-2x2+x 3)(x + 2) 51. Factoring is rewriting expression as a product of 2 or more factors E.g. If c = ab, then a and b are factors of c Refer to page 21 of textbook for list of rules for factoring 52. Common Factors Prob.5 Prob.5 (Sec. 0.5) Factor the following expressions completely 8a3bc - 12ab3cd + 4b4c2d2 12abAnswer: 4bc(2a3 - 3ab2d + b3cd2)bc(2 53. Factoring Trinomials Prob.9 Prob.9 (Sec. 0.5) Factor the following expressions completely p2 + 4p + 3Answer: (p+1)(p+3 (p+1)(p+3) 54. Algebraic expressions which are fractions can be simplified multiplying and dividing both numerator and denominator of a fraction by the same non- non-zero quantity 55. Simplifying Fractions Prob.3 Prob.3 (Sec. 0.6) Simplify the expressions Answer: 56. Multiplying and Dividing Fractions Rule for multiplying with isRule for dividing with where c 0 is 57. Multiplying and Dividing Fractions Prob.11 Prob.11 (Sec. 0.6) Simplify the expressionsAnswer: 58. Rationalizing the Denominator Prob.53 Prob.53 (Sec. 0.6) Simplify and express your answer in a form that is free of radicals in the denominator 59. Addition and Subtraction of Fractions Prob.29 Prob.29 (Sec. 0.6) Perform the operations and simplify as much as possible Answer: 60. Addition and Subtraction of Fractions Prob.39 (Sec. 0.6) Prob.39 Perform the operations and simplify as much as possible 61. Addition and Subtraction of Fractions Prob.47 Prob.47 (Sec. 0.6) Factor the following expressions completelyAnswer: 62. Sec. 0.1 4, 5, 7, 8 Sec. 0.2 3, 4, 10, 2210, Sec. 0.3 55, 58, 65, 85,55, 58, 65, 85, 90 Sec. 0.4 18, 37, 39,18, 37, 39, 50 Sec. 0.5 38, 40, 46,38, 40, 46, 50 Sec. 0.6 10, 34, 46,10, 34, 46, 59 Sec. 0.7 30, 37, 46, 59, 71, 75, 85, 92,30, 37, 46, 59, 71, 75, 85, 92, 105 Sec. 0.8 25, 32, 40, 47, 54, 74,25, 32, 40, 47, 54, 74, 84 63. Mathematical Systems,groups and fields a set of elementsOne or more operations defined on this setDefinitions and rules for applying the operationson the set.Theorems can be deduced from the givendefinitions and rules. 64. The set G is closed under the operation * The operation * is associative There is an identity element e of G for * There is an inverse element for every element of G 65. A Field is a mathematical system that consisting a set F and two operations thatsatisfy 11 properties: 66. Domain a set of all possible replacement values for a given variable. Ex. D= { x| x R} Quantifier a word or phrase that describes in general terms the part of the domain for which a sentence/ statement is true. Ex. - , 67. Universal Quantifier a statement that has the same truth value for every element of the domain.X = > For all/ every x Existential Quantifier a phrase that describes a statement as being true for some or at least one element from the domain.: X : x + 3 = 7 = > There exist a value forx such that x + 3 = 7 68. Negating Quantified StatementsThe negation of a universally quantified statement p is an existentially quantified statement of the negation of p (~ p). The negation of x p is expressed as x ~p The negation of an existentially quantified statement p is a universally quantified statement of the negation of p (~ p). The negation of x p is expressed as x ~p 69. X :x= 2X :x> 7X ~ kSome animals can flySome rectangles are squares 70. Set - a collection of objects or elements. 2 types - Finite & Infinite eg. { 1 , 2 , 3 } { 2 , 4 , 6 , 8 } { all w omen < 21 }21} { }Subset - a set whose entire contents also belongs to another set 71. Define each of the following and show how they are represented? Empty Set Union of sets I ntersection of sets Universal set - Complement of a set 72. Empty Seta set with no members - { } orUnion of sets A set containing the members of both/all given given sets Union of sets A set containing all all the members of both/all setsIntersection of sets a set containing only members that are elements of BOTH/ALL setsUniversal set - The complete set or groups of elements from which solution variables/subsets can be chosen. Normally the Universal set is also the Domain.Complement of a set If a subset A of elements is identified within Universal set U,elements is identified within Universal set U, the complement A is all the elements that are NOT in the identified set, but are if the universal set.ex. if the universal set is the set of natural numbers, and the set of even numbers is identified, then the complement of that set is the set of odd numbers.