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					Answer the following on your card:

1. What’s your full name? What do
you want me to call you?

2. Write a short description of
yourself that would help me learn
your name.
3. List high school math classes
you completed with C or higher.

4. What college math classes have
you completed with a C or higher?

5. What is your local phone
number ?
On the back of your card:

6. Why are you taking this class?
How does it fit into (or lead you to)
your future plans? What do you
hope to gain by taking this
particular course?
7. Any other comments? Is there
anything else you would like me to
know about you?
Welcome!

This is
Math 108
Intermediate Algebra

Kathy Stover
Chapter 1

Set – a collection of
   objects called elements
 Special sets of numbers:
       Real Numbers
    Irrational Numbers
       Rational numbers
       Integers
       Whole numbers
       Natural numbers
  Additive inverse (opposite)
   Additives inverses are on
   opposite sides of 0 on the
   number line…so just
   change the sign to get the
   additive inverse.
Examples: 3 and –3
            x and –x
  Absolute Value
   Absolute value measures
   the distance from 0 on the
   number line. It is always +

Examples: 5= 5
            6= 6
  Inequality symbols
Graph on a number line
3 ways to write sets:
   Roster method
   Set-builder method
   Interval notation
    is not a number
Operations on sets:
   Union
   Intersection


 See p. 14, # 73-106
 Know all the rules for
   + and – signs

 Rules to note for division:
 Fractions: add, subtract,
   multiply, divide

 Decimals: add, subtract,
   multiply, divide
Complex fractions

 Do operations above and below
  the main fraction bar first
Exponential notation
Order of operations:
1. Within Parens or other
   grouping symbols
2. Exponents
3. Multiply and Divide from
   left to right
4. Add and Subtract
PEMDAS
Properties of real numbers
 The most important to
 remember by name are:
  Commutative
  Associative
  Distributive
Evaluate variable
expressions


 Remember the order of
   operations!
  Geometric formulas
 Look inside the front cover
 of your book.
Geometric formulas
 You‟ll need to add this:

    Volume of pyramid is
    V = 1/3 (base area)(height)

    Use appropriate labels on
          all answers!
Translating verbal
expressions into symbols



  See p. 38 - 39
Solving equations in one variable


  The answers to an equation
   are called roots or solutions.
   They are values which will
   make the given equation true.
Combine like terms on each side of the
 = , then do “opposite operations” to
 put the equation into the form:
 variable = constant
If the terms contain fractions,
 multiply each side of the = by the
 common denominator to clear the
 equation of fractions, then solve.
  Stamp and Coin problems
• Make a table; let x = the number
  of one type of coin or stamp

• Write the number of coins or
  stamps as well as the value of
  those coins or stamps.
Stamp and Coin problems

• Write an equation using the
  relationships given in the
  problem.

• Solve and check.
  Integer Problems


The sum of two integers is
 given…… for example
If sum is 25, let
 one # = x
 the other = 25-x
  Integer Problems


 Consecutive integers: n, n+1,
  n+2, etc.
 Consecutive even integers:
  n, n+2, n+4
 Consecutive odd integers are
  also: n, n+2, n+4
1. Let a variable represent one of the
  integers, then express the others in terms
  of that same variable.

2. Write an equation using the
 relationships given in the problem.

3. Solve.
Value Mixture Problems

    In these problems, you need to
   combine ingredients to make one
                 blend
Make a table !
Write an equation:

Sum of values of ingred. = value of mix
Uniform Motion Problems
        Use the formula:
     Rate • Time = Distance
     Again, a table will help you
      organize the information.
Problems involving %
For investment problems,
1. Use the formula:
    Principal • Rate = Interest
2. Make a table
Note: change % rate to a decimal!
3. Write an equation using
info from the problem.

4. Solve
For % Mixture Problems
1. Make a table.
2. Write an equation.
Solving Inequalities


The solution is a whole
set of numbers:
If you add any #, subtract
any #, multiply by a +, or
divide by a +, the inequality
symbol stays the same…..
But if you divide (or
multiply) both sides by a
negative, you must reverse
the inequality symbol!!!
In compound inequalities
remember that

   „and‟ means intersection

   „or‟ means union
If the compound
inequality has „x‟ in the
middle section, do
opposite operations to
isolate it in the middle.
Important concepts and
formulas for section 3.1
 1. Rectangular coordinate
     system
 2. Ordered pair solutions to
    equations
3. Pythagorean Theorem

4. Distance formula

5. Midpoint formula

6. Scatter diagrams
Main terms and processes of
section 3.2
Relation

Function

Function notation and
 evaluating functions
Domain and range

Excluding values from the
 domain
Linear equations and functions

Linear equations have x
 and/or y with + or –, but
 never xn , never „xy‟, and

 never division by x or y
Every nonvertical line is a
 function and the equation can
 be put into the form

Y = mx+b or f(x) = mx+b
Ways to graph lines:
1. Table of random xy values
2. x-intercept and
     y-intercept
3. Recognize special lines
4. Using slope and   y-
   intercept…..
    next section!
Slope


Slope is a measure of the
 steepness of a line. There are
 several ways to remember the
 formula:
To graph the slope,

Top #  go up if + , down if –
Bottom #  go to right
Draw graphs
of lines
using slope
Finding the equation of a line

Two ways:
1. Using y = mx+b
2. Using y – y1 = m(x – x1)

Put your answer as y =
Parallel lines
   same slope                    (if
 not vertical)

Perpendicular lines
 Slopes are negative
 reciprocals (if not vertical)
(product of the slopes is -1)
Chapter 4

 System of equations: 2 or
  more equations considered
  together
 The solution will usually be
  an ordered pair (or triple)
  which satisfies all the
  equations.
Ways to solve systems of
equations

1. Graphing
   a. Graph line #1
   b. Graph line #2
   c. Name the point where the
        lines cross
Special cases
1. If lines are parallel….
     No solution or Ø

  The system is called
   inconsistent
2. If both equations make
   the same line….
   Write a generic ordered
   pair (x, expression) to
   represent all the points
The system is called
   dependent
Another way to solve a system

2. Substitution method
    a. Rewrite one equation as
        x = or as y =
    b. Put this expression into
        the other equation
   c. Solve to get the value of
       one variable
   d. Put this value into the
           x = or y =
equation            from step 1
to find the             value of
the second              variable
3. Addition Method
 a. Multiply one or more
    equations by constants to
    make the coefficients of one
    variable equal …but
    opposite in sign…like 6x
    and –6x, or 12y and – 12y
 b. Add the equations
c. Solve to get the value of one
    variable
d. Put this value into either
    equation to find the value
    of the other variable.
Some systems have 3
 equations and 3 variables

See handout!!!
Matrix

 Rectangular array of
  numbers (any size)
 The numbers are called
  elements of the matrix
 Size: rows x columns
 Every square matrix has a
  numerical value called its
  determinant.
 Evaluate 2x2 and 3x3
  determinants
Cramer‟s Rule uses
 determinants to solve
 systems of equations.

(This is our
           4 thmethod for
 solving systems of
 equations!)
Cramer‟s Rule is very useful
 when you don‟t have to solve
 for all the unknowns…..
 circuits, chemistry, physics,
 etc.
5 th
   way to solve a system of
equations:

 Augmented matrices and
  “row operations”
 See handout!
Chapter 5
 Monomial – a number, a
  variable, or the product of
  numbers and variables
 Degree of a monomial – add
  the exponents on the variables
 A constant is degree 0
Rules for Exponents

 See summary on p. 257

 Scientific notation p. 261-262
Scientific Notation
Used to write really large
 and very small numbers
 in compact form
    1. 2.4 x 10 –4

    2. 1.7 x 105
Polynomials
 Monomial
 Binomial
 Trinomial
 Degree of a polynomial: the
  greatest of the degrees of any
  of the terms
 Leading coefficient: coef. of
  the term with highest degree
  (not necessarily the first term)

 Constant
 Evaluate polynomial
  functions

 Graph polynomial functions

 Add and subtract polynomials
Multiplying Polynomials

 Monomial • trinomial

 Binomial • binomial

 Binomial • trinomial
Multiplying Polynomials
 Special products:
(a + b)(a – b)
 (a +b) 2

(a – b) 2

 Application problems
Multiplying Polynomials
 Monomial • trinomial

 Binomial • binomial

 Binomial • trinomial
Multiplying Polynomials
Special products:
   (a + b)(a – b)
   (a +b)  2

   (a – b)  2



Application problems
Division of polynomials
 Long division

 Synthetic Division
Remainder Theorem: If you
 divide P(x) by (x – a)… so
 that “a” is outside the
 division box, the
 remainder will always =
 P(a)
    constant is +
signs are both + if middle term
is + ; signs are both – if middle
term is –
    constant is –
one binomial has +, the other
has –
An important fact is that
if the terms of the
trinomial do not have a
common factor, then you
cannot have a common
factor within either
binomial.
Special Factoring
a2 + b2 is not factorable

a2 – b2 = (a – b) (a + b)

a3 – b3 = (a – b)(a2 + ab + b2)

a3 + b3 = (a + b)(a2 – ab + b2)
Special Factoring
Some trinomials are factorable,
 even though not degree 2.
Checklist for the factoring
process (p. 304):
1. Is there a common factor?
2. Only 2 terms? Is it a2 - b2 or
    a3 - b3 or a3 + b3?
    Use the memorized
    patterns!
3. Trinomial ? Make 2
binomials! Check by FOIL
4. 4 terms ? Make 2 groups
of 2 terms then factor each
group watching for a
common factor to pull
out in front.
5. Are all factors prime or
can they be factored more?
Using Factoring to Solve
Equations
Method is based on the
“principle of zero products”.
If ab = 0, then a = 0 or b = 0
Important:
Equation must be set = 0
Then factor
Then set each factor = 0 and
solve
Application problems:
1.The sum of the squares of
  two consecutive odd
  integers is 130. Find the
  two integers.
Remember
For consecutive even or
 consecutive odd integers
 use x, x + 2, x + 4
For consecutive integers,
 use x, x + 1, x + 2
2. The length of a rectangle is
    5 inches longer than the
    width. The area of the
    rectangle is 66 in2. Find the
    width and the length of the
    rectangle.
3. If f(x) = – x – 2, find two
               x 2

 values of c in the domain of
 f(x) for which f(c) = 4

(see p. 312)
Rational Expression:
a fraction with polynomials
in the top and bottom
Some problems are
  review…☻☺☻☺
 Function notation
 Evaluate rational
  functions
 Find the domain
Operations on Rational
Expressions (see handout)

1. Simplify
 – Factor first!
 – Divide out (cancel)
    common factors
2. Multiply
 – Factor everything!
 – Divide out common
    factors from top and
    bottom
 – Multiply straight across
    leaving in simplified
    factored form
3. Divide
 – Factor everything!
 – Invert second fraction and
 multiply by recip
 – Divide out common
    factors
 – Multiply straight across...
    again leave parens
4. Add and Subtract
 – Factor denom and find
   common denom!
 – Change all fractions to
   common denom
– Add / subtract tops and
put that answer      over
the common       denom
– Simplify (factor then
  cancel)
Important!!!
Never cancel across
   +, –, ÷, =
Rational Expression:
a fraction with polynomials
in the top and bottom
Some problems are
  review…☻☺☻☺
 Function notation
 Evaluate rational
  functions
 Find the domain
Operations on Rational
Expressions (see handout)

1. Simplify
 – Factor first!
 – Divide out (cancel)
    common factors
2. Multiply
 – Factor everything!
 – Divide out common
    factors from top and
    bottom
 – Multiply straight across
    leaving in simplified
    factored form
3. Divide
 – Factor everything!
 – Invert second fraction and
 multiply by recip
 – Divide out common
    factors
 – Multiply straight across...
    again leave parens
4. Add and Subtract
 – Factor denom and find
   common denom!
 – Change all fractions to
   common denom
– Add / subtract tops and
put that answer      over
the common       denom
– Simplify (factor then
  cancel)
Important!!!
Never cancel across
   +, –, ÷, =
Complex Fractions have at
    least one fraction
    within a fraction.
To simplify:
 1. Find the smallest common
    denominator (LCD) of all
    the denominators in the
    top and bottom of the
    fraction
2. Multiply the entire
    numerator and the
    entire denominator by
    the LCD. This should
    clear the fractions from
    the top and the bottom
    of the “main” fraction.
3. Factor the new top and
    bottom, then cancel
    common factors to
    simplify.
Rational equations (equations
containing fractions):
A. Multiply both sides of the =
   by the common denom. of
   all the fractions . This
   should clear all the
   denominators!
If you have just one fraction
    on each side of the = you
    can just cross multiply.
B. Solve this “fraction free”
    equation
C. Check your answer into the
    original equation and
    reject any answer that
    makes a denom = 0
Work Problems
See handout!
Uniform Motion Problems




See handout!
Proportion –      equation with 2
rates or 2 fractions set =
Set up the pattern in words,
 then create the equation!
To solve, do the 2 cross
 products and set them =
Be sure that both fractions have
 the same set up:

  lb = lb       tax = tax
cost cost     total  total

etc.
  Variations

K = constant of proportionality

Direct variation: y = kx
So      y = constant = k
        x
In a direct variation, both
values increase or both
decrease.
weight & postage
cost of item & sales tax
income & income tax
In an inverse variation,
   y = k
         x
So xy = constant = k
In every inverse variation,
one quantity increases but the
other decreases.
volume of gas & pressure
light intensity & distance
    from bulb
rate of speed & time
In every variation, determine
the pattern first and write it
down, then plug in the given
numbers.
In a joint variation, one
quantity varies directly as a
product....
But there are none in
exercises!!
Literal equations have more
than one variable.
You will have to rewrite the
given equation so that a
specified variable is isolated.
To do this:
1. Clear fractionsand
   get rid of parens.
2. Do opposite operations to
   isolate the specified
   variable on one side of
   the =
Note: If 2 terms have the
needed variable, put them
on the same side of the =,
then factor before dividing.
  Chapter 7
 Our same rules for exponents
  from Chapter 5 apply even if
  the exponents are fractions.
 Remember a negative
  exponent does not cause a
  negative # …and do not leave
  any negative exponent!
Using fraction exponents
 Apply the rules
 Evaluate expressions
 Change to radicals
 Change radicals back to
    fraction exponents
Facts about radicals

 Principal √




Can be simplified! Start with
 innermost

or use fraction exponents.
Operations with radicals
 Simplify

 Multiply (need same index)

 Add / Subtract (need same
 index and same radicand
 Divide (need same index)
 Do not leave a fraction
  under any radical
 Do not leave any radical in
  the denominator!
      Monomial in denom.
      Binomial in denom.
Complex Numbers




i =–1
  2
  Complex Numbers
 a + bi

real   imaginary
part     part
  Complex Numbers
 Simplify
 Always rewrite the
   with “i” first thing!
 Add / subtract
  Complex Numbers
Multiply

 Divide

   Conjugate of a + bi
      is a – bi
31. One printer can print the paychecks
 for the employees of a company in 54
 min. A second printer can print the
 checks in 81 min. How long would it
 take to print the checks with both
 printers operating? 32.4 min
32. A mason can construct a retaining
  wall in 18 h. The mason's apprentice
 can do the job in 27 h. How long would
  it take to construct the wall if they
    worked together? 10.8 h
33. One solar heating panel can raise the
  temperature of water l' in 30 min. A
 second solar heating panel can raise the
  temperature l' in 45 min. How long
  would it take to raise the temperature of
  the water l' with both solar panels
  operating? 18 min
34. One member of a gardening team
  can landscape a new lawn in 36 h. The
 other member of the team can do the
  job in 45 h. How long would it take to
  landscape a lawn if both gardeners
  worked together? 20 h
35. One member of a telephone crew can
  wire new telephone lines in 5 h. It takes
  7.5 h for the other member of the crew
  to do the job. How long would it take to
  wire new telephone lines if both
  members of the crew worked together?
  3h
36. A new printer can print checks three
  times faster than an old printer. The
 old printer can print the checks in 30
  min. How long would, it take to print
 the checks with both printers operating?
  7.5 min
37. A new machine can package
  transistors four times faster than an
  older machine. Working together, the
  machines can package the transistors in
 8 h. How long would it take the new
  machine, working alone, to package
 the transistors? 10 h
38. An experienced electrician can wire a
  room twice as fast as an apprentice
 electrician. Working together, the
  electricians can wire a room in 5 h. How
 long would it take the apprentice,
  working alone, to wire a room? 15 h
39. The larger of two printers being used
  to print the payroll for a major
  corporation requires 40 min to print the
  payroll. After both printers have been
  operating for 10 min, the larger printer
  malfunctions. The smaller printer
  requires 50 more minutes to complete
  the payroll. How long would it take the
  smaller printer, working alone, to print
  the payroll? 80 min
40. An experienced bricklayer can work
  twice as fast as an apprentice brick-
 layer. After they worked together on a
  job for 8 h, the experienced bricklayer
  quit. The apprentice required 12 more
  hours to finish the job. How long would
  it take the experienced bricklayer,
  working alone, to do the job? 18 h
41. A roofer requires 12 h to shingle a roof.
  After the roofer and an apprentice
 work on a roof for 3 h, the roofer moves
  on to another job. The apprentice
 requires 12 more hours to finish the job.
  How long would it take the apprentice,
  working alone, to do the job? 20 h
42 . A welder requires 25 h to do a job.
  After the welder and an apprentice
  work on a job for 10 h, the welder quits.
  The apprentice finishes the job in 17 h.
  How long would it take the apprentice,
  working alone, to do the job? 45 h
43 . Three computers can print out a task
  in 20 min, 30 min, and 60 min,
 respectively. How long would it take to
  complete the task with all three
  computers working? 10 min
44. Three machines fill soda bottles. The
  machines can fill the daily quota of soda
  bottles in 12 h, 15 h, and 20 h,
  respectively. How long would it take
 to fill the daily quota of soda bottles with
  all three machines working?

  5h
45. With both hot and cold water running,
  a bathtub can be filled in 10 min. The
  drain will empty the tub in 15 min. A
  child turns both faucets on and leaves
  the drain open. How long will it be
  before the bathtub starts to overflow?
  30 min
46. The inlet pipe can fill a water tank in
  30 min. The outlet pipe can empty the
  tank in 20 min. How long would it take
  to empty a full tank with both pipes
  open? 60 min
47. An oil tank has two inlet pipes and
  one outlet pipe. One inlet pipe can fill
 the tank in 12 h, and the other inlet pipe
  can fill the tank in 20 h. The outlet pipe
  can empty the tank in 10 h. How long
  would it take to fill the tank with all three
  pipes open? 30 h
48. Water from a tank is being used for
  irrigation at the same time as the tank
 is being filled. The two inlet pipes can
  fill the tank in 6 h and 12 h, respectively.
  The outlet pipe can empty the tank in 24
  h. How long would it take to fill the tank
  with all three pipes open? 4.8 h
49. An express bus travels 320 mi in the
 same amount of time it takes a car to
 travel 280 mi. The rate of the car is 8
 mph less than the rate of the bus. Find
 the rate of the bus. 64 mph
50. A commercial jet travels 1620 mi in
  the same amount of time it takes a cor-
 porate jet to travel 1260 mi. The rate of
  the commercial jet is 120 mph greater
  than the rate of the corporate jet. Find
  the rate of each jet.
 Commercial: 540 mph;
   corporate: 420 mph
51. A passenger train travels 295 mi in
  the same amount of time it takes a
 freight train to travel 225 mi. The rate of
  the passenger train is 14 mph greater
  than the rate of the freight train. Find
  the rate of each train.
Passenger: 59 mph; freight: 45 mph
52. The rate of a bicyclist is 7 mph more
  than the rate of a long-distance runner.
  The bicyclist travels 30 mi in the same
  amount of time it takes the runner to
  travel 16 mi. Find the rate of the runner.
  8 mph
53. A cyclist rode 40 mi before having a
  flat tire and then walking 5 mi to a
  service station. The cycling rate was
  four times the walking rate. The time
 spent cycling and walking was 5 h. Find
  the rate at which the cyclist was riding.
  12 mph
54. A sales executive traveled 32 mi by
  car and then an additional 576 mi by
 plane. The rate of the plane was nine
  times the rate of the car. The total
 time of the trip was 3 h. Find the rate of
  the plane. 288 mph
55. A motorist drove 72 mi before running
  out of gas and then walking 4 mi to a
  gas station. The driving rate of the
  motorist was twelve times the walking
  rate. The time spent driving and
  walking was 2.5 h. Find the rate at
  which the motorist walks. 4 mph
56. An insurance representative traveled
  735 mi by commercial jet and then
 an additional 105 mi by helicopter. The
  rate of the jet was four times the rate of
  the helicopter. The entire trip took 2.2
  h. Find the rate of the jet. 525 mph
57. An express train and a car leave a
  town at 3 P.m. and head for a town
 280 mi away. The rate of the express
  train is twice the rate of the car. The
 train arrives 4 h ahead of the car. Find
  the rate of the train. 70 mph
58. A cyclist and a jogger start from a
  town at the same time and head for a
 destination 18 mi away. The rate of the
  cyclist is twice the rate of the jogger.
  The cyclist arrives 1.5 h before the
  jogger. Find the rate of the cyclist.
  12 mph
59. A single-engine plane and a
  commercial jet leave an airport at 10
  A.M. and head for an airport 960 mi
  away. The rate of the jet is four times
  the rate of the single-engine plane. The
  single-engine plane arrives 4 h after the
  jet. Find the rate of each plane.
Jet: 720 mph; single-engine: 180 mph
60. A single-engine plane and a car start
  from a town at 6 A.m. and head for a
 town 450 mi away. The rate of the
  plane is three times the rate of the car.
 The plane arrives 6 h before the car.
  Find the rate of the plane. 150 mph
61. A motorboat can travel at 18 mph in
  still water. Traveling with the current
 of a river, the boat can travel 44 mi in
  the same amount of time it takes to
 go 28 mi against the current. Find the
  rate of the current. 4 mph
62. A plane can fly at a rate of 180 mph in
  calm air. Traveling with the wind,
 the plane flew 615 mi in the same
  amount of time it took to fly 465 mi
 against the wind. Find the rate of the
  wind. 25 mph
63. A tour boat used for river excursions
  can travel 7 mph in calm water. The
 amount of time it takes to travel 20 mi
  with the current is the same
 amount of time it takes to travel 8 mi
  against the current. Find the rate of
 the current. 3 mph
64. A canoe can travel 8 mph in still
  water. Traveling with the current of a
 river, the canoe can travel 15 mi in the
  same amount of time it takes to travel
  9 mi against the current. Find the rate of
  the current. 2 mph
Solving Equations Containing
Radicals
1. Isolate the radical. If there
   are 2 radicals, isolate one of
   them.
2. Raise both sides of the = to
   the same power as the index
   on the radical. Use FOIL as
   needed!
3. If you still have a radical,
   repeat steps 1 and 2. (Isolate
   the radical and square again!)
4. Solve
5. Check results into the original
   equation.
Application Problems

 Use Pythagorean Theorem
 (see pages 412-413)

 Use formula given in the
   problem
 (see p. 414)
Quadratic Equations

Standard form:
    ax2 + bx + c = 0

Quadratic equations are always
 degree 2
Quadratic Equations

Solve by factoring:
  1. Set equation = 0
  2. Factor
  3. Set each factor = 0 and
   solve
Quadratic Equations

 Given solutions, write the
  quadratic equation with integral
  coefficients
Solve by taking square roots:

  1.          ) 2
       Write ( = #
  2.   Take       of both sides
         (Remember )
  3.   Isolate the variable
Ways to Solve Quadratic
Equations

1.Factoring
2.Taking
3. Completing the square
  Solve by completing the square
1. Divide all terms by the coef of
   the x2 so that you have just x2
2. Move constant to the right of =
3. Add (1/2 coef.of x)2 to both side
   of = to “complete the square”
4. Rewrite as (  ) 2 =#
5. Take       ….Remember 
6. Isolate the variable
Ways to Solve Quadratic
Equations

1.Factoring
2.Taking
3. Completing the square
4. Using the Quadratic
   Formula
Use the Quadratic Formula
1. Write the equation = 0
    ax 2 + bx + c = 0

2. Use the formula
    X = –b 
          2a
3. Simplify
 Use the discriminant to decide
 the types of solutions

Discriminant =   b2   – 4ac
If b2 – 4ac < 0, there are 2
  complex (i) solutions
If b2 – 4ac  0, there are 2 real
  (no i) solutions
If b 2 – 4ac = 0, there is 1 real

  solution (called a double root)
Equations that are    in
Quadratic Form

The exponent on one variable
 will be ½ of the exponent on
 the other variable term.
 Set = 0
 Factor
 Set factors = 0 and solve
  Radical Equations
 Isolate the radical
 Square both sides of the =
  (Use FOIL when needed)
 If there is still a   repeat
  steps 1 and 2
 Set = 0 and solve using
  factoring, the quad.
  formula, or completing the
  square
 Reject any       solution
  that     causes
Equations with fractions
 Multiply by the common
  denominator to clear
  fractions
 Set = 0 then solve by
  factoring, completing the
  square, or quad. Formula
 Reject any solution that
  makes denom = 0
Nonlinear Inequalities
Two types: See handout!
1. Polynomial inequalities

2. Rational (fraction)
 inequalities
Quadratic function
   f(x) = ax2 + bx + c or
        y = ax2 + bx + c
 The graph is a parabola
    opens up if a  0
    opens down if a  0
    axis of symmetry is
             x = -b/2a
Vertex
Y – intercept (let x = 0)
X – intercepts:
    let y = 0 and solve the
  equation by factoring,
  quadratic formula or
  completing the square
The parabola will have       0, 1,
  or 2 x – intercepts.

You can tell how many         x–
  intercepts from the graph, from
  the solving process, or from the
  discriminant (b  2 – 4ac)
 Domain: all real numbers

 Range: y values used
          (look at graph!)
Graphing functions
 Knowing the general shape
  will help you graph it!

  1. straight line: y = mx + b

  2. parabola: y = ax2
3. Cubic: f(x) =   ax 3



4. Absolute value

5. Radical
 Domain and range can be
  estimated from the graph and
  named in set notation or
  interval notation
Some graphs are functions…
 and some are not.



Use the vertical line test to
 determine whether a graph is
 a function.
Operations on Functions
 Add
 Subtract
 Multiply
 Divide
Operations on Functions
 Composition of functions
 Inverse of a function
 1. Interchange x and y
 2. Solve for the “new” y if it
    is an equation
 3. Notation for inverse is f -1
One-to-one functions
 Every 1 – 1 function passes the
  horizontal line test as well as
  the vertical line test.
 If a function is 1 – 1, it has an
  inverse that is also a function.
Test today... or
tomorrow!!!!

 Are you ready???
Test today... or
tomorrow!!!!

 Don’t forget!!!!
Today is the last day
for the Chapter
Test….
Don’t forget!!!!
The Chapter 4 Test
opens today…..
Are you ready????
The Chapter 6 Test
opens tomorrow…..
Are you ready????
The Chapter 6 Test
opens tomorrow…..
Are you ready????
The Chapter 7 Test
opens tomorrow…..
Are you ready????
Chapter 10….our last chapter!
  Exponential functions:
    Format
    Evaluate

    Graph




      Special function, f(x) = ex
Logarithmic Form

  For b > 0, y = logb x is
    equivalent to by = x

  Rewrite log form and
    exponential form
Logarithmic Form
 Evaluate logs

 Solve for x in various
   positions
Special logs
 log x means log10 x

 ln x means loge x
Notice that
  bp
If =   bn,then p = n
This can be used to solve
 equations.
Properties of logs

1. If logb x = logb y,
   then x = y
2. logb xy =
     logb x + logb y
Properties of logs

3. logb x/y =
     logb x – logb y

4. logb   x p   = p logb x
Properties of logs

5. logb 1 = 0
6. logb b = 1
7. logb bx = x
Evaluate logs using a
calculator
Use the “change of base”
 formula:
loga n = log n or ln n
          log a ln a
In any y = logb x , you
must have x > 0 and b > 0

log 7 –49 is undefined!
log 10 0 is undefined!
log -2 8 is undefined!
2 ways to graph
   f(x) = logb x
1. Rewrite as b y = x and make
   an x/y roster of points.

2. Use calculator and the
   change of base formula to
   get points.
Solving exponential equations

Two types:
A. Sometimes you can rewrite
  as bx = by and     then x =
  y
Solving exponential equations

B. Sometimes you need to take
   the log or ln of both sides
   and move the exponents
   down.
   Then solve.
Solving log equations
Two types:
A. Logs on just one side of =
 1. Use properties to rewrite with

    one log on left
 2. Rewrite in exponential form

 3. Solve
  Solving log equations
B. Logs on both sides of the =
  1. Use properties to make

     one log on each side of the
     =
  2. Drop the logs

  3. Solve
 Know what your instructor
  requires!
     Read your syllabus;
     keep it for future reference.
 Don't fall behind! Math skills must
  be learned immediately and
  reviewed often. Keep up-to date
  with all assignments.
 Most instructors advise students
  to spend two hours outside of
  class studying for every hour
  spent in the classroom. Do not
  cheat yourself of the practice you
  need to develop the skills taught
  in this course!
 Take the time to find places
  that promote good study
  habits. Find a place where
  you are comfortable and can
  concentrate. (library, quiet
  lounge area, study lab)
 Survey each chapter                ahead of
  time.
     Read the chapter title, section headings and
      the objectives listed to get an idea of the
      goals and direction for the chapter.
 Take careful notes and write
  down examples.
 The book provides material to
  read and examples for each
  objective studied. It also has
  answers to the odd-numbered
  exercises in the back of the
  book so that you can check your
  answers on assignments.
 Be sure to read the Chapter
  summary and use the Chapter
  Review and Chapter Test
  exercises to prepare for each
  Chapter exam. (All answers are
  in the back for these)
 Spaced practice is generally
  superior to massed practice.
  You will learn more in 4
  half-hour study periods than
  in one 2 hour session.
 Review material often
  because repetition is
  essential for learning. You
  remember best what you
  review most. Much of what
  we learn is soon forgotten
  unless we review it.
 Attending class is vital if you are
  to succeed in any math course.
 Be sure to arrive on time…. and
  stay the entire class period!
 You are responsible for
  everything that happens in class,
  even if you are absent.
If you must be absent :
 1. Deliver due assignments to
  instructor as soon as possible (even
  ahead of time if you know in advance).
 2. Copy notes taken by a classmate
  while you were absent.
 3. Ask about announcements,
  assignments or test changes made in
  your absence.
 If you have trouble in this
  course – seek help!
    1. Instructor

    2. Tutors

    3. Video Tapes

    4. Computer Tutoring
Study Tips: Preparing for Tests
 Try the Chapter Test at the end
  of each chapter before the actual
  exam. Do these exercises in a
  quiet place and pretend you are
  in class taking the exam.
Study Tips: Preparing for Tests

 If you missed questions on the
  practice test, review the material,
  practice more problems of the
  same type, get help as needed.
Try these strategies of successful
test takers:
 1. Skim over the entire test
  before you start to solve any
  problems.
 2. Jot down any rules, formulas
  or reminders you might need.
 3. Read directions carefully.
 4. Do the problems that are
  easiest for you first.
 5. Check your work to be sure
  you haven't made any careless
  errors.

				
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