0
Int roduc t ory Mat hem at ic al Analysis for
 B 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
              ...
Review of algebra

More Algebra
25 POINTS: 5 QUIZES

25 POINTS: FINAL

50 POINTS: CALULAS (after the
mid-term)
Review of the Subsets of the
   Real Number System
N

- counting numbers

-
WHOLE NUMBERS (W
              (W
               )
INTEGERS (Z)

               ,
RATIONAL NUMBERS (Q
                  (Q
                   )

numbers that can be expressed as a
 quotient a/b, where a a...
c

Any number that
cannot be written as a       3 1.732050808
ratio (fraction)         3
                             2 1....
Set of all rational and irrational
numbers.
                          c


A Real Number is any number that can be
graphed ...
.   Graph the numbers   0.2, 7 , 1, 2 , and        4 on a
    number line.             10



                             ...
I
M   c               Q
A
G
                Z
I
N           W
A
R
Y
        N
Properties of Addition &
Multiplication

Properties of Equivalence or
Equality

Properties of Order or Inequality
CLOSURE PROPERTY

Given real numbers a and b,
Then,
or
Is the set of integers CLOSED
with respect to:
    1. Addition
    2. Subtraction
    3. Multiplication
    4. Division
Given real numbers a and b,
Addition:
Multiplication:
Given real numbers a, b and c,
Addition:


Multiplication:
Given real numbers a, b and c,
Given a real number a,
Addition:
Multiplication:
Given a real number a,
Addition:
Multiplication:      -1

 a   0
For any real numbers a, b, and c
                   Addition        Multiplication
Commutative
                a b b a    ...
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, i...
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 n...
3- If a < b then a+c < b+c c R

4- If a < b and c > 0 ,then ac < bc
                        and a     b
                  ...
5- If a < b and a = c , then c < b

6- If 0 < a < b or a < b < 0, then
                1    1
                a    b
7-   If 0 < a < b and n > 0 , then
                    an < bn    and n a < n b
- is an element of
  ~ - not or negation of
    - Union
    - Intersection
    - is a subset of
    - and (conjunctive/ in...
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 )...
n


        1
    n   n
4
n




one +ve root    no real
one -ve root     roots        one real
one +ve root   no +ve roots
                         ...
1
3   3       3

        1
4       4       4
2 2         2
4




    4         2 2



              2
3       2 3       2
3       6




3   3       3       3         3
Review basic laws of exponents and
radicals on page 10
Simplify and express all answers in terms of
positive exponents




Answer:
Simplify the expressions




Answer:
Write the expression in terms of positive
exponents only. Avoid all radicals in the final
form.



Answer:
Write the exponential forms involving radicals



Answer:
Simplify the expressions. Express all answers
in terms of positive exponents. Rationalize the
denominator where necessary ...
Simplify the expressions. Express all answers
in terms of positive exponents. Rationalize the
denominator where necessary ...
Algebraic expressions are numbers
represented by symbols which are
combined by any or all of the arithmetic
operations suc...
Algebraic expressions
  with exactly one term : monomials
  with exactly two terms: binomials
  with exactly three terms: ...
Adding Algebraic Expression
Prob.7
Prob.7 (Sec. 0.4) Perform the indicated
operations and simplify
  (6x2 + 10xy + 2) (2z ...
Subtracting Algebraic Expression
Prob.13
Prob.13 (Sec. 0.4) Perform the indicated
operations and simplify
  3(x2 + y2) x(y...
Removing Grouping Symbols
Prob.15
Prob.15 (Sec. 0.4) Perform the indicated
operations and simplify
  2[3[3(x2 + 2) 2(x2 5)...
Special Products
Refer to page 18 of textbook for list of
rules for special products
Prob.19 (Sec. 0.4) Perform the indica...
Multiplying Multinomials
Prob.35
Prob.35 (Sec. 0.4) Perform the indicated
operations and simplify
  (x2 - 4)(3x2 + 2x - 1)...
Dividing a Multinomial by a Multinomial
Prob.47
Prob.47 (Sec. 0.4) Perform the indicated
operations and simplify




Answe...
Long Division
Prob.51
Prob.51 (Sec. 0.4)
Perform the indicated
operations and simplify
 (3x3-2x2+x 3)÷(x + 2)
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...
Common Factors
Prob.5
Prob.5 (Sec. 0.5) Factor the following
expressions completely
  8a3bc - 12ab3cd + 4b4c2d2
          ...
Factoring Trinomials
Prob.9
Prob.9 (Sec. 0.5) Factor the following
expressions completely
  p2 + 4p + 3

Answer:
(p+1)(p+3...
Algebraic expressions which are
fractions can be simplified multiplying
and dividing both numerator and
denominator of a f...
Simplifying Fractions
Prob.3
Prob.3 (Sec. 0.6)   Simplify the expressions




Answer:
Multiplying and Dividing Fractions
Rule for multiplying   with   is



Rule for dividing   with where c   0 is
Multiplying and Dividing Fractions
Prob.11
Prob.11 (Sec. 0.6)   Simplify the expressions



Answer:
Rationalizing the Denominator
Prob.53
Prob.53 (Sec. 0.6) Simplify and express your
answer in a form that is free of radica...
Addition and Subtraction of Fractions
Prob.29
Prob.29 (Sec. 0.6) Perform the operations and
simplify as much as possible

...
Addition and
Subtraction of
Fractions
Prob.39 (Sec. 0.6)
Prob.39
Perform the operations
and simplify as much
as possible
Addition and Subtraction of Fractions
Prob.47
Prob.47 (Sec. 0.6) Factor the following
expressions completely
             ...
Sec. 0.1   4, 5, 7, 8
Sec. 0.2   3, 4, 10, 22
                 10,
Sec. 0.3   55, 58, 65, 85,
           55, 58, 65, 85, 9...
Mathematical Systems,groups and fields




a set of elements
 One or more operations defined on this set
 Definitions and ...
The set G is closed under the operation *
The operation * is associative
There is an identity element e of G for *
There i...
A Field is a mathematical system that
consisting a set F and two operations that
           satisfy 11 properties:
Domain a set of all possible
replacement values for a given variable.
Ex.   D= { x| x   R}
Quantifier a word or phrase tha...
Universal Quantifier a statement that has
the same truth value for every element of the
domain.      X = > For all/ every ...
Negating Quantified Statements

The negation of a universally quantified
statement p is an existentially quantified
statem...
X   :x= 2

X   :x> 7

X   ~ k

Some animals can fly

Some rectangles are squares
Set - a collection of objects or elements.
  2 types - Finite & Infinite
  eg. { 1 , 2 , 3 } { 2 , 4 , 6 , 8 } { all w ome...
Define each of the following and show
how they are represented?

           Empty Set

           Union of sets

         ...
Empty Set    a set with no members - { } or

Union of sets A set containing the members of both/all given given sets
Union...
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  1. 1. Int roduc t ory Mat hem at ic al Analysis for B 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. 2. Review of algebra More Algebra
  3. 3. 25 POINTS: 5 QUIZES 25 POINTS: FINAL 50 POINTS: CALULAS (after the mid-term)
  4. 4. Review of the Subsets of the Real Number System
  5. 5. N - counting numbers -
  6. 6. WHOLE NUMBERS (W (W )
  7. 7. INTEGERS (Z) ,
  8. 8. RATIONAL NUMBERS (Q (Q ) numbers that can be expressed as a quotient a/b, where a and b are integers. terminating or repeating decimals
  9. 9. c Any number that cannot be written as a 3 1.732050808 ratio (fraction) 3 2 1.492106248 Any number whose e1 2.718281828 decimal 3.141592654 representation neither repeats or stops.
  10. 10. Set of all rational and irrational numbers. c A Real Number is any number that can be graphed in the number line
  11. 11. . Graph the numbers 0.2, 7 , 1, 2 , and 4 on a number line. 10 7 4 1 0.2 2 10 4 3 2 1 0 1 2 3 4
  12. 12. I M c Q A G Z I N W A R Y N
  13. 13. Properties of Addition & Multiplication Properties of Equivalence or Equality Properties of Order or Inequality
  14. 14. CLOSURE PROPERTY Given real numbers a and b, Then, or
  15. 15. Is the set of integers CLOSED with respect to: 1. Addition 2. Subtraction 3. Multiplication 4. Division
  16. 16. Given real numbers a and b, Addition: Multiplication:
  17. 17. Given real numbers a, b and c, Addition: Multiplication:
  18. 18. Given real numbers a, b and c,
  19. 19. Given a real number a, Addition: Multiplication:
  20. 20. Given a real number a, Addition: Multiplication: -1 a 0
  21. 21. For any real numbers a, b, and c Addition Multiplication Commutative a b b a a b b a Associative (a b) c a (b c) (a *b)* c a *(b * c) Identity a+0=a=0+a a a a If a in not zero, then Inverse a ( a) 0 ( a) a 1 1 a *a 1 a *a
  22. 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 and a-c=b-c Multiplication For any reals a, b, and c, if and Division a=b then a*c=b*c, and, if c is not zero, a/c=b/c
  23. 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=b 2- If a < b and b < c then a < c
  24. 24. 3- If a < b then a+c < b+c c R 4- If a < b and c > 0 ,then ac < bc and a b c c If a < b and c < 0, then ac > bc and a b c c
  25. 25. 5- If a < b and a = c , then c < b 6- If 0 < a < b or a < b < 0, then 1 1 a b
  26. 26. 7- If 0 < a < b and n > 0 , then an < bn and n a < n b
  27. 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. 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. 29. n 1 n n
  30. 30. 4
  31. 31. n one +ve root no real one -ve root roots one real one +ve root no +ve roots root, 0 one -ve root no -ve roots
  32. 32. 1 3 3 3 1 4 4 4
  33. 33. 2 2 2 4 4 2 2 2
  34. 34. 3 2 3 2 3 6 3 3 3 3 3
  35. 35. Review basic laws of exponents and radicals on page 10
  36. 36. Simplify and express all answers in terms of positive exponents Answer:
  37. 37. Simplify the expressions Answer:
  38. 38. Write the expression in terms of positive exponents only. Avoid all radicals in the final form. Answer:
  39. 39. Write the exponential forms involving radicals Answer:
  40. 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. 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. 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. 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. 44. Adding Algebraic Expression Prob.7 Prob.7 (Sec. 0.4) Perform the indicated operations and simplify (6x2 + 10xy + 2) (2z xy + 4) 10xy Answer: (6x2 + 10xy + 2) (2z xy + 4) (6 10xy = 6x2 + 10xy + 2 2z + xy - 4 10xy = 6x2 + 10xy + xy - 2z + 2 4 10xy = 6x2 + 11xy - 2z + 2 4 11xy
  45. 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. 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 + 96 16] 48]
  47. 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. 48. Multiplying Multinomials Prob.35 Prob.35 (Sec. 0.4) Perform the indicated operations and simplify (x2 - 4)(3x2 + 2x - 1) )(3 Answer: x2(3x2 + 2x 1) 4(3x2 + 2x 1) = 3x4 + 2x3 - x2 - 12x2 - 8x + 4 12x = 3x4 + 2x3 - 13x2 - 8x + 4 13x
  49. 49. Dividing a Multinomial by a Multinomial Prob.47 Prob.47 (Sec. 0.4) Perform the indicated operations and simplify Answer:
  50. 50. Long Division Prob.51 Prob.51 (Sec. 0.4) Perform the indicated operations and simplify (3x3-2x2+x 3)÷(x + 2)
  51. 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. 52. Common Factors Prob.5 Prob.5 (Sec. 0.5) Factor the following expressions completely 8a3bc - 12ab3cd + 4b4c2d2 12ab Answer: 4bc(2a3 - 3ab2d + b3cd2) bc(2
  53. 53. Factoring Trinomials Prob.9 Prob.9 (Sec. 0.5) Factor the following expressions completely p2 + 4p + 3 Answer: (p+1)(p+3 (p+1)(p+3)
  54. 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. 55. Simplifying Fractions Prob.3 Prob.3 (Sec. 0.6) Simplify the expressions Answer:
  56. 56. Multiplying and Dividing Fractions Rule for multiplying with is Rule for dividing with where c 0 is
  57. 57. Multiplying and Dividing Fractions Prob.11 Prob.11 (Sec. 0.6) Simplify the expressions Answer:
  58. 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. 59. Addition and Subtraction of Fractions Prob.29 Prob.29 (Sec. 0.6) Perform the operations and simplify as much as possible Answer:
  60. 60. Addition and Subtraction of Fractions Prob.39 (Sec. 0.6) Prob.39 Perform the operations and simplify as much as possible
  61. 61. Addition and Subtraction of Fractions Prob.47 Prob.47 (Sec. 0.6) Factor the following expressions completely Answer:
  62. 62. Sec. 0.1 4, 5, 7, 8 Sec. 0.2 3, 4, 10, 22 10, 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. 63. Mathematical Systems,groups and fields a set of elements One or more operations defined on this set Definitions and rules for applying the operations on the set. Theorems can be deduced from the given definitions and rules.
  64. 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. 65. A Field is a mathematical system that consisting a set F and two operations that satisfy 11 properties:
  66. 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. 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 for x such that x + 3 = 7
  68. 68. Negating Quantified Statements The 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. 69. X :x= 2 X :x> 7 X ~ k Some animals can fly Some rectangles are squares
  70. 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. 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. 72. Empty Set a set with no members - { } or Union 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 sets Intersection of sets a set containing only members that are elements of BOTH/ALL sets Universal 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.
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