Real Numbers and Algebra

1

• Chapter 1
• Real Numbers and Algebra

2
1.1 Describing Data with Set of numbers

• Natural Numbers are counting numbers and
  can be expressed as
• N 1, 2, 3, 4, 5, 6, .  
• Set braces , are used to enclose the
  elements of a set.
• A whole numbers is a set of numbers, is given
  by
• W 0, 1, 2, 3, 4, 5,
•  

3
Continued

• The set of integers include both natural and
  the whole numbers and is given by  
• I , -3, -2, -1, 0, 1, 2, 3, .
•  
• A rational number is any number can be
• written as the ratio of two integers p/q,
  where q 0. Rational numbers can be written
  as fractions and include all integers.
• Some examples of rational numbers are 
• 8/1, 2/3, -3/5, -7/2, 22/7, 1.2, and 0.
•  

4
continued

• Rational numbers may be expressed in decimal form
  that either repeats or terminates. 
• The fraction 1/3 may be expressed as 0.3, a
  repeating decimal, and the fraction ¼ may be
  expressed as 0.25, a terminating decimal. The
  overbar indicates that 0.3 0.3333333.
• Some real numbers cannot be expressed by
  fractions. They are called irrational numbers.
• 2, 15, and ? are examples of irrational
  numbers.

5

• Identity Properties 
• For any real number a,  
• a 0 0 a a,
• 0 is called the additive identity and
• a . 1 1 . a a,
• The number 1 is called the multiplicative
  identity.
• Commutative Properties
• For any real numbers a and b,
• a b b a (Commutative Properties of
  addition)
• a.b b.a (Commutative Properties of
  multiplication)
•  

6
Continued

• Associative Properties 
• For any real numbers a, b, c,
• (a b) c a (b c) (Associative
  Properties of addition)
• (a.b) . c a . (b . c) (Associative Properties
  for multiplication)
• Distributive Properties
• For any real numbers a, b, c,
• a(b c) ab ac
  and
• a(b- c) ab - ac

7
1.2 Operation on Real Numbers

• The Real Number Line

-3 -2 -1 0 1 2
3 Origin
-2 2
-2 2 Absolute value
cannot be negative 2 2
-3 -2 -1 0 1 2
3 Origin
8
Continued

• If a real number a is located to the left of a
  real number b on the number line, we say that a
  is less than b and write altb.
• Similarly, if a real number a is located to the
  right of a real number b, we say that a is
  greater than b and write agtb.
• Absolute value of a real number a, written a ,
  is equal to its distance from the origin on the
  number line. Distance may be either positive
  number or zero, but it cannot be a negative
  number.

9
Arithmetic Operations

• Addition of Real Numbers
• To add two numbers that are either both positive
  or both negative, add their absolute values.
  Their sum has the same sign as the two numbers.
• Subtraction of real numbers
• For any real numbers a and b, a-b a (-b).
• Multiplication of Real Numbers
• The product of two numbers with like signs is
  positive. The product of two numbers with unlike
  signs is negative.
• Division of Real Numbers
• For real numbers a and b, with b 0, a/b a .
  1/b
• That is, to divide a by b, multiply a by the
  reciprocal of b.

10
1.3 Bases and Positive Exponents

• Squared 4
  Cubed

4
4
4
4
4 4 . 4 42 4 .
4. 4 43

Exponent Base
11
Powers of Ten
12

• 1.3 Integer Exponents
• Let a be a nonzero real number and n be a
  positive integer. Then  
• an a. a. a. aa (n factors of a ) 
• a0 1, and  a n 1/an
• a -n b m
• b -m a n
• a -n b n
• b a
•  

13
cont

• The Product Rule
•   For any non zero number a and integers m and n,
• am . an a mn
• The Quotient Rule
• For any nonzero number a and integers m and
  n,
• am
• a m n
• a n

14

• Raising Products To Powers
• For any real numbers a and b and integer n,
• (ab) n a n b n
• Raising Powers to Powers
• For any real number a and integers m and n,
• (am)n a mn
• Raising Quotients to Powers
• For nonzero numbers a and b and any integer
  n.
• a n an

b
bn
15
Continued

• A positive number a is in scientific notation
  when a is written as b x 10n, where 1 lt b lt 10
  and n is an integer. 
• Scientific Notation
• Example 52,600 5.26 x 104 and 0.0068 6.8 x
  10 -3
•  

16
1.4 Variables, Equations , and Formulas 

• A variable is a symbol, such as x, y, t, used to
  represent any unknown number or quantity.
• An algebraic expression consists of numbers,
  variables, arithmetic symbols, paranthesis,
  brackets, square roots.
• Example 6, x 2, 4(t 1) 1,

X 1
17
cont

• An equation is a statement that says two
  mathematical expressions are equal. 
• Examples of equation 
• 3 6 9, x 1 4,  d 30t, and x y
  20
•  
• A formula is an equation that can be used to
  calculate one quantity by using a known value of
  another quantity.
• The formula y x/3 computes the no. of yards
  in x feet. If x 15, then y15/3 5.

18
Square roots

• The number b is a square root of a number a if b2
  a.
• Example - One square root of 9 is 3 because 32
  9. The other
• square root of 9 is 3 because (-3)2 9. We use
  the symbol to
• 9 denote the positive or principal square root
  of 9. That
• is, 9 3. The following are examples of how
  to evaluate
• the square root symbol. A calculator is sometimes
  needed to approximate square roots,
• 4 2
• -
• The symbol is read plus or minus. Note
  that 2
• -
• represents the numbers 2 or 2.
•  

19
Cube roots

•  
• The number b is a cube root of a number a if b3
  a
• The cube root of 8 is 2 because 23 8, which
  may
• be written as 3 8 2. Similarly 3 27 -3
  because
• (- 3)3 - 27. Each real number has exactly one
  cube root.

20
1.5 Introduction to graphing  

• Relations is a set of Ordered pairs.
• If we denote the ordered pairs in a relation
  (x,y),
• then the set of all x-values is called the
• domain of the relation and the set of all y
• values is called the range.

21
Example 1.

•  
• Find the domain and range for the relation given
  by
• S ( -1, 5), (0,1), (2, 4), (4,2), (5,1)
  Solution
•  
• The domain D is determined by the first element
  in each ordered pair, or
• D -1, 0, 2, 4,5
•  
• The range R is determined by the second element
  in each ordered pair, or R 1,2,4,5

22
The Cartesian Coordinate System
Quadrant II y Quadrant I
y
(1, 2)
2 1 0 -1 -2
Origin
2 1 -1 -2
x
x
-2 -1 1 2
-2 -1 1 2
Quadrant III Quadrant IV The xy
plane
Plotting a point
23
Scatterplots and Line Graphs

• If distinct points are plotted in the xy- plane,
  the resulting graph is called a scatterplot.

Y
7 6 5 4 3 2 1
(4, 6)
(3, 5)
(5, 4)
(2, 4)
(6, 3)
(1, 2)
X
1 2 3 4 5 6 7
0
24
Using Technology
25
Ex 9 page 44
Make a table for y x 2 / 9, starting at x 10
and incrementing by 10 and compare The table for
example 4 ( pg 41)
Go to Y and enter x 2 /9 Go to 2nd then
table set and enter Go to 2nd then table
26
Viewing Rectangle ( Page 57 )
Ymax
Ysc1
Xmax
Xmin
Xsc1
Ymin
-2, 3, 0.5 by -100, 200, 50
27
Making a scatterplot with a graphing
calculator Plot the points (-2, -2), (-1, 3), (1,
2) and (2, -3) in -4, 4, 1 by -4, 4, 1
(Example 10, page 58)
Go to 2nd then stat plot
Go to Stat Edit then enter points
Scatter plot
-4, 4, 1 by -4, 4, 1
28
Example 11Cordless Phone Sales
Enter line graph
Go to Stat edit and enter data
Hit graph
Enter datas in window
1985, 2002, 5 by 0, 40, 10
29

• Chapter 2
• Linear Functions and Models

30
Ch 2.1 Functions and Their Representations

• A function is a set of ordered pairs (x, y),
  where each x-value corresponds to exactly one
  y-value.
•  

Input x
Output y
Function f
(x, y) Input
Output
31
continued

• y is a function of x because the output y is
  determined by and depends on the input x. As a
  result, y is called the dependent variable and x
  is the independent variable
• To emphasize that y is a function of x, we
  use the notation y f(x) and is called a
  function notation.
•   A function f forms a relation between inputs
  x and outputs y that can be represented verbally
  (Words) , numerically (Table of values) ,
  symbolically (Formula), and graphically (Graph).
•  
•  

32
Representation of Function
y
Table of Values
Graph
20 16 12 8 4

y 3x
x
0 4 8 12 16 20 24


Numerically
Graphically
33
Diagrammatic Representation

• Function
  Not a function

x y
x
y

3 6 9
(1, 3), (2, 6), (3, 9)
1 2 3
4 5 6
1 2
1 2 3
4 5
(1, 4), (2, 5), (2, 6)
(1,4), (2, 4), (3, 5)
Example 1, pg - 77
34
Domain and Range Graphically

• The domain of f is the set of all x- values, and
• the range of f is the set of all y-values

y
3 2 1
Range R includes all y values
satisfying 0 lt y lt 3
x Domain D includes all x
values Satisfying 3 lt x lt 3
Range
-3 -2 -1 0 1 2 3
Domain
Pg 79 Ex-5
35
Vertical Line Test
If each vertical line intersects the graph at
most once, then it is a graph of a function

• 5
• 4
• 3
• 2
• 1
• -1
• -2
• -3

Not a function
(-1, 1)
-4 -3 -2 -1 0
1 2 3 4
(-1, -1)
36
Continued

Not a function
4 3 2 1 -1 -2 -3
(-1, 1)
-3 -2 -1 0 1
2 3
(1, -1)
Example 9 page - 81
37
Using Technology
Graph of y 2x - 1
Hit Y and enter 2x - 1
Hit 2nd and hit table and enter data
- 10, 10, 1 by - 10, 10, 1
38
2.2 Linear Function

• A function f represented by f(x) ax b,
  where a and b are constants, is a linear
  function.

100 90 80 70 60
100 90 80 70 60
f(x) 2x 80
0 1 2 3 4 5
6 0 1 2
3 4 5 6
Scatter Plot
A Linear Function
Ex- 1, 2, 3, 4, 5 Pg 91
39
Modeling data with Linear Functions

• Example 7

1500 1250 1000 750 500 250 0
Cost (dollars)

• 8 12 16 20
  x
• Credits

Symbolic Representation f(x) 80x
50 Numerical representation 4 8
12 16
370 690 1010 1330
40
Using a graphing calculator

• Example 5
• Give a numerical and graphical representation
  f(x) ½ x - 2
• Numerical representation Y1 .5x 2 starting
  x -3
• Graphical representation

-10, 10, 1 by -10, 10, 1
41
2.3 The Slope of a line
Y

• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

Cost (dollars)
Rise 3
Slope Rise 3 Run
2
Run 2
1 2 3 4 5 6

x Gasoline (gallons ) Cost
of Gasoline Every 2 gallons purchased the cost
increases by 3
42
2.3 Slope
rise y2 - y1 m run x2
- x1  
y2
(x2, y2)


y2 y1 y1 (x1, y1)
x2 x1
Rise

• The Slope m of the line passing through the
  points (x1 y1 ) and (x2, y2) is
• m y2 y1/x2 x1
• Where x1 x2. That is, slope equals rise over
  run.
•  

Run
Ex- 1, 2 pg 104
43
4 3 2 1 0 -1 -2
4 3 2 1 -1 -2 -3
2
-1
m - ½ lt 0
m 2 gt 0
2
-4 -2 1 2 3 4
- 4 -2 1 2
Positive slope
Negative slope
m 0
m is undefined
Zero slope
Undefined slope
44
(0, 4)Example 2 - Sketch
a line passing through the point (0, 4) and
having slope - 2/3y - valuesdecrease 2
units each times x- values increase by 3(0 3,
4 2) (3, 2)

• 4
• 3
• 2
• 1
• 1
• 2
• 3
• 4

( 0, 4)
Rise -2
( 3, 2)
- 4 - 3 - 2 1 0 1 2
3 4
45
Slope-Intercept Form 

• The line with slope m and y intercept b is
  given by
•  
• y mx b
•  
• The slope- Intercept form of a line

Example 3, and 4, 5, 6, 7 pg - 106
46
Example - 3
3 2 1 -1 -2 -3
Y ½ x
Y ½ x 2
-3 -2 -1 1 2
Y ½ x - 2
47
Analyzing Growth in Walmart

• Example 9

Year 1997 1999 2002
2007 Employees 0.7 1.1 1.4 2.2
3.0 2.5 2.0 1.5 1.0 0.5
Employees (millions)
m3
m2
m1
Years
0 1999 2003 2007
m1 1.1 0.7 0.2
m2 1.4 - 1.1 0.1 and 1999
1997
2002 1999 m3
2.2 - 1.4 0.16
2007 - 2002
Average increase rate
48
2.4 Point- slope form

• The line with slope m passing through the
  point (x1 , y1 ) is given by
• y m ( x - x1 ) y1
• Or equivalently,
• y y1 m (x x1)
• The point- slope form of a line

(x, y)
y y1
(x1, y1)
x x1
m y y1 / x x1
Ex 1, 2,3 pg 117
49
Horizontal and Vertical Lines
Equation of vertical line

y
y
x h
x
x
b
h
y b
Equation of Horizontal Line
50
Continued

• Parallel Lines
• Two lines with the same slope are parallel.
• m1 m2
• Perpendicular Lines
• Two lines with nonzero slopes m1 and m2 are
• perpendicular
• if m1 m2 -1

Examples 7, 8 Page - 123
51
m2 -1
m2 - 1/m1
m1 1
m2 - 1/2
m1 2
m1
52

• Chapter 3
• Linear Equations and Inequalities

53
3.1 Linear Equation in One Variable

• A Linear Equation in one variable is an equation
  that can be written in the form
•  
•   ax b 0
•  Where a 0
• A linear function can be written as f(x) ax b
  
• Examples of Linear equation.
• 2x 1 0, -5x 10 x, and 3x 8 5  

54
Properties of Equality

• Addition Property of Equality
• If a, b, c are real numbers, then
• a b is equivalent to a c b c.
• Multiplication Property of Equality
• If a, b, c are real numbers with c 0, then
• a b is equivalent to ac bc.

Example 1,2,3,4 pg 146
55
Example 7 Solving a Linear equation
graphicallyusing technology

• -6, 6, 1 by -4, 4, 1

56
Example 8

1984, 1991, 1 by 0, 350, 50 in 1987 In 1987
CD and LP record sales were both 107 million
57
Standard form of a line

• An equation for a line is in standard form when
  it is written as
• ax b c, where a, b, c are
  constants
• With a, b and c are constants with a and b not
  both 0
• To find x-intercept of a line, let y 0 in the
  equation and solve for x
• To find y-intercept of a line, let x 0 in the
  equation and solve for x

58
3.2 Linear Inequality in One Variable 

• A linear inequality in one variable is an
  inequality that can be written in the form
•  
• ax b gt 0, where a 0. ( The symbol gt may be
  replaced with gt, lt, or gt )
• There are similarities among linear functions,
  equations, and inequalities. A linear function is
  given by f(x) ax b, a linear equation by ax
  b 0, and a linear inequality by ax b gt 0.

59

• Examples of linear inequalities are
• 2x 1lt 0, 1-x gt 6, and 5x 1 lt 3 2x
• A solution to an inequality is a value of the
  variable that makes the statement true. The set
  of all solutions is called the solution set. Two
  inequalities are equivalent if they have the same
  solution set.
• Inequalities frequently have infinitely many
  solutions. For example, the solution set to the
  inequality x- 5gt 0 includes all real numbers
  greater than 5, which can be written as x gt 5.
  Using set builder notation, we can write the
  solution set as x x gt 5 .
• Meaning
• This expression is read as the set of all real
  numbers x such that x is greater than 5.

60
3.2 Solving a problem

• Using a variable to an unknown quantity
• Number problem
• n is the smallest even integers
• Three consecutive three even integers
• n, n 2, n 4
• n is the smallest odd integers
• Three consecutive odd integers
• n, n 2, n 4

61
Mixing acid ( Ex 8 pg 163)
20 2 liters
60 x liters
50 x 2 liters

• Step 1 x liters of 60 sulphuric acid
• x 2 Liters of 50 sulphuric
  acid
• Step 2 Concentration Solution Amount
  Pure Acid
• 0.20 (20)
  2 0.20(2)
• 0.60 (60)
  x 0.60x
• 0.50(50)
  x 2 0.50(x 2)
• Equation 0.20(2) 0.60x
  0.50(x 2)
• (pure acid in 20 sol.) (pure acid
  in 60 sol.) (pure acid in 50 sol.)
• Step 3 Solve for x
• 0.20(2) 0.60x 0.50(x
  2)
• 2 (2) 6x 5(x 2)
  Multiply by 10
• 4 6x 5x 10
  (Distributive Property) Subtract 5x and 4 from
  each side x 6
• Six liters of the 60 acid solution should be
  added to the 2 liters of 20 acid solution.
• Step 4 If 6 liters of 60 acid solution are
  added to 2 liters of 20 solution, then there

62
Ex 46 (Pg 166) Anti freeze mixture

• A radiator holds 4 gallons of fluid
• x represents the amount of antifreeze that is
  drained and replaced
• The remaining amount is 4 x
• 20 of 70 of solution of 50
  of solution
• 4 x gallons x gallons
  4 gallons
• 0.20(4-x) 0.70x 0.50(4)
• 0.8 0.2x 0.7x 2
• 0.5x 1.2
• x 2.4
• The amount of antifreeze to be drained and
  replaces is 2.4 gallons

63
Geometric Formulas
h

• Perimeter of triangle P abc unit
• Area of triangle
• A ½ bh sq.unit
• Area of Rectangle LW sq.unit
• Perimeter P 2(L W) unit
• Area of Parallelogram
• A bh sq.unit
• Area Volume of cylinder
• A 2 rh sq.unit
• V r2 h cu.unit
• Area Volume of a cube
• A 6a2 sq.unit
• V a3 cu.unit

a

c
b
W
L
h b
h
r

a
a
a
64
3.3 Properties of Inequalities

• Let a, b, c be real numbers. 
• a lt b and ac lt bc are equivalent
• ( The same number may be added to or subtracted
  from both sides of an inequality.)
• If c gt 0, then a lt b and ac lt bc are equivalent.
• (Both sides of an inequality may be multiplied or
  divided by the same positive number)
• If clt 0, then a lt b and ac gt bc are equivalent.
• Each side of an inequality may be multiplied or
  divided by the same negative number provided the
  inequality symbol is reversed.
•  

65
Fig. 3.14 Graphical Solutions (Pg 172)
Distance (miles)

y1 y2
y2 y1
x 2 0 1 2 3
4 Time ( hours ) Distances of two
cars

• 350
• 300
• 250
• 200
• 150
• 100
• 50

Distance ( milesDistance
When x 2 , y1 y2, ie car 1 and car 2 both are
150 miles From Chicago
y1 lt y2 when xlt 2 car 1 is closer to Chicago
than car 2 y1 is below the graph of y2 y1 gt y2
when x gt 2 Car 1 is farther from Chicago than
Car 2 Y1 above the graph of y2
66
Continued

• Ex 5 (Pg 173) Solving an inequality
  graphically
• Solve 5 3x lt x 3
• y1 5 3x and y2 x 3 Intersect at the
  point (2, -1)

5 4 3 2 1
X 2
y1
y2
-4 -3 -2 -1 0 1 2 3
4
y1 y2 when x 2
-1 -2 -3
(2, -1)
y1 lt y2 when x gt 2 , y1 is below the graph of
y2 Combining the above result y1 lt y2 when x
gt 2
Thus 5 3x lt x 3 is satisfied when x gt 2. The
solution set is x / xgt 2
67
Using Technology

Hit window
Hit Graph Hit Y Enter inequality
Enter
-5, 5, 1 by -5, 5, 1
68
Ex 80 ( Pg 178) Sales of CD and LP records
Hit Y , enter equations Enter table
set Enter window
Hit Table
Hit graph
1987 or after CD sales were greater than or equal
to LP records
69
3.4 Compound inequalities

• A compound inequality consists of two
  inequalities joined by the words and or or.
•  
• The following are two examples of compound
  inequalities. 
• 2x gt -3 and 2x lt 5
• 2(1) gt -3 and 2(1) lt 5 1
  is a solution
• True True
• 5 2 gt 3 or 5 1 lt -5  5 is
  a solution
• x 2 gt 3 or x 1 lt -5
• True False

Example 1
Example 2
70
. Cont.

• If a compound inequality contains the word and,
• solution must satisfy both inequalities.
• For example, x 1 is a solution of the first
  compound inequality because
•  2 (1) gt -3 and 2 (1) lt 5
• True
  True
•  are both true statements.
•  
• If a compound inequality contains the word or,
• solution must satisfy atleast one of the two
• inequalities. Thus x 5 is a solution to the
  second
• compound inequality.
•   5 2 gt 3 or 5 1 lt -5
• True False

71
Symbolic Solutions and Number Lines

x lt 6 xgt - 4 x lt 6 and x gt - 4
-8 -6 -4 -2 0 2 4
6 8
(
-8 -6 -4 -2 0 2
4 6 8

(
-8 -6 -4 -2 0 2 4
6 8 - 4lt x lt 6
72
Three-part inequality

• Sometimes compound inequality containing the word
  and can be combined into a three part inequality.
  For example, rather than writing
• x gt 5 and x lt 10
• We could write the three-part inequality
• 5 lt x lt 10

(

-1 0 1 2 3 4 5 6 7 8
9 10 5 lt x lt 10
73
Example 5Page 184

• Solve x 2 lt -1 or x 2 gt 1
• x lt -3 or x gt -1 (
  subtract 2 )
• The solution set for the compound inequality
  results from taking the union of the first two
  number lines. We can write the solution, using
  builder notation, as
• x x lt - 3 U x x gt - 1 or
• x x lt - 3 or x gt - 1
• x lt - 3
• x gt - 1
• x lt - 3 or x gt -1

)
- 4 -3 -2 -1 0 1
2 3 4
(
- 4 -3 -2 -1 0 1
2 3 4
)
(
- 4 -3 -2 -1 0 1
2 3 4
74
Interval NotationTable 3.5 (Page 186)

• Inequality Interval Notation
  Number line Graph
• - 1 lt x lt 3 ( - 1, 3)
• - 3 lt x lt 2 ( - 3, 2
• - 2 lt x lt 2 - 2, 2
• x lt - 1 or x gt 2 ( - ? , - 1) U (2, ? )
• x gt - 1 ( - 1, ? )
• x lt 2 ( - ?, 2

(
)
-4 -3 -2 -1 0 1 2 3 4

(
-4 -3 -2 -1 0 1 2 3 4 -4 -3
-2 -1 0 1 2 3 4 -4 -3 -2
-1 0 1 2 3 4 -4 -3 -2 -1
0 1 2 3 4 -4 -3 -2 -1 0 1
2 3 4


(
)
(

75
Example 6 (Page 185)

• Solving a compound inequality numerically and
  graphically Using Technology
• Tution at private colleges and universities from
  1980 to 1997
• Can be modelled by f(x) 575(x 1980) 3600
• Estimate when average tution was between 8200
  and 10,500.

Hit Y and enter equation Hit Window
and enter Hit 2nd and Table

1980, 1997, 1 by 3000, 12000, 3000
Hit 2nd and calc and go to Intersect and enter 4
times to get intersection
76
Ex 85 (Page 189)Solve graphically and
numerically. Write your answer in interval
notationx 1 lt -1 or x 1 gt 1 Y1 -1,
Y2 x 1, Y3 1
-5, 5, 1 by -5, 5, 1
x 1 lt - 1 or x 1 gt 1 Solution in
interval notation is ( - ? -2) U (0, ? )
77
School EnrollmentExample 92 (Pg 190)

• 70
• 60
• 55
• 50

Enrollment (millions)

• 1980 1990 2000
• Year

78

• Chapter 4
• Systems of Linear Equations

79
4.1 Systems of Linear Equations in two variables

• The system of linear equation with two variables.
  Each equation contains two variables, x and y.
• Example x y 4
• x y 8
• An ordered pair (x, y) is a solution to linear
  equation
• if the values for x and y satisfy both equations.
• The standard form is
• ax by c
• dx ey k
• Where a, b, c, d, e, k are constants.

80
Types of Linear Equations in two variables

• A system of linear equations in two variables can
  be represented graphically by two lines in the
  xy-plane  
• The lines intersect at a single point, which
  represents a unique solution. Consistent system
• The equations are called independent equation
• If the two lines are parallel it is an
  inconsistent system and no solution
• If two lines are identical and every point on
  the line represents solution and give infinitely
  many solutions. The equations are called
  dependent equations

81
..cont.

• Example
• The equations x y 1 and 2x 2y 2 are
  equivalent.
• If we divide the second equation by 2 we obtain
  the first equation. As a result, their graphs are
  identical and every point on the line represents
  a solution.Thus there are infinitely many
  solutions, and the system of equations is a
  dependent system.

y
y
y x
x

x
Unique
dependent
Inconsistent
82
The Substitution Method

• Consider the following system of equations.
•  2x y 5
• 3x 2y 4
•  It is convenient to solve the first equation for
  y to obtain y 5 2x. Now substitute ( 5 2x)
  for y into the second equation.
•  
• 3x 2(y) 4 Second equation
• 3x 2(5-2x)
  4, Substitute
•  to obtain a linear equation in one variable.
•  
• 3x
  2(5 2x) 4
• 3x 10
  4x 4 Distributive property
• 7x 10 4
  Combine like terms
• 7x 14
  Add 10 to both sides
• x 2
  Divide both sides by 7
• To determine y we substitute x 2 into y 5
  2x to obtain
• y 5 2(2) 1
•  The solution is (2, 1).

83
Elimination Method

• The elimination method is the second way to solve
  linear systems symbolically. This method is based
  on the property that equals added to equals are
  equal. That is, if
• a b and c d
• Then a c b d
• Note that adding the two equations eliminates the
  variable y
• Example
• 2x y 4
• x y 1
• 3x 5
• or x 5/3 and solve for x
• Substituting x 5/3 into the second equation
  gives
• 5/3 y
  1 or y - 2/3
• The solution is (5/3, - 2/3)

84
Solve the system of equations using
Technology4.1 Pg 226

Ex 43 Ex 44
No
solution Infinitely
many solutions
85
Burning caloriesEx 74 pg 239

• During strenuous exercise an athelete can burn on
  
• Rowing machine Stair
  climber
• 10 calories per minute 11.5 calories per
  minute
• In 60 minute an athelete burns 633 calories by
  using both exercise machines
• Let x minute in rowing machine, y minute in stair
  climber
• The equations are
• x y 60
• 10x 11.5 y 633
• Find x and y
• -10x - 10 y -600 (Multiply by -10)
• 10x 11.5 y 633
• Add 1.5y 33, y 33/1.5 22 minute in
  stair climber
• and x 38 minute in rowing machine

86
Mixing acids Ex 76 ( Pg 239 )

• x represents the amount of 10 solution of
  Sulphuric acid
• y represents the amount of 25 solution of
  Sulphuric acid
• According to statement x y 20
• 10 of x 25 of
  y 0.18(20)
• 0.10x .25 y
  3.6
• .1x .25y
  3.6 Multiply by 10
• x 2.5y
  36
• x y 20
• Subtract 1.5 y 16
• y
  16/1.5 10.6
• x
  20 y 20 10.6 9.4
• Mix 9.4 ml of 10 acid with 10.6 ml of 25 acid

87
River current ( Ex 84, pg 240 )

• x
  Speed of tugboat
• y
  Speed of current
• Distance Speed x Time
• 165 (x y) 33 Upstream x y
  5
• 165 (x y) 15 Downstream x y
  11
• By elimination method 2x
  16, x 8
• 
  y 3
• The tugboat travels at a rate of 8 mph and river
  flows at a rate of 3 mph

88
Solving Linear inequalities in Two variables


x lt 1 x gt 1
-2 -1 0 1 2 3 4
- 2 -1 1 2 3
- 2 -1 1 2

• 1
• - 2

-1
x 2y lt 4
y lt 2x - 1
x intercept 4 (4, 0) y intercept -2
(0, -2)
Choose a test point Let x 0, y 0
x - 2y lt 4 0 - 2(0) lt 4 0 lt 4 which is true
statement Shade containing (0, 0)
89
Solving System of Linear Inequalities ( Pg 244)x
y lt 4y gt x
Testing point x y lt 4 x 1, y 2
1 2 lt 4 ( True )

• 4
• 3
• 2
• 1

Shaded region Testing point 2 gt 1 (
True)
(1, 2)
(1, 2)
y gt x
-4 -3 -2 -1 0 1 2 3 4
Shaded region
Shaded region
To solve inequalities
( 1, 2)
90
Modeling target heart rates (Ex 5 Pg 246 )For
Aerobic Fitness
A persons Maximum heart rate ( MHR) 220 - A
200 175 150 125 100 75 50 25
Heart Rate Beats Per minute
T - 0.8A 196 ( Upper Line ) T - 0.7 A
154 (Lower Line )

• 0 20 30 40 50 60 70 80
• Age in years
• 
  (30, 150) is a
  solution

• 0.8 (40) 196 lt 165 When A 40 yrs
• - 0.7 (40) 196gt 125

150 lt - 0.8 (30) 196 172 True 150 gt - 0.7
(30) 196 133 True
91
Solving a system of linear inequalities with
technologyEx 6 ( Pg 247 )

• Shade a solution set for the system of
  inequalities, using graphing calculator
• - 2x y gt 1 or ygt 2x 1

2x y lt 5 or y lt 5 2x
- 15, 15, 5 by - 10, 10, 5

- 15, 15, 5 by - 10, 10, 5
Hit 2nd and Draw then go to Shade
92

• Linear Functions and Polynomial Functions
•  
• Every linear function can be written as
• f(x) ax b and is an example of a polynomial
  function. However, polynomial functions of degree
  2 or higher are nonlinear functions. To model
  nonlinear data we use polynomial functions of
  degree 2 or higher.
•  

93

• Chapter 5
• Polynomial Expressions and Functions

94
5.1 Polynomial Functions

• The following are examples of polynomial
  functions.
•  
• f(x) 3 Degree 0
  Constant
• f(x) 5x 3 Degree 1
  Linear
• f(x) x2 2x 1 Degree 2
  Quadratic
• f(x) 3x3 2x2 6 Degree 3
  Cubic

95
Polynomial Expressions

• A term is a number, a variable, or a product of
  numbers
• and variables raised to powers.
• Examples of terms include
• 15, y, x4, 3x3 y, x-1/2 y-2 , and 6 x-1 y3
• If the variables in a term have only nonnegative
  integer
• exponents, the term is called a monomial.
  Examples of
• monomials include
• 4, 5y, x2, 5x 2 z4, - x y4 and 6xy4

96
Monomials
x
x
x
x

• x

x
y
y
y
Volume x3
Total Area xy xy xy 3xy
97
Modeling AIDS cases in the United StatesEx 11 (
Pg 308 )
600 500 400 300 200 100
Aids Cases (Thousands)
0 4 6 8 10 12 14

Year ( 1984
1994 )
f(x) 4.1x2 - 25x 46 f(7) 4.1(7) 2-
25.7 46 71.9 f(17) 4.1 (17) 2 25.7 46
805.9
98
Modeling heart rate of an athelete ( ex 12 )
250 200 150 100 50
Heart Rate (bpm)
0 1 2 3 4 5 6 7 8
Time (minutes )
P(t) 1.875 t2 30t 200
Where 0 lt t lt 8
P(0) 1.875(0) 2 30(0) 200 200 (Initial
Heart Rate) P(8) 1.875(8) 2 30(80 ) 200 80
beaqts per minute (After 8 minutes)
The heart rate does not drop at a constant rate
rather, it drops rapidly at first and then
gradually begins to level off
99
A PC for all(Ex 102 Pg 312)
200 150 100 50

• Worldwide computer

0 1998 2000 2002 2004
Year
f(x) 0.7868 x2 12x 79.5 x 0
corresponds to 1997 x 1 corresponds to 1998 and
so on x 6 corresponds to 2003 f(6) 0.7868 x
6 2 12 x 6 79.5 179.8248 180
million(approx)
100
5.2 Review of Basic Properties

• Using distributive properties
• Multiply
• 4 (5 x) 4.5 4. x 20 4x

20
4x
4

• x
• Area 20 4x

101
Using Technology
-6, 6, 1 by -4, 4, 1 -6, 6,
1 by -4, 4, 1
102
.Cont

• Multiplying Binomials
• Multiply (x 1)(x 3)
• Geometrically
• Symbolically

a) Geometrically x 1
1 x
3
x2 3x

x 3
x 3 Area (x 1)(x
3) Area x2
4x 3 b) Symbolically apply distributive
property (x 1)(x 3) (x 1)(x) (x 1)(3)
x.x 1.x x.3 1.3 x 2 x 3x 3 x2
4x 3
x
103
Some Special products

• (a b) (a - b) a 2 - b 2
• Sum
• (a b) 2 a 2 2ab b 2
• Difference
• (a b) 2 a 2 - 2ab b 2

104
Squaring a binomial

• (a b) 2 a2 ab ab b2 a2 2ab b2

a a2
ab b ba
b2
a b
(a b)2 a2 2ab b2
105
5.3 Factoring Polynomials

• Common Factors
• Factoring by Grouping
• Factoring and Equations ( Zero Product Property)

106
Zero- Product Property

•  
• For all real numbers a and b, if ab 0, then a
  0 or b 0 ( or both)
•  

107
Ex 5 Solving the equation 4x x2 0
graphically and symbolically
Graphically

• Numerically
• x y
• -1 -5
• 0 0
• 3
• 4
• 3
• 0
• 5 -5

• Symbolically
• 4x x 2 0
• x(4 x) 0 (Factor out x )
• x 0 or 4 x 0 ( Zero Product property)
• x 0 or x 4

4 3 2 1
-3 -2 -1 1 2 3
4 5 6
108
5.4 Factoring TrinomialsFactoring x2 bx
cx2 bx c , find integers m and n that
satisfy m.n c and m n bx2 bx c (x
m)(x n)
109
5.4 Factoring Trinomials ax 2 bx c by
grouping

• Factoring Trinomials with Foil
• 3x 2 7x 2 ( 3x 1 ) ( x
  2 )

x 6x 7x
If interchange 1 and 2 (3x 2)(x 1) 3x2
5x 2 which is incorrect
2x 3x 5x
110
5.5 Special types of Factoring

• Difference of Two Squares
• (a b) (a b) a2 - b2
• (a b) 2 a2 2ab b2
• (a b) 2 a2 - 2ab b2
• Sum and Difference of Two Cubes
• (a b)(a2 ab b2) a3 b3
• (a b)(a2 ab b2 ) a3 b3
• Verify
• ( a b) (a2 ab b2) a. a2 a . ab a. b2
  b. a2 -b . ab b. b2
• 
  a3 - a2 b a b2 a2 b - a b2 b3
• a3 b3

Perfect Square
111
5.6 Polynomial Equations

• Solving
• Quadratic Equations
• Higher Degree Equations

112
Using Technology
-4.7, 4.7, 1 by -100, 100, 25

Y 16x4 64x3 64x2
x 0 , y 0, x 2 y 0
113

• Chapter 6
• Rational Expressions and Functions

114

• 6.1 Rational Expressions and Functions
• Rational Function
•  
• Let p(x) and q(x) be polynomials. Then a
  rational function is given by
•  
• f(x) p(x)/ q(x)
•  The domain of f includes all x-values such
  that q(x) 0
• Examples - 4 , x
  , 3x2 6x 1
• x x
  5 3x - 7
•  

115
Identify the domain of rational function(Ex 2
pg 376)

• g(x) 2x
• x2 - 3x 2
• Denominator x2 - 3x 2 0
• (x 1)(x 2) 0 Factor
• x 1 or x 2
  Zero product property
• Thus D x / x is any real number except 1 and
  2

116
Using technology( ex 57, pg 384 )
-4.7 , 4.7 , 1 by -3.1, 3.1, 1
(ex 64, pg 384)
117

• Highway curve ( ex 72, page 384 )
• R(m) 1600/(15m 2)

500 400 300 200 100
Radius
0 0.2 0.4 0.6 0.8
slope
a) R(0.1) 1600 / (15(0.1) 2) 457 About
457 a safe curve with a slope of 0.1 will have
a minimum radius of 457 ft b) As the slope of
banking increases , the radius of the curve
decreases c) 320 1600/(15m 2) , 320( 15m
2) 1600 , 4800m 640 1600 4800m 960, m
960/4800 0.2
118
Evaluating a rational function Ex 4, Pg
-377Evaluate f(-1), f(1), f(2)

• Numerical value x -3 -2 -1
  0 1 2 3
• y 3/2
  4/3 1 0 __ 4 3
• f(x) 2x
• x - 1

4 3 2 1
-4 -3 -2 -1 1 2 3
Vertical asymptote
f(-1) 1 f(1) undefined and f(2)
4
119

• 6.2 Products and Quotients of Rational
  Expression 
• To multiply two rational expressions, multiply
  numerators and multiply denominators. 
• A/B. C/D AC /BD B and D are nonzero.
• Example 2/3 . 5/7 10/21 
• To divide two rational expressions, multiply by
  the reciprocal of the divisor. 
• A/B C/D AC/BD B, C, and D are
  nonzero
• Example 3/4 - 2/4
  3.4/4.5 3/5

120

• 6.3 Sums and Differences of Rational Expressions
•  
• To add (or subtract) two rational expressions
  with like denominators, add (or subtract) their
  numerators. The denominator does not change.
•  A/C B/C (A B)/C
• Example - 1/5 2/5 (1 2) / 5 3/5
• A/C B/C (A B) /C 
• 3/5 - 2/ 5 (3 2)/5 1/5

121
6.4 Solving rational equations graphically and
numerically ( Ex- 3 (a) pg 409 )

• 1/2 x/3 x/5
• Solution- The LCD for 2,3, and 5 is their
  product, 30.
• 30( 1/2 x/3) x/5 . 30 Multiply by the LCD.
• 30/2 30x/3 30x/5 Distributive
  property
• 15 10x 6x Reduce
• 4x -15 Subtract 6x
  and 15
• x -15/4 Solve
• Graphically Y1 1/2 x/3 Y2 X/5
  

-9, 9, 1 by -6, 6, 1
122
Determining the time required to empty a pool (
pg 415, no.68)

• A pump can empty a pool in in 40 hours. It can
  empty 1/40 of the pool in 1 hour.
• In 2 hour, can empty a pool in 2/40 th of the
  pool
• Generally in t hours it can empty a pool in t/40
  of the pool.
• Second pump can empty the pool in 70 hours. So it
  can empty a pool in t/70 of the pool in t hours.
• Together the pumps can empty
• t/40 t/70 of the pool in t hours.
• The job will complete when the fraction of the
  pool is empty equals 1.
• The equation is
• t/40 t/ 70 1
• Multiply (40)(70)
• (40)(70) t/40 t/ 70 1 (40)(70)
• 70t 40t 2800
• 110t 2800
• t 2800/110 25.45 hr Two pumps can empty a
  pool in 25.45 hr

123
Example 6( pg 412)

• x speed of slower runner
• x 2 the speed of the winner
• d rt , t d/r
• The time for slower runner 3/x, ran 3 miles
  at x miles per hour
• The winning time is 3/(x 2 ) , the winner
  ran 3 miles at x 2 miles per hr
• Add 3 minutes 3/60 1/20 hr to the winners
  time, as finishes race 3 minutes ahead of another
  runner which equals the slower runners time
• 
  3/(x2) 1/20 3/x
• Multiply each side by the LCD, which is 20x(x
  2)
• 20x(x 2)( 3/x 2) 1/20 20x(x 2)3/x
• 60x x(x 2) 60(x 2) Distributive
  property
• 60x x 2 2x 60x 120 ,,
• x 2 2x 120 0
• (x 12)(x 10) 0 Factor
• x - 12 or x 10 Zero
  product property
• Running speed cannot be negative. The slower
  runner is running at 10 miles per hour, and the
  faster runner is running at 12 miles per hour.

124
Ex 73 Pg 416

• A tugboat can travel 15 miles per hour in still
  water
• 36 miles upstream ( 15 x) Total
  time 5 hours
• downstream (15 x)
• t d/r
• So the equation is 36/(15 x) 36/ (15 x)
  5
• The LCD is (15-x)(15 x)
• Multiply both sides we get
• (15 x)(15 x)36/(15 x) 36/ (15 x) 5
  (15 x)(15 x)
• 540 36x 540 - 36x 1125 5x 2
• 5x2 45 0
• 5x2 45
• x 9, x 3 mph

125
Modeling electrical resistance (Ex- 8, pg 413)
R1 120 ohms R2 160 ohms
R
1/R 1/R1 1/R2 1/120 1/ 160
1/120 . 4/4 1/160 . 3/3 LCD 480
4/480 3/480 7/480 R 480/7 69
ohms
126
6.6 Proportion

• a c is equivalent to ad bc
• b d
• Example 6 8
• 5 x
• 6x 40
• or x 40/6 20/3

h/44 6/4 h 6.44/4 66 feet

• 6 feet h
  feet

4 feet 44
feet
127
Modeling AIDS cases
Y 1000 (x 1981)2
1980, 1997, 2 by -10000, 800000, 100000
128

• Chapter 7
• Radical Expressions and Functions

129
Chapter 7

• Square Root
• The number b is a square root of a if b2 a
• Example 100 102 10
• radical sign
• Under radical sign the expression is called
  radicand
• Expression containing a radical sign is called a
  radical expression.
• Radical expressions are 6, 5 x 1 ,
  and 3x
• 
  2x - 1

130
Cube Root

• The number b is a cube root of a if b3 a
• Example Find the cube root of 27
• 3 27 3 33 3

131
Estimating a cellular phone transmission distance
R
The circular area A is covered by one
transmission tower is A R2

2 The total area covered by
10 towers are 10 R , which must equal
to 50 square miles Now solve R R 1.26, Each
tower must broadcast with a minimum radius of
approximately 1.26 miles
132
Expression

• For every real number
• If n is an integer greater than 1, then a1 n
  n a
•  Note If a lt 0 and n is an even positive
  integer, then
• a1 n is not a real number.
• If m and n are positive integer with m/n in
  lowest terms, then
• a m n n a m ( n a ) m
•  Note If a lt 0 and n is an even integer, then a
  m n
• is not a real number.
• If m and n are positive integer with m/n in
  lowest terms, then
• a - m n 1/ a m n a 0

133
Properties of ExponentLet p and q be rational
numbers. For all real numbers a and b for which
the expressions are real numbers the
followingproperties hold. Page -467

• a p . a q a p q Product rule
• a - p 1/ a p
  Negative exponents
•   a/b -p b a p Negative
  exponents for quotients
• a p a p-q
  Quotient rule for exponents
• a q
• a p q a pq Power
  rule for exponents
• ab p a p b p Power rule
  for products
• a p a p Power
  rule for products
• b b p
  Power rule for quotients

1 2 3 4 5 6 7
134

• Product Rule for Radical Expressions
•  
• Let a and b be real numbers, where n a and n b
  are both defined. Then
•  
• n a . n b n a. b  
• To multiply radical expressions with the same
  index, multiply the radicands.

135
Quotient Rule for Radical Expressions 

• Let a and b be real numbers, where a and b are
  both defined and b 0.
•  
• a/b a/b
•  
• The Expression a
•  
• If agt0, then a a
•  
•  

136
Square Root Property

• Let k be a nonnegative number. Then the solutions
  to
• the equation.
• x2 k
• are x k. If k lt 0. Then this equation has
  no real
• solutions.

Ex 2, 3, 4, 5, 6
137
Technology
5, 13, 1 by 0, 100, 10
138
To find cube root technologically
139
Technologically

• Y1 x2

-6, 6, 1 by -4, 4, 1
140
7.2 Let a and b be real numbers whereare both
defined
,

• Product rule for radical expression (Pg 472)

Quotient rule for radical expression where b 0
(Pg 475)

141
Pg -476

• Rationalizing Denominators having square roots

142
7.3 Operations on Radical Expressions

• Addition
• 10 4 (10 4) 14
• Subtraction

10 - 4 (10 - 4) 6
Rationalize the denominator (Pg 484)
143
7.6 Complex NumbersPg 513
x 2 1 0 x 2 -1 x
Square root property
- 1
Now we define a number called the imaginary unit,
denoted by i
Properties of the imaginary unit i
- 1
i
A complex number can be written in standard form,
as a bi, where a and b are real numbers. The
real part is a and imaginary part is b
144
Pg 513
a ib

• Complex Number -3 2i 5 -3i -1
  7i - 5 2i 4 6i

Real part a - 3 5 0
-1 -5 4
Imaginary Part b 2 0 -3
7 -2 6
145
Complex numbers contains the set of real numbers

• Complex numbers
• a bi a and b real

Real numbers a bi b0
Imaginary Numbers a bi b 0
Rational Numbers -3, 2/3, 0 and 1/2
Irrational numbers 3 And -
11
146
Sum or Difference of Complex Numbers 

• Let a bi and c di be two complex numbers.
  Then
•  
• Sum
• ( a bi ) (c di) (a c) (b d)i
• Difference
• (a bi) (c di) (a - c) (b d)i
  

147

• Chapter 8
• Quadratic Functions and Equations

148
QuadraticFunction

• A quadratic equation is an equation that can be
  written as 
• f(x) ax2 bx c ,
• where a, b, c are real numbers, with a 0.

Axis of symmetry
(0, 2)
-2 1 -1
2 1 0
(0, 0)
-2 -1 0 1 2
(2, -1)
Vertex