Chapter P
Prerequisites
P.1
Real Numbers
LESSON OBJECTIVES
To be able to Represent Real
Numbers.
 To be able to Order real numbers
using interval notation

… and why
These topics are fundamental in the study of
Write these
mathematics and science.
In your
notebooks
Slide P- 3
Please discuss in pairs and match the following
vocabularies with their meanings
Terminate
Notation
Any number that can be written as a
decimal
A whole number, a number that is not
fraction
Has no end, endless, limitless
Real numbers
Bring to an end, stop, close, finish
Integer
Not expressed as a ratio of two
integers eg (pie)
Can be expressed as a ratio or
repeated decimals
System of symbols
Slide P- 4
irrational
Infinite
rational
In pairs please do the following Review quiz
(4min)
1. List the positive integers between -4 and 4. 1,2,3
2. List all negative integers greater than -4.
-3,-2,-1
3. Use a calculator to evaluate the expression
2  4.5   3
. Round the value to two decimal places.  2.73
2.3  4.5
4. Evaluate the algebraic expression for the given values
of the variable. x  2 x  1, x  1,1.5
3
-4,5.375
5. List the possible remainders when the positive integer
n is divided by 6. 1,2,3,4,5
Slide P- 5
Copy the following into your note books
A real number is any number that can be written
as a decimal.
Subsets of the real numbers include:
n The natural (or counting) numbers:
{1,2,3…}
n The whole numbers: {0,1,2,…}
n
The integers: {…,-3,-2,-1,0,1,2,3,…}
Slide P- 6
Copy the following in your note books
Rational numbers can be represented as a
ratio a/b where a and b are integers and b ≠
0.
The decimal form of a rational number
either terminates or is indefinitely
repeating.
Slide P- 7
The Real Number Line
Pin the worksheet
in your note
Slide P- 8
In pairs group the following numbers as either
rational, irrational, integers, infinite repeating
decimal OR terminal decimal
n
⅚, -3, √6, π, ⅚, 4.34,
0.363636…, 0, 9, 35, 1.75,
0.625, 0.11111….. ⅛, 2π,
5
,
5√25, 6√7, ⅜, ⅞, 2 ⅘,
2
,
10 ⅒, -5, ⅔, 25, ⅐
Slide P- 9
Pin
the worksheet
your
Order
of Real note
Numbers
books
in
Let a and b be any two real numbers.
Symbol
Definition
Read
a>b
a<b
a≥b
a≤b
a – b is positive
a – b is negative
a – b is positive or zero
a – b is negative or zero
a is greater than b
a is less than b
a is greater than or equal to b
a is less than or equal to b
The symbols >, <, ≥, and ≤ are inequality symbols.
Slide P- 10
Write the following
in your note books

Trichotomy Property
Let a and b be any two real numbers.
Exactly one of the following is true:
a < b, a = b, or a > b.
Slide P- 11
Class activity to be done in pairs
Describe the graph of x > 2.
Slide P- 12
Example Interpreting Inequalities
Describe the graph of x > 2.
The inequality describes all real numbers greater than 2.
In pairs show the
aboe statement on
the graph min
Slide P- 13
Bounded Intervals of Real Numbers
Let a and b be real numbers with a < b.
Interval Notation
Inequality Notation
[a,b]
a≤x≤b
(a,b)
a<x<b
[a,b)
a≤x<b
(a,b]
a<x≤b
The numbers a and b are the endpoints of each
interval.
Pin in you
Notebook
Slide P- 14
Unbounded Intervals of Real
Numbers
Pin in you
Notebook
Let a and b be real numbers.
Interval Notation
Inequality Notation
[a,∞)
x≥a
(a, ∞)
x>a
(-∞,b]
x≤b
(-∞,b)
x<b
Each of these intervals has exactly one endpoint,
namely a or b.
Slide P- 15
In groups Graph the following Inequalities
(class competition 15 min)
x>2
(2,)
x < -3
(-,-3]
-1< x < 5 (-1,5]
Slide P- 16
HOMEWORK:
PRECALCULUS TEXT
BOOK PAGE  #1EXIT QUIZ:

answer the
question on the
worksheet
NEXT LESSON
before you
Basic properties
exit and hand
Slide P- 17
Day
2Basic
Properties of
Algebra
Slide P- 18
Learning objectives
1To be
able to
describe
basic
properties
of algebra
Slide P- 19
Please show on the mwb the words that have
same meaning. Use numbers only…..
1.Cumulative
2.Associative
3.Inverse
4.Distributive
1. opposite, reverse,
contrary
2. divide, partition,
share
3. connected, related,
like
4. increasing, growing,
colective
Slide P- 20
Please in pairs discuss in English the following
review questions.
Ealuate the algebraic
expression for the gien
alues of the ariables
Answers
Slide P- 21
Pin the following worksheets in your
notebooks……..
Properties of Algebra
Let u , v, and w be real numbers, variables, or algebraic expressions.
1. Communative Property
Addition: u  v  v  u
Multiplication uv  vu
2. Associative Property
Addition: (u  v)  w  u  (v  w)
Multiplication: (uv) w  u (vw)
3. Identity Property
Addition: u  0  u
Multiplication: u 1  u
Slide P- 22
Properties of Algebra
Let u , v, and w be real numbers, variables, or algebraic expressions.
4. Inverse Property
Addition: u  (-u )  0
1
 1, u  0
u
5. Distributive Property
Mulitiplication: u 
Multiplication over addition:
u (v  w)  uv  uw
(u  v) w  uw  vw
Multiplication over subtraction:
u (v  w)  uv  uw
(u  v) w  uw  vw
Slide P- 23
Properties of the Additive Inverse
Let u , v, and w be real numbers, variables, or algebraic expressions.
Property
Example
1.  (u )  u
 (3)  3
2. (  u )v  u (v)  uv
(  4)3  4(3)  12
3. (u )(v)  uv
(6)(7)  42
4. (1)u  u
(1)5  5
5.  (u  v)  (u )  (v)
 (7  9)  (7)  (9)  16
Slide P- 24
Group Activity
n
Write the basic properties of
algebra on the manila paper
provided and hang it on the
classroom noticeboard.
Slide P- 25
Class
actiity
In pairs work out questions on
page 34 #37-46
Slide P- 26
CLASS ACTIVITIES
ANSWERS
Slide P- 27
ANSWERS
Slide P- 28
ANSWERS
Slide P- 29
Slide P- 30
ANSWERS
Slide P- 31
Please write this in your notebooks…..
Exponential Notation
Let a be a real number, variable, or algebraic expression and n
a positive integer. Then a  aaa...a, where n is the
n
n factors
n
exponent, a is the base, and a is the nth power of a,
read as "a to the nth power."
Slide P- 32
Properties of Exponents
Let u and v be a real numbers, variables, or algebraic expressions
and m and n be integers. All bases are assumed to be nonzero.
Property
Example
1. u u  u
m
n
5 5  5  5
mn
3
m
u
2.
u
u
3. u  1
1
4. u 
u
5. (uv)  u v
m
mn
u
u
7.   
v
v
m
m
m
1
y 
y
(2 z )  2 z  32 z
-3
n
6. (u )  u
5
0
-n
m
x
94
4
0
n
7
9
x
x
x
8 1
mn
n
m
3 4
4
3
m
5
5
5
(x )  x  x
2
3
23
a a
  
b b
7
5
6
7
7
Slide P- 33
Example Simplifying Expressions
Involving Powers
2
3
1
2
uv
Simplify
.
u v
2
3
2
1
3
1
2
2
3
5
uv
uu u


u v
vv
v
Slide P- 34
In pairs workout the following ( discussion
must be in English….)
1. Convert 0.0000345 to scientific notation.
0.0000345  3.45 10
-5
Slide P- 35
Converting from Scientific Notation
2. Convert 1.23 × 105 from scientific notation.
123,000
Slide P- 36
n
HOMEWORK: page 34 #47-51
precalculus textbook
n
Exit quiz: answer the question on the
worksheet provide and hand in before
you leave.
n
Next lesson: Cartesian coordinate
system.
Slide P- 37
Day 3
n
Cartesian coordinate
system
Slide P- 38
For everyone: work out the following on
the mwb and swap with your neighbors
to compare the answers….
-5
3
and . 2.75
4
2
Use a calculator to evaluate the expression. Round answers
1. Find the distance between
to two decimal places.
2. 8  6
2
-12  8
2
4. 3  5
2
10
3.
2
5.
-2
2
5.83
 2  5  1  3
2
2
3.61
Slide P- 39
Learning Objectives
To be able to draw a Cartesian Plane
n To be able to evaluate the Absolute
Value of a Real Number
n To be able to find Distance and
midpoint between two points.
n
… and why
These topics provide the foundation for the material that will be
covered in this textbook.
Write the lesson objective in your note books
Slide P- 40
Take 3 min to discuss in your groups the
meaning of the following key words..!
n
Congruent
n
Cartesian
plane
Coordinate
Quadrant
n
n
n
Absolute
value
identical in form; coincide/fix
exactly when put together.
two perpendicular number
lines: the x-axis and the y-axis
where x-value and y-value meet
four quarters of the coordinate
plane
the distance of a number on the number value
line from 0 without considering
which direction from zero the number lies.
Slide P- 41
For everyone: Draw the following in your
notebooks…..
The Cartesian Coordinate Plane
Slide P- 42
For everyone: Draw the following in the
notebooks….
Quadrants
Slide P- 43
Class Activity (5 min)
n
In groups Draw the 4
quadrants on the manila
paper and pin it on the
classroom notice board.
Slide P- 44
Copy this in your notebooks please…
(3min)
Absolute Value of a Real Number
The absolute value of a real number a is
a, if a  0

|a |  a if a  0.
0, if a  0

Slide P- 45
Copy this in your notebook please…….
(3 min)
Properties of Absolute Value
Let a and b be real numbers.
1. | a | 0
2. | -a || a |
3. | ab || a || b |
a |a|
4.

, b0
b |b|
Slide P- 46
Pin the worksheet for distance formulas in your
notebooks……… (2min)
Distance Formula (Number Line)
Let a and b be real numbers. The distance between a and b is | a  b | .
Note that | a  b || b  a | .
Distance Formula (Coordinate Plane)
The distance d between points P(x , y ) and Q(x , y ) in the
1
coordinate plane is d 
x  x
1
2

2
1
y  y
1
2
2
.
2
2
Slide P- 47
Class
Actiity 1
The Distance Formula using the
Pythagorean Theorem
Slide P- 48
Pin the worksheet for distance formulas
in your notebooks……… (2min)
Midpoint Formula (Number Line)
The midpoint of the line segment with endpoints a and b is
ab
.
2
Midpoint Formula (Coordinate Plane)
The midpoint of the line segment with endpoints (a,b) and (c,d ) is
ac bd 
,

.
2 
 2
Slide P- 49
Class Actiity 1
Distance and Midpoint
Find the distance and midpoint for the line segment joined
by A(-2,3) and B(4,1).
d ( A, B)  (2  4)  (3  1)
2
A(-2,3)
2
B(4,1)
d ( A, B)  (6) 2  (2) 2
d ( A, B)  40
 4 10
 2 10
 (2  4) (3  1) 
,


2
2 

2 4
Show all these steps in
,

 = (1,2)
Your notebook
2 2
Slide P- 50
Class Activity
Show2 all
Show that A(4,1), B(0,3), and
C(6,5) are vertices of an
isosceles triangle.
these step
Your notebook
C(6,5)
d ( A, B)  (4  0) 2  (1  3) 2  2 5
d ( B, C )  (0  6)  (3  5)  2 10
2
2
B(0,3)
A(4,1)
d ( A, C )  (4  6) 2  (5  1)2  2 5
Since d(AC) = d(AB) , ΔABC is isosceles
Slide P- 51
Class Activity 3
Sole this problem a
Swap your notebook
P is a point on the y-axis that is 5
units from the point Q (3,7). Find P.
Q(3,7)
d ( P, Q)  (3  0) 2  ( y  7) 2  5
P (0,y)
9  y 2  14 y  49  5
y 2  14 y  58  5
y 2  14 y  58  25
y 2  14 y  33  0
( y  11)( y  3)  0
y = 3, y = 11
The point P is (0,3) or (0,11)
Slide P- 52
Class Activity 4
Coordinate Proofs
Discuss this question
In your groups the
Swap answers
Prove that the diagonals of a
rectangle are congruent.
B(0,a)
C(b,a)
A(0,0)
D(b,0)
Given ABCD is a rectangle.
Prove AC = BD
dAC  (b  0) 2  (a  0) 2  b 2  a 2
dBD  (0  b) 2  (a  0) 2  b 2  a 2
Since AC= BD, the diagonals of a square are congruent
Slide P- 53
n
HOMEWORK: page 41 #1-28
precalculus textbook
n
Exit quiz: answer the question on the
worksheet provide and hand in before
you leave.
n
Next lesson: Equation of a circle
Slide P- 54
Day 4
Slide P- 55
Slide P- 56
Standard Form Equation of a Circle
The standard form equation of a circle with center (h, k )
and radius r is ( x  h)  ( y  k )  r .
2
2
2
Slide P- 57
Standard Form Equation of a Circle
Slide P- 58
Example Finding Standard Form
Equations of Circles
Find the standard form equation of the circle with center
(2,  3) and radius 4.
( x  h)  ( y  k )  r
2
2
2
where h  2, k  3, and r  4.
Thus the equation is ( x  2)  ( y  3)  16.
2
2
Slide P- 59
P.3
Linear Equations and Inequalities
Quick Review
Simplify the expression by combining like terms.
1. 2 x  4 x  y  2 y  3x
3x  3 y
2. 3(2 x  2)  4( y  1)
6 x  4 y  10
Use the LCD to combine the fractions. Simplify the
resulting fraction.
3 4 7

x x
x
x  2 x 7x  6
4.

4
3
12
2
2y  2
5. 2 
y
y
3.
Slide P- 61
What you’ll learn about
n
n
n
n
Equations
Solving Equations
Linear Equations in One Variable
Linear Inequalities in One Variable
… and why
These topics provide the foundation for algebraic
techniques needed throughout this textbook.
Slide P- 62
Properties of Equality
Let u, v, w, and z be real numbers, variables, or algebraic expressions.
1. Reflexive
uu
2. Symmetric
If u  v, then v  u.
3. Transitive
If u  v, and v  w, then u  w.
4. Addition
If u  v and w  z, then u  w  v  z.
5. Multiplication
If u  v and w  z , then uw  vz.
Slide P- 63
Linear Equations in x
A linear equation in x is one that can be
written in the form ax + b = 0, where a and b
are real numbers with a ≠ 0.
Slide P- 64
Operations for Equivalent Equations
An equivalent equation is obtained if one or more of the following
operations are performed.
Operation
Given Equation
1. Combine like terms,
2x  x 
3
9
Equivalent Equation
3x 
1
3
reduce fractions, and
remove grouping symbols
2. Perform the same
operation on both sides.
(a) Add (  3)
x3  7
(b) Subtract (2x )
5x  2x  4
x4
3x  4
(c) Multiply by a
nonzero constant (1/3)
3 x  12
x4
(d) Divide by a constant
nonzero term (3)
3 x  12
x4
Slide P- 65
Example Solving a Linear Equation
Involving Fractions
Solve for y.
10 y  4 y
 2
4
4
10 y  4 y
 2
4
4
 10 y  4   y

4


2
 
4
 4  4

10y  4  y  8
Distributive Property
9 y  12
Simplify
y
Multiply by the LCD
4
3
Slide P- 66
Linear Inequality in x
A linear inequality in x is one that can be written in the form
ax  b  0, ax  b  0, ax  b  0, or ax  b  0, where a and b are
real numbers with a  0.
Slide P- 67
Properties of Inequalities
Let u , v, w, and z be real numbers, variables, or algebraic expressions,
and c a real number.
1. Transitive
If u  v, and v  w, then u  w.
2. Addition
If u  v then u  w  v  w.
If u  v and w  z then u  w  v  z.
3. Multiplication
If u  v and c  0, then uc  vc.
If u  v and c  0, then uc  vc.
The above properties are true if < is replaced by  . There are
similar properties for > and  .
Slide P- 68
P.4
Lines in the Plane
Quick Review
Solve for x.
1.  50 x  100  200 x  2
2. 3(1  2 x)  4( x  2)  10
x
1
2
Solve for y.
2x  5
3
x
4. 2 x  3( x  y )  y y 
4
72
5
5. Simplify the fraction.

10  (3)
7
3. 2 x  3 y  5
y
Slide P- 70
What you’ll learn about
n
n
n
n
n
n
Slope of a Line
Point-Slope Form Equation of a Line
Slope-Intercept Form Equation of a Line
Graphing Linear Equations in Two Variables
Parallel and Perpendicular Lines
Applying Linear Equations in Two Variables
… and why
Linear equations are used extensively in applications involving
business and behavioral science.
Slide P- 71
Slope of a Line
Slide P- 72
Slope of a Line
The slope of the nonvertical line through the points ( x , y )
1
1
y y  y
and ( x , y ) is m 

.
x x  x
2
2
1
2
1
2
If the line is vertical, then x  x and the slope is undefined.
1
2
Slide P- 73
Example Finding the Slope of a Line
Find the slope of the line containing the points (3,-2) and (0,1).
y  y 1  (2) 3
m


 1
x x
03
3
2
1
2
1
Thus, the slope of the line is  1.
Slide P- 74
Point-Slope Form of an Equation of a Line
The point - slope form of an equation of a line that passes through
the point ( x , y ) and has slope m is y  y  m( x  x ).
1
1
1
1
Slide P- 75
Point-Slope Form of an Equation of a Line
Slide P- 76
Slope-Intercept Form of an Equation of a
Line
The slope-intercept form of an equation of a line with slope m
and y-intercept (0,b) is y = mx + b.
Slide P- 77
Forms of Equations of Lines
General form:
Slope-intercept form:
Point-slope form:
Vertical line:
Horizontal line:
Ax + By + C = 0, A and B not both zero
y = mx + b
y – y1 = m(x – x1)
x=a
y=b
Slide P- 78
Graphing with a Graphing Utility
To draw a graph of an equation using a grapher:
1. Rewrite the equation in the form y = (an
expression in x).
2. Enter the equation into the grapher.
3. Select an appropriate viewing window.
4. Press the “graph” key.
Slide P- 79
Viewing Window
Slide P- 80
Parallel and Perpendicular Lines
1. Two nonvertical lines are parallel if and only if their
slopes are equal.
2. Two nonvertical lines are perpendicular if and only
if their slopes m and m are opposite reciprocals.
1
2
1
That is, if and only if m   .
m
1
2
Slide P- 81
Example Finding an Equation of a
Parallel Line
Find an equation of a line through (2,  3) that is parallel to
4 x  5 y  10.
Find the slope of 4 x  5 y  10.
5 y  4 x  10
or y = mx + b
4
y  xb
5
4
 3   ( 2)  b
5
8
3 b
5
1
4
1
b
y x
5
5
5
4
4
y   x  2 The slope of this line is  .
5
5
Use point-slope form:
4
y  3    x  2
5
Slide P- 82
Example
Determine the equation of the line (written in
standard form) that passes through the point (-2, 3)
and is perpendicular to the line 2y – 3x = 5.
Slide P- 83
P.5
Solving Equations Graphically,
Numerically, and Algebraically
Quick Review Solutions
Expand the product.
1.  x  2 y 
2
x  4 xy  4 y
2
2.  2 x  1 4 x  3
2
8x  2 x  3
2
Factor completely.
3. x  2 x  x  2
3
2
4. y  5 y  36
4
2
 x  1 x  1 x  2 
 y  9  y  2  y  2 
2
5. Combine the fractions and reduce the resulting fraction
to lowest terms.
x
2

2x 1 x 1
x  5x  2
 2 x  1 x  1
2
Slide P- 85
What you’ll learn about
n
n
n
n
n
Solving Equations Graphically
Solving Quadratic Equations
Approximating Solutions of Equations Graphically
Approximating Solutions of Equations Numerically with
Tables
Solving Equations by Finding Intersections
… and why
These basic techniques are involved in using a graphing utility to
solve equations in this textbook.
Slide P- 86
Example Solving by Finding x-Intercepts
Solve the equation 2 x  3 x  2  0 graphically.
2
Slide P- 87
Example Solving by Finding x-Intercepts
Solve the equation 2 x  3 x  2  0 graphically.
2
Find the x-intercepts of y  2 x  3x  2.
2
Use the Trace to see that (  0.5, 0) and (2,0) are x-intercepts.
Thus the solutions are x  0.5 and x  2.
Slide P- 88
Zero Factor Property
Let a and b be real numbers.
If ab = 0, then a = 0 or b = 0.
Slide P- 89
Quadratic Equation in x
A quadratic equation in x is one that can be written in
the form ax2 + bx + c = 0, where a, b, and c are real
numbers with a ≠ 0.
Slide P- 90
Completing the Square
To solve x  bx  c by completing the square, add (b / 2) to
2
2
both sides of the equation and factor the left side of the new
equation.
b
b
x  bx     c   
2
2
2
2
2
b
b

x  c
2
4

2
2
Slide P- 91
Quadratic Equation
The solutions of the quadratic equation ax  bx  c  0, where
2
a  0, are given by the quadratic formula
b  b  4ac
x
.
2a
2
Slide P- 92
Example Solving Using the Quadratic
Formula
Solve the equation 2 x  3x  5  0.
2
a  2, b  3, c  5
b  b  4ac
x
2a
(2 x  5)( x  1)  0
2
3  3  4  2  5 
2 x  5  0 or x  1  0
2

2  2
3  49

4
3  7

4
5
x   or x  1.
2
5
x   or x  1
2
Slide P- 93
Solving Quadratic Equations Algebraically
These are four basic ways to solve quadratic equations
algebraically.
1. Factoring
2. Extracting Square Roots
3. Completing the Square
4. Using the Quadratic Formula
Slide P- 94
Agreement about Approximate Solutions
For applications, round to a value that is
reasonable for the context of the problem. For
all others round to two decimal places unless
directed otherwise.
Slide P- 95
Example Solving by Finding Intersections
Solve the equation | 2 x  1| 6.
Slide P- 96
Example Solving by Finding Intersections
Solve the equation | 2 x  1| 6.
Graph y | 2 x  1| and y  6. Use Trace or the intersect feature
of your grapher to find the points of intersection.
The graph indicates that the solutions are x  2.5 and x  3.5.
Slide P- 97
P.6
Complex Numbers
Quick Review
Add or subtract, and simplify.
1. (2 x  3)  ( x  3)
x6
2. (4 x  3)  ( x  4)
3x  7
Multiply and simplify.
3. ( x  3)( x  2)


x  x6
4. x  3 x  3
5. (2 x  1)(3 x  5)
2

x 3
2
6 x  13 x  5
2
Slide P- 99
What you’ll learn about
n
n
n
n
Complex Numbers
Operations with Complex Numbers
Complex Conjugates and Division
Complex Solutions of Quadratic Equations
… and why
The zeros of polynomials are complex numbers.
Slide P- 100
Complex Numbers
Find two numbers whose sum is 10 and whose
product is 40.
n
n
n
x
= 1st number
10 – x = 2nd number
x(10 – x) = 40
Slide P- 101
Complex Numbers
n
n
n
n
n
x(10 – x) = 40
10x – x2 = 40
x2 – 10x
= -40
x2 – 10x + 25 = -40 +25
(x – 5)2 = -15
x  5    15
x  5   15, 5   15
Slide P- 102
Complex Numbers
(5   15 )  (5   15 )  10
(5   15 ) (5   15 ) 
25  5  15  5  15  (15)  40
Slide P- 103
Complex Numbers
n
The imaginary number i is the
square root of –1.
i  1
i  1
2
 16   1  16  4i
 10  i 10
Slide P- 104
Complex Numbers
n
Imaginary numbers are not real numbers, so all
the rules do not apply.
Example: The product rule does not
apply:
 5   20  100
 5   20  i 5  i 20  i
2
100
 1  10  10
Slide P- 105
Complex Numbers
n
If a and b are real numbers, then
a + bi is a complex number.
n
n
n
a is the real part.
bi is the imaginary part.
The set of complex numbers consist of all the
real numbers and all the imaginary numbers
Slide P- 106
Complex Numbers
A complex number is any number that can be
written in the form a + bi, where a and b are
real numbers. The real number a is the real
part, the real number b is the imaginary part,
and a + bi is the standard form.
Slide P- 107
Complex Numbers
n
Examples of complex numbers:
3 + 2i
n 8 - 2i
n 4 (since it can be written as 4 + 0i).
The real numbers are a subset of the complex
numbers.
n -3i (since it can be written as 0 – 3i).
n
Slide P- 108
Complex Numbers
i  1
2
i 


2
 1  1
i  i  i  1  i   i
4
2
2 2
i  i    1  1
3
2
Slide P- 109
Complex Numbers
i  i  i  1 i  i
5
4
i  i  i  1  1  1
6
4
2
i  i  i  1  i  i
8
4
4
i  i  i  11  1
7
4
3
Slide P- 110
Complex Numbers
*
i
-1
-i
1
i
-1
-i
1
i
-1
-i
1
i
-1
-i
1
i
1
i
-1
-1
-i
-i
1
Slide P- 111
Complex Numbers
Evaluate:
i
35
i
24
   i  1  i  i
 i   1  1
 i
4 8
4 6
8
3
6
Slide P- 112
Addition and Subtraction of Complex
Numbers
If a + bi and c + di are two complex numbers, then
Sum: (a + bi ) + (c + di ) = (a + c) + (b + d)i,
Difference: (a + bi ) – (c + di ) = (a - c) + (b -d)i.
Slide P- 113
Example Multiplying Complex Numbers
Find  3  2i  4  i  .
Slide P- 114
Example Multiplying Complex Numbers
Find  3  2i  4  i  .
 3  2i  4  i 
 12  3i  8i  2i
2
 12  5i  2( 1)
 12  5i  2
 14  5i
Slide P- 115
Complex Conjugate
The complex conjugate of the complex number z  a  bi is
z  a  bi  a  bi.
Slide P- 116
Discriminant of a Quadratic Equation
For a quadratic equation ax  bx  c  0, where a, b, and c are
2
real numbers and a  0.
if b  4ac  0, there are two distinct real solutions.
2
if b  4ac  0, there is one repeated real solution.
2
if b  4ac  0, there is a complex pair of solutions.
2
Slide P- 117
Example Solving a Quadratic Equation
Solve x  x  2  0.
2
Slide P- 118
Example Solving a Quadratic Equation
Solve x  x  2  0.
2
a  b  1, and c  2.
1  1  4 1 2
2
x
2 1
1  7

2
1  i 7

2
So the solutions are x 
1  i 7
1  i 7
and x 
.
2
2
Slide P- 119
Complex Numbers
n
When dividing a complex number by a real
number, divide each part of the complex
number by the real number.
6  8i
6 8i
3
 
  2i
4
4 4
2
Slide P- 120
Complex Numbers
n
n
n
The numbers (a + bi ) and (a – bi ) are
complex conjugates.
The product (a + bi )·(a – bi ) is the real
number a 2 + b 2.
Show: (3 + 2i) (3 – 2i) = 3 2 + 2 2.
Slide P- 121
Complex Numbers
n
Show: (3 + 2i) (3 – 2i) = 3 2 + 2 2.
(3 + 2i) (3 – 2i) = 3.3 + 3(-2i) + 2i .3 + 2i (-2i)
= 3 2 – 6i + 6i – 2 2i 2
= 3 2 – 2 2(-1)
= 32 + 22
=9+4
= 13
Slide P- 122
Complex Numbers
n
When dividing a complex number by a complex
number, multiply the denominator and numerator by
the conjugate of the denominator.
2  3i
1 i

2  3i  1  i 


1  i  1  i 
2  2 i  3 i  3i

2
1 i
2
2  i  3 1

1   1
Slide P- 123
Complex Numbers
2  i  3 1
1   1
5i

2
5 1
  i
2 2
Slide P- 124
Complex Numbers
Slide P- 125
P.7
Solving Inequalities Algebraically
and Graphically
Quick Review
Solve for x.
1.  3  2 x  1  9
2. | 2 x  1| 3
2 x  4
x  2 or x  1
3. Factor completely. 4 x  9
2
 2 x  3 2 x  3
x  49
4. Reduce the fraction to lowest terms.
x  7x
x
x2
5. Add the fractions and simplify.

x 1
x
2
2
x7
x
2 x  3x  2
x x
2
2
Slide P- 127
What you’ll learn about
n
n
n
n
Solving Absolute Value Inequalities
Solving Quadratic Inequalities
Approximating Solutions to Inequalities
Projectile Motion
… and why
These techniques are involved in using a graphing
utility to solve inequalities in this textbook.
Slide P- 128
Solving Absolute Value Inequalities
Let u be an algebraic expression in x and let a be a real number
with a  0.
1. If | u | a, then u is in the interval ( a, a ). That is,
|u | a if and only if  a  u  a.
2. If | u | a, then u is in the interval ( ,  a ) or ( a, ). That is,
|u | a if and only if u  a or u  a.
The inequalities < and > can be replaced with  and ,
respectively.
Slide P- 129
Solving Absolute Value Inequalities
Solve
2x – 3 < 4x + 5
-2x < 8
x > -4
n
n
n
-5 -4 -3
Solve
|x – 2| < 1
-1 < x – 2 < 1
1<x<3
n
n
n
0 1 2 3 4
Slide P- 130
Solving Absolute Value Inequalities
Solve
-1 < 3 – 2x < 5
-4 < -2x < 2
2 > x > -1
-1 < x < 2
n
n
n
n
Solve
|x – 1| > 3
-3 > x – 1 or x – 1 > 3
-2 > x or x > 4
x < -2 or x > 4
n
n
n
n
-2 -1 0 1 2 3
-2 -1 0 1 2 3 4
Slide P- 131
Solving Absolute Value Inequalities
•|2x – 6| < 4
•-4 < 2x – 6 < 4
•2 < 2x < 10
•1< x < 5
•|3x – 1| > 2
•3x – 1 < -2 or 3x – 1 > 2
•3x < -1 or 3x > 3
•x < -1/3 or x > 1
(
)
-1 0 1 2 3 4 5
] [
-1 0 1 2 3 4 5
Slide P- 132
Example Solving an Absolute Value
Inequality
Solve | x  3 | 5.
| x  3 | 5
5  x  3  5
8  x  2
As an interval the solution in (  8, 2).
Slide P- 133
Example Solving a Quadratic Inequality
Solve x  3 x  2  0
2
( x  2)( x  1)  0
x  2 or x  1.
+++0--------0+++
-2
-1
Use these solutions and a sketch of the equation
y  x  3 x  2 to find the solution to the inequality
2
in interval form (  2, 1).
Slide P- 134
Example Solving a Quadratic Inequality
Solve x2 – x – 20 < 0
1. Find critical numbers (x + 4)(x - 5) < 0
x = -4, x = 5
2. Test Intervals (-∞,-4) (-4,5) and (5, ∞)
3. Choose a sample in each interval
x = -5 (-5)2 – (-5) – 20 = Positive
x = 0 (0)2 - (0) - 20 = Negative
x = 6 (6)2 – 3(6)
= Positive
+++0-------0+++
-4
5
Solution is (-4,5)
Slide P- 135
Example Solving a Quadratic Inequality
Solve x2 – 3x > 0
1. Find critical numbers x(x - 3) > 0
x = 0, x = 3
2. Test Intervals (-∞,0) (0,3) and (3, ∞)
3. Choose a sample in each interval
x = -1 (-1)2 – 3(-1) = Positive
x = 1 (1)2 - 3(1) = Negative
x = 4 (4)2 – 3(4) = Positive
+++0------0+++
0
3
Solution is (-∞,0) or (3, ∞)
Slide P- 136
Example Solving a Quadratic Inequality
Solve x3 – 6x2 + 8x < 0
1. Find critical numbers x(x2 – 6x + 8) < 0
x(x – 2)(x – 4) x = 0, x = 2, x = 4
2. Test Intervals (-∞,0) (0,2) (2,4) and (4, ∞)
3. Choose a sample in each interval
x = -5 (-5)3 – 6(-5)2 + 8(-5) = Negative
x = 1 (-1)3 – 6(-1)2 + 8(-1) = Positive
x = 3 (3)3 – 6(3)2 + 8(3) = Negative
x = 5 (5)3 – 6(5)2 + 8(5) = Positive
-----0++0----0+++
0
2
4
Solution is (-∞,0] U [2,4]
Slide P- 137
Projectile Motion
Suppose an object is launched vertically from a point
so feet above the ground with an initial velocity of vo
feet per second. The vertical position s (in feet) of the
object t seconds after it is launched is
s = -16t2 + vot + so.
Slide P- 138
Chapter Test
1. Write the number in scientific notation.
The diameter of a red blood corpuscle is about 0.000007 meter. 7  10
-6
2. Find the standard form equation for the circle with center (5,  3)
and radius 4.  x  5    y  3  16
2
2
3. Find the slope of the line through the points (  1, 2) and (4,  5). 
3
5
4. Find the equation of the line through (2,  3) and perpendicular
to the line 2 x  5 y  3. y 
5
x 8
2
x2 x5 1
5. Solve the equation algebraically.


3
2
3
9
x
5
1
3
6. Solve the equation algebraically. 6 x  7 x  3 x  or x  
3
2
2
Slide P- 139
Chapter Test
1
7. Solve the equation algebraically. | 4 x  1 | 3 x  or x  1
2
 2 
8. Solve the inequality. | 3 x  4 | 2 (, 2]    ,  
 3 
9. Solve the inequality. 4 x  12 x  9  0
2
 ,  
10. Perform the indicated operation, and write the result
in standard form. (5  7i )  (3  2i)
2  5i
Slide P- 140