Download Chapter 1

Chapter 1
1
Introduction – part 1
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Course Syllabus (Reminder: prepare a folder for homework)
Prepare a 3-ring binder with dividers for class notes, homework,
quizzes, tests, or related information
Homework
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Check the answers for odd numbers on the back of the book, show your
work. No credit will be awarded if no work is shown.
Use 3 holes line papers with your name on the right upper corner with
your ID; write down the homework # and assignments.
Place your homework on the homework folder and is due Monday,
unless the test in on Wednesday (then it is due Wednesday).
A Scientific calculator is required.
Homework log – for your own record
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Write the date next to the homework number and homework log should be in
your homework folder with your homework
Failure to comply with instructions, will result point deduction
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Introduction – part 2
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Lecture notes will be on line, please print it out before the
class
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Lecture notes may only contain the summary of lectures,
examples will be shown on the whiteboard, thus taking notes
is essential part of the learning.
Beginning each class , there may be a warm-up (may be
counted as class participation points).
At the end of each class, there may be an exit slip which
contains in-class work (may be counted as class participation
points).
Quizzes (counted as your grade) may be given randomly
unannounced, may contains vocabularies, homework
problems.
Familiar with syllabus, and keep track your total score on your
own.
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Chapter 1 Review of Real Numbers and 1.1 – 1
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Chapter 1 Objectives:
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Inequalities and absolute value, set operations and interval notation
Operations on integers and rational numbers, exponential expressions, order of
operations
Properties of real numbers, evaluate and simplify variable expressions
Translate a verbal expressions to a variable expression, and applications.
Definitions:
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Set – a collection of objects; notation: { }, e.g. A = {dog, cat, dolphin, panda} is a
set.
Elements – the objects of a set, e.g. dog is an element of the set A
Natural numbers – the set of counting numbers, sometimes we call it counting
numbers. Natural numbers = {1, 2, 3, 4, ….}
Prime number - A natural number that is greater than 1 that is divisible (evenly) only
by itself and 1, e.g. 2, 3, 5, 7, 11 and 13 are the first six prime numbers.
Composite number – A natural number that is greater than 1 and is not a prime
number is a composite number, e.g. the numbers 4, 6, 8, 9 are the first four composite
numbers.
Note that 1 is neither a prime nor a composite number.
Whole numbers – a set of natural numbers and zero, thus whole number s = {0, 1, 2,
……}.
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Review of Real Numbers(1.1 – 2)
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Definitions
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Integers – a set of positive integers and negative integers, i.e. integers =
{…, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, ….}. An integer is either positive
or negative and zero is neither a positive nor a negative integer.
Rational numbers – is a set of numbers that can be expressed
as a fraction
p
(or ratio of two integers), i.e. rational numbers = { q , where p and q are
integers and q  0}. E.g 2/3, 9/2, 5/1. Note that all integers n are rational
numbers, since n = n/1.
Is 4/ a rational number?
A rational number written as a fraction can be written in a decimal
notation by dividing the numerator by the denominator. E.g. 3/8 = 0.375,
2/15 = 0.133…. = 0.13 (3 is a repeating decimal)
Some numbers cannot be written as terminating or repeating decimals are
called irrational numbers, e.g. 7  2.64575131..
Real numbers is a set of all rational numbers and irrational numbers. i.e.
real numbers = {rational numbers and irrational numbers}
The graph of a real number is made by placing a heavy dot on a number
line directly above the number, examples on the board.
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Review of Real Numbers(1.1 – 3)
Definitions:
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A variable is a alphabet used to represent a real number.
 A domain of the variables is a set of the real numbers that the variable can represent. E.g. if
x is an element of the set {0, 2, 4, 6} means x can be replaced by 0, 2, 4, or 6, thus the set {0,
2, 4, 6} is the domain of x. The symbol  mean “is an element of”, and  means “is not an
element of}, e.g. 2  {1, 2, 4, 8}, and 0  {1, 2, 4, 8}.
 Inequality symbols < and >
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If a and b are real numbers, and a is less than b (when graph them, a is to the left of b), then we write a <
b, e.g. 5 < 10
If a and b are real numbers, and a is greater than b (when graph them, a is to the right of b), then we write
a > b, e.g. 10 > 5
Symbols  or  are important,  means “less than or equal to” and  means “greater than or equal to”.
Thus 4  2 is a true statement, since 4 > 2, and 5  5 is also a true statement, since 5 = 5.
When graph the numbers 5 and 5, you notice that they are same distance from 0 and are on
the opposite site of 0 on the number line. We call 5 and 5 “additive inverse” or “opposite”
of each other. 4 is opposite of positive 4, and (6) means opposite of negative 6.
 The absolute value of a number is a measure of its distance from zero on the number line,
with the symbol | |. The absolute value of a positive number is the number itself. The
absolute value of a negative number is the opposite of the negative number. The absolute
value of zero is zero. E.g. |5| = 5, |5| = 5, |5| = 5
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Review of Real Numbers(1.1 – 4)
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Example 1
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Example 2
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Let a  {12, 0, 4}. Determine a, the opposite of a, for each element of the set.
Let z  { 11, 0, 8}. Determine |z| for each element of the set.
Set notation: roster method is to write all elements in the braces { } to list the
elements of the set, two kinds: infinite and finite
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Let y  {7, 0, 6}. For which values of y, the inequality y < 4 is a true statement?
Let z  { 10, 5, 6}. For which value of z, the inequality z > 5 is a true statement?
Example of infinite set {1, 2, 3, ….} which is the set of natural numbers
Example of finite set A {2, 4, 6, 8} which is the set of even numbers less than 10.
Empty set (or null set) :  or { } is the set with no elements.
Set builder notation is useful to describe a set, e.g. { x | x > 3, x integers} and is read as “the set of
all x such that x is greater than 3 and x is an integer”
Example 1: write set-builder notation to write the set of integers less than 7.
Set operations:
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Union: the union of two sets, write as A  B = { x | x  A or x  B}, i.e. x is either in A or B.
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Intersection: the intersection of two sets, write as A  B = { x | x  A and x  B}
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Example: A = { 1, 2, 3}, B = {0, 2, 4, 5},then A  B = {0, 1, 2, 3, 4, 5}
Example: A = { 1, 2, 3, 5}, B = {0, 2, 4, 5},then A  B = {2, 5}
If two sets have nothing in common, then A  B =  (empty set)
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Review of Real Numbers(1.1 – 5)
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The graph of real numbers
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Example
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The graph of real numbers is denoted by the number line, and mark the
area specified with either parenthesis: ( for > and ) for < or bracket [ for 
and ] for .
1. Graph {x | x  3}
2. graph {x | x > 3}
Graph representation of union and intersection of sets
1. Graph the set {x | x  1}  {x | x > 3}, same as {x | x  1 or x > 3}
 2. Graph the set {x | x > 2}  {x | x > 4}, same as {x | x > 2}
 Graph the set {x | x > 2}  {x | x < 5}, same as {x | 2  x <5}
 Graph the set {x | x  4}  {x | x < 3}, same as {x | x < 3}
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Interval notation
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Closed interval: if it includes both end points, [ ]
 Open interval: if it does not include either end point, ( )
 Half – open: if one endpoint is included and the other is not, ( ] or [ )
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Review of Real Numbers(1.1 – 5) and HW#1
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Infinity symbol  means the interval extends forever in one or both
directions using interval notation; Positive infinity : + ; Negative infinity: 
Set-builder notation
Interval Notation
Graph
{x | 0 < x  5}
{x | 8  x < 1}
(, 9]
[5, )
{x | x > 1}
(1, )
{x | x  1}
[1, )
{x | x < 1}
(, 1)
{x | x  1}
(, 1]
{x |  < x < }
(, )
(, 3 )  [1, )
(, 2 )  (1, )
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Homework #1, pp 10 – 14, #1, 3, 9 – 12, 13, 17, 23 – 33 Odd, 43 – 55 odd, 59, 67,
71, 75, 83 –91odd, 97, 101, 105, 119, 121.
Operations on Rational Numbers (1.2 – 1)
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Objectives:
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3.
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Operations on integers
Operations on rational numbers
Exponential expressions
The order of operations agreement
Operations on integers, rules – review yourself; watch out “sign”
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Addition, e.g. 27 + (25); – 65 + ( 48)
Subtraction, use a – b = a + (b); e.g. 43 – (23); 2425
Multiplication, the product of two numbers with same sign is positive; the product of two
numbers with different sign is negative; e.g. 4(9); 8(5);
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Division, use a  b = a  (1/b) [b  0], i.e. same as multiplying the reciprocal of b; e.g.
(63)/7; (21)(3);
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Properties of zero and one in division:
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any number divided by one is the number itself , i.e. a/1 = a ;
zero divided by any nonzero number is 0, i.e. 0/a = 0 (where a  0)
Any number (including zero) divided by zero is undefined
Any nonzero number divided by itself is 1, i.e. a/a = 1 (where a  0)
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Multiplication inverse of a nonzero number a is (1/a) : called reciprocal of a.
Examples: (a) evaluate 3(5)9; (b) 6 (8)10 ; (c) 5|6|(15); (d)
42/(3); (e) 2|7|(8) (f) (36)/(3)
Operations on Rational Numbers (1.2 – 2)
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Operations on rational numbers; rules – review yourself; watch out “sign”
Remember decimal numbers are rational numbers (even the repeated decimal)
Addition of fractions:
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If they have the same denominator, then just add the numerators
If they don’t have the same denominator, then convert them to same denominator by using the
LCD(lowest common denominator) which is actually the LCM (least common multiple) of the
denominator
Subtraction of fractions: treat it as adding the negative of the 2nd fraction.
Multiplication of fractions: simply multiply the numerators and over the product of
the denominators; (a/b)(c/d) = (ac)/(bd); Need to make the answer the simplest
fraction by dividing them with GCF (greatest common factor)
Division of fractions: (a/b)/(c/d) = (a/b) (d/c) = ad/bd
Examples (a) 15.23 + (18.1); (b) 18.42(10.42); (c) 5/6 + (7/8); (d)
(5/12)(8/15); (e) (3/8)/(9/16)
Complex fraction is a fraction whose numerator or denominator (or both) contains
one or more fractions. Examples: (a) simplify (3/4 – ½)/ (2/3+ ¼) (b) simplify 5  2
9 6
11
3
8
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7
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Operations on Rational Numbers (1.2 – 3) and HW #2
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Exponential expressions – is used to represent repeated multiplication of same
factor; the exponent indicate how many times the factor (base) is multiplied.
Example of the exponential expressions is 26 which means 2 is the base, and 6 is the
exponent and multiply 2 six times.
nth power of a is denoted as an = aaaa, with n times where a is a real number
and n is a positive integer. E.g. evaluate (3)4 and 34
The order of operations agreement:
1.
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3.
4.
Perform operations inside grouping symbols: ( ), [ ], the absolute symbol | |. And fraction bar.
Simplify exponential expressions
Multiplication and division as they occur from left to right
Do addition and subtraction as they occur from left to right
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12  2 2
8
2
4 1
(b) 5[2 (3 + 7)4]
(d)
9 6
2
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7
6
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Examples: (a)
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HW #2, pp26 – 31, 7, 15, 23, 31, 51, 59, 67, 75, 79, 87 – 103 EOO, 113 – 133
EOO
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(c)
32  9
18  3
2 5
3
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Variable Expressions (1.3 – 1) and HW #1
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Properties of the real numbers
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Commutative property of
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Addition a + b = b + a
Multiplication a  b = b  a
Associative property of
Addition (a + b) + c = a + (b + c)
Multiplication (a  b)  c = a  (b  c)
Addition property of zero
a+0=0+a
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Multiplication property of one
a  1= 1  a
Inverse property of addition a + (a) = (a) + a = 0
Inverse property of multiplication a  (1/a) = (1/a) a = 1, a  0
Distributive property a (b + c) = a b + a c
Evaluate variable expressions – a variable expression is a mathematical expression with
a variable x, y, z, or others in it. E.g. 3x2y + 2z – x + 4 is a variable expression.
A coefficient a is a real number that multiply the variable, e.g. 3 is a coefficient for 3x2y.
(Note that if you substitute x and y with values, then 3 is a multiplication factor.)
To evaluate an expression is to replace the variables with values, e.g.
evaluate 3 – 2|3x -2y2| , when x = 1 and y = 2
Example 1: The radius of the base of a cylinder is 3 in. and the height is 6 in. Find the
volume of the cylinder (round to the nearest hundredth) (A. 169.95 in3)
Example 2: Find the surface area of a right circular cone with a radius of 5cm and a slant
height of 12 cm. Give both the exact area and an approximation to the nearest hundredth.
(A. 85 cm2, or 267.04 cm2)
Variable Expressions (1.3 – 2) and HW #3
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Simplify variable expressions – use combine the like terms.
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Examples
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Like terms – the terms with the same variable part. E.g. 2x and ax are like terms, 3xy and 5xy are
like terms.
Combine like terms is to add the coefficient of the like terms by using the Distributive Property.
Simplify 2(x + y) +3(y – 3x)
Simplify (2x +xy – y) – (5x – 7xy + y)
Simplify 4y – 2[ x – 3(x + y) – 5y
Simplify 2x – 3[y – 3(x – 2y +4)]
Homework # 3: pp 36 – 49, 19 – 29 odd, 37 – 61 EOO, 67, 71, 75 (see last page for
formulas of the geometric figures), 85 – 117 EOO
Variable Expressions and variable expressions (1.4 – 1)
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Translate a verbal expression into a variable expression – use the following
keywords
Operation
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Key word
Example
Expression
Addition
More than, added to, the sum of ,
the total of, increased by
8 more than w, x is
added to 9, the sum of z
and 9, the total of r and
s, x increased by 7
w + 8, 9 + x, z + 9, r + x,
x+7
Subtraction
Less than, the difference between,
minus, decreased by
12 less than b, the
difference between x
and 1, z minus 7, 17
decreased by a
b – 12, x – 1, z – 7,
12 – a
Multiplication
Times, the product of, multiplied
by, of, twice
Negative 2 times c, the
product of x and y, 3
multiplied by n, threefourth s of m, twice d
– 2c, xy, 3n, (3/4)m, 2d
Division
Divided by, the quotient of, ratio
v divided by 15, the
quotient of y and 3, the
ratio of x to 7
v/15, y/3, x/7
Power
The square of or the second power
of, the cube of or the third power
of, the fifth power of
The square of x, the
cube of r, the fifth
power of a
x2 , r 3 , a 5
Variable Expressions and variable expressions (1.4 – 2) & HW 4-6
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Example 1: Translate and simplify “the total of five times a number and twice the difference between
the number and three”
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Example 2: Translate and simplify “a number decreased by the difference between eight and twice the
number”
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Example 3: Translate and simplify “fifteen minus one-half the sum of a number and ten”
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Example 4: Translate and simplify “the sum of three-eights of a number and five-twelfths of the
number”
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Example 5: A cyclist is riding at a rate that is twice the speed of a runner: Express the speed of the
cyclist in terms of the speed of the runner
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Example 6: A mixture of candy contains 3 lb more of milk chocolate than of caramel. Express the
amount of milk chocolate in the mixture in terms of the amount of caramel in the mixture.
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Example 7: The length of a rectangle is 2 ft more than 3 times the width. Express the length of the
rectangle in terms of the width.
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HW #4 pp 45 – 47, #5 – 21 EOO, 27 – 33 odd.
Test 1 (chapter 1) on Jan 19. Test guide: read p. 51 – p.54 chapter 1 summary and HW #5 and #6
HW#5 Chapter 1 Review, p 54 – 55, 1 – 8, 9 – 37 odd, HW #6, Chapter 1 Test, p. 55 – 56, all.
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