Transcript Math 108

Chapter 1
Set: a collection of objects
called elements
Important sets of numbers:
Natural numbers
Whole numbers
Integers
Use the ‘roster method’ to
write a set
Inequality symbols
Additive inverses
Absolute value
Addition of integers
• SAME SIGNS
1. Add the absolute values of the
numbers ( ignore the signs
and add)
2. Attach the common sign
• DIFFERENT SIGNS
1. Find the difference of the
absolute values (ignore the
signs and subtract)
2. Attach the sign of the
number with the larger
absolute value.
Subtraction of integers
1. Rewrite the “—” as “+
the opposite of the
number”
2. Follow the rules for
addition.
Subtraction of integers
• We might say this
subtraction process as
“Change to the opposite
then add”
Multiplication of integers
• The product of two numbers
with the same sign is positive.
positive · positive = +
negative · negative = +
Multiplication of integers
• The product of two numbers
with different signs is negative.
positive · negative = negative
negative · positive = negative
When multiplying or
dividing:
ODD # of negative signs
makes the answer negative.
EVEN # of negative signs
makes the answer positive.
Division of integers (The rules
are like those for multiplication)
• The quotient of two numbers
with the same sign is positive.
positive ÷ positive = +
negative ÷ negative = +
• The quotient of two numbers
with different signs is negative
positive ÷ negative = negative
negative ÷ positive = negative
• If a fraction is negative,
the ‘–’ sign can be place in
any of three different
positions, and all are
considered equivalent.
• 0 ÷ any number = 0
• Division by 0 is not defined!
• To compute the arithmetic
mean or average:
Operations with rational numbers
• Rational numbers are fractions
and decimals which repeat or
stop.
• Reduce fractions to simplest
form
• Change fractions into decimals
which repeat or stop
• Add, subtract, multiply and
divide fractions
• Add, subtract, multiply and
divide decimals
• Don’t forget to use our rules
for + and - signs!
Working with %
• Change % to a decimal
• Change a decimal to a %
• Change a fraction to a %
• Change a % to a fraction
Order of Operations
1. Do operations within
grouping symbols first.
2. Exponents
3. Multiply and divide from
left side of problem toward
right side.
4. Add and subtract last!
Working with Exponents
4
2
= (2)(2)(2)(2)
4
–2)
(
is not the same
4
as – 2
A good way to remember the
order is
PEMDAS
Remember the “parens” can
also be brackets, absolute
value symbols, or a fraction
bar.
Chapter 2
• Variable: a letter used to
stand for a quantity
• A variable expression is
made up of terms
Types of terms:
variable terms
constant terms
A variable term has
a coefficient
When evaluating variable
expressions, remember
PEMDAS to get the right
order of operations.
Also, watch
your signs!
Like terms – terms which
have the exact same
variable part
Constant terms are also
“like terms”
Only ‘like’ terms can be
added or subtracted!!!
Be able to use the
distributive
property
Translating Verbal
Expressions
See p. 67 for a list of
common phrases…..
When you see the
phrase “in terms of”…
let the part which
follows be x
Chapter 3
Solve an equation means to
find the value which makes
the equation true.
We want to end with :
Variable = constant
• To do this, perform ‘opposite
operations’ to both sides of
the equation.
• On your assignments, please
show the process!
• To check whether a value is
really a solution to an
equation, put it in place of the
variable and see if it makes a
true statement.
% Problems
1.
2.
3.
4.
5.
Change % to a decimal
Of  multiply
Is  =
Write an equation
Solve the equation
Solving Equations
1. First simplify each side of the
equation. Distribute to get rid
of parens and combine like
terms
2. Add or subtract terms to
move all constants to one side
and all variable terms to the
other side of the =
3. Divide or multiply to get rid
of the coefficient of the
variable.
End with :
= constant
variable
3.4 Translating Sentences into
Equations
These words or phrases are
replaced with an = sign:
equals
is
is equal to
amounts to
represents
Steps:
1. Assign a variable to the
unknown quantity.
2. Translate the words into math
symbols. Write the equation.
3. Solve the equation.
4. Check your answer.
Recall the integers are the positive
and negative whole numbers:
{… -4, -3, -2, -1, 0, 1, 2, 3, 4 …}
An even integer is an integer that is
divisible by 2 like 12, -4, 28, 0
An odd integer is not divisible by 2
like 33, -27, and 5
Consecutive Integers (none in
exercises)
5, 6, 7 or -11, -10, -9 or n, n + 1, n + 2
Consecutive Even Integers
-12, -10, -8 or 4, 6, 8 or n, n + 2, n + 4
Consecutive Odd Integers
5, 7, 9 or -13, -11, -9 or n, n + 2, n + 4
Find three consecutive even
integers such that three times
the second is four more than
the sum of the first and the
third.
Five times the first of two
consecutive odd integers
equals three times the second
integer. Find the integers.
Translate “three more than
twice a number is the number
plus six” into an equation.
Four less than one-third of a
number equals five minus
two-thirds of the number.
Find the number
The sum of two numbers is
sixteen. The difference
between four times the
smaller number and two is
two more than twice the
larger number. Find the two
numbers.
The sum of two numbers is
twelve. The total of three
times the smaller number and
six amounts to seven less
than the product of four and
the larger number. Find the
two numbers.
The difference between a
number and twelve is twenty.
Find the number.
A board 20 ft long is cut into
two pieces. Five times the
length of the smaller piece is
2 ft more than twice the
length of the longer piece.
Find the length of each piece.
A company makes 140
televisions per day. Three
times the number of black
and white TV’s made equals
20 less than the number of
color TV’s made. Find the
number of color TV’s made
each day.
Translating Sentences into
Equations
1. Assign a variable or an expression to the
unknown quantity or quantities.
2. Translate the verbal expressions into math
symbols. We want two expressions equal to
each other.
3. Solve the equation.
4. Check your answer.
Chapter 4
• Monomial….a number, a
variable, or the product
(mult.) of numbers and
variables
• Polynomial…2 or more
monomials added or
subtracted (The monomials
are then called terms.)
Special polynomials
• Binomial: 2 terms
• Trinomial: 3 terms
• To add or subtract
polynomials, you can use a
vertical or a horizontal
format.
• When subtracting, remember
to distribute the ‘–’ to all
terms inside the parens.
Basic Rules for exponents
1.
n
ax
n
+bx
=
n
(a+b)x
In ‘like’ terms, add coefficients
but not the exponents.
• =
n
p
n
+p
3. (ax ) • (bx ) = (ab)x
2.
n
x
p
x
n
+p
x
(Mult. coef. and add expon.)
n
p
(x )
np
x
4.
=
n
p
p
np
5. (ax ) = a x
(Do coef. to power, but mult.
expon.)
n
b
p
np
bp
6. (x y ) = x y
(Multiply exponents)
Using the rules…..
1. monomial • monomial
2. monomial • polynomial
3. binomial • binomial
4. binomial • trinomial
5. Special products
Rules for exponents
1.
n
ax
n
+bx
=
n
(a+b)x
In ‘like’ terms, add coefficients
but not the exponents.
2.
n
(ax )
•
p
(bx )
= (ab)x
n +p
(Mult. coef. and add expon.)
n
p
(ax )
p
np
a x
3.
=
(Do coef. to power, but mult.
expon.)
n
b
p
(x y )
np
bp
x y
4.
=
(Multiply exponents)
5. Rule for division:
Divide coefficients.
Do top exponent minus bottom
exponent to get new
exponent….or cancel equal
amount from top and bottom.
6. Fraction to a power:
Apply exponent to top and
bottom
7.
0
a =
1 if a  0
8. x
-p
ax
-p
p
x
=1/
and
p
= a/x
A negative exponent makes
that part move but does not
make it negative!
9. a
/x
-p
=
p
ax
-p
10. (fraction) =
p
(reciprocal of fraction )
Negative exponent inverts
fraction…but does not
make it negative!
Simplest form is written with
no negative exponents!
Scientific notation
Used to write really large and
very small numbers in
compact form
2.4  10
4
2. 1.7  10
1.
-3
• 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
time.
ahead of
– 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.
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????
Finding and Factoring out
Common Factors
 Factor 15
 Factor 30
 To factor a number into prime
numbers, we break it into the
prime numbers which would
multiply to equal the original
number
• Factor a polynomial, means
the same thing….break it
into monomials, binomials,
polynomials, etc. that are
prime and would multiply
together to equal the original
polynomial.
Always look for a common factor
first thing!
Sometimes the common factor is a
monomial…….
• Sometimes the common factor
is a binomial…………..
• And sometimes you have to
factor out a -1 in order to
have a common binomial.
If you have a polynomial with 4
terms:
1. Make 2 groups of 2 terms each
2. Factor the groups
3. Find the common binomial
factor and write it in parens.
4. Write the second set of parens.
Factoring Trinomials like x2
+ bx +c
1. Factor out all common
factors and write them in
front of the parens.
2. If the coefficient of x2 is -1,
factor out the -1; put the
terms in descending order.
3. If the constant is + , make 2
binomials with the same sign
as the middle term.
4. If the constant is - , one
binomial will have a + , the
other will have a -.
5. Outside product + inside
product must = middle term.
6. Check your binomial factors by
using the FOIL process.
5.3 Factoring trinomials
like ax2 + bx + c
1. Factor out any common factors
2. Make sets of 2 binomials for
each combination of factors of
the coefficient of x2.
3. Use the signs of the
constant and middle term to
determine the signs within the
binomials:
constant is +
signs are both + if middle term
is + ; signs are both - if middle
term is constant is -
one binomial has +, the other
has -
4. Remember from FOIL that
the outside product + the inside
product must = the middle
term.
5. 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.
We will not do factoring by
grouping….ignore it in the
book!
5.4 Special factoring situations
a2 + b2 is nonfactorable over
the integers
a2 – b2 is called the difference
of perfect squares and it is
factorable!
2
2
a – b = (a + b)(a - b)
Some trinomials in this section
will be perfect squares:
2
x
+ 4x + 4
4x2– 20x + 25
2
9y
+ 30y + 25
16a2 – 8a + 1
2
x–
6x + 9
This section will also have some
polynomials with four terms
like we had earlier :
x3 – 3x2 – 4x + 12
2
2
ab
–
2
49a
–
2
b
+ 49
Good checklist for the factoring
process on p. 229:
1. Is there a common factor?
2. Only 2 terms?
Try (a + b)(a – b)
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
If ab = 0, then a = 0 or b = 0
This is called the “principle of
zero products”.
STEPS
1. Make the equation = 0
2. Then factor
3. Then set each factor = 0
and solve.
1. (x – 3)(x – 5) = 0
2. 3x(x + 2) = 0
3. (2x + 3)(3x – 1) = 0
4. 2x2 + x – 6 = 0
5.
2
y
– 8y = –15
6. 2x2 – 50 = 0
7. (x + 2)(x – 7) = 52
Application problems:
1. The sum of the squares of
two consecutive positive
integers is 61. Find the two
integers.
2. A rectangle has a width of
x inches. The length is 4
inches longer than twice
the width. The area of
2
the rectangle is 96 in .
Find the width and the
length of the rectangle.
3. The sum of two
numbers is six. The
sum of the squares of
the two numbers is
twenty. Find the two
numbers.
4. Sometimes a formula
is given for the
problem….see page 241
Rational expression:
a fraction with a polynomial
in the numerator and / or
the denominator
To simplify:
1. Factor the top and
bottom
2. Divide out common
factors from the top and
the bottom
To Multiply:
1. Factor the top and bottom
2. Divide out common
factors from the top
and the bottom
3. Multiply straight across,
leaving the top and
bottom in simplified
factored form
To Divide:
1. Flip the second fraction
and multiply by that
reciprocal
2. Factor and cancel as in
multiplication
To add or subtract rational
expressions:
1.Factor all denominators
2. Change all fractions to a
common denominator by
multiplying the top and
bottom of each fraction
by the factors it needs to
match the common
denominator.
3. Add / subtract the tops and
put that answer over the
common denominator
4. Simplify by factoring the top
and canceling common
factors from the numerator
and the denominator.
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 whole
numerator and the
whole denominator by
this LCD. This should
clear the fractions from
the top and the bottom
of the “main” fraction.
3. Simplify by factoring the
new top and bottom and
canceling common factors
To solve an equation containing
fractions:
1. Factor all denominators and
find the common denom
2. Multiply each side of the
equation by the LCD. This
should clear all the
denominators!
3. Solve this “fraction free”
equation
4. Check your answer into the
original equation and
reject any answer that
makes a denom = 0
Proportion: an equation
stating that 2 fractions are
equal
To solve a proportion,
1. Multiply both sides of
the = by the common
denom
OR
2. Cross multiply and set
the cross products =
Application problems
Write the fraction’s pattern
in words first!
Application problems
1. Sixteen tiles are needed
to tile an area that is 9 ft2.
At this rate, how many
square feet can be tiled
using 256 tiles?
2. A monthly
loan
payment is $29.75
for each $1000
borrowed. At this
rate, find the
monthly payment for
a $9800 car loan.
3. Package directions say that
3.5 ounces of a certain
medication are required
for a 150 lb. adult. At this
rate, how many additional
ounces are needed for a
210 lb. adult?
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 fractions
by
multiplying both sides
of the equation by the
LCD and get rid of
parens.
2. Use opposite operations to
move all terms with the
specified variable to one
side of the = and all
terms without that
variable to the other side
of the =
3. Isolate the specified
variable (usually by
division…but if 2 or
more terms have the
specified variable, you
will have to factor first,
then divide)
Important concepts and
formulas for sections 7.1, 7.2
1. Rectangular coordinate
system
2. Ordered pairs
3. Graph ordered pairs
4. Ordered pair solutions to
equations
5. Make relation from given
data
6. Is a relation a function?
7. Function notation…f(x)
8. Evaluate functions
Graphing lines using x|y roster
★ Find 3 solution points
★You can choose any x-value,
then put it into the equation to
get the y-value
Special Points
★ x-intercept and y-intercept
★ Find them
★ Use them to draw the graph
Let x = 0 to get the y-intercept.
Y-intercept is on the y-axis!
Let y = 0 to get the x-intercept.
X-intercept is on the x-axis!
Slope
Slope is a measure of the
steepness of a line. There are
several ways to find the slope:
Ways to find slope:
1. From the graph
2. From 2 points
3. From the equation
To graph the slope,
Top #  go up if + , down if –
Bottom #  go to right
Draw graphs
of lines
using slope and
intercept
y-
Set….a collection of objects
Naming sets:
1. Roster method
2. Set-builder notation
Inequalities
• Symbols
• Graph on a number line
Operations on sets
Union (A  B)
Set of all elements of A
together with all elements
of B. They are in A or in
B… all are included!
☺☺☺☺☺☺☺
Operations on sets
Intersection (A  B)
Set of all elements that are
common to A and B.
They are in both A and in
Some are left out!
B
☹☹☹☹☹☹☹☹☹☹
Solving Inequalities
Do operations exactly like
solving equations, but if you
divide both sides by a
negative number or multiply
both sides by a negative #,
you must reverse the
inequality symbol.
constant
• Put answer as
Variable < constant
OR
Variable  constant
Chapter 10
Operations with Square
Roots
x=
the number whose
square is x
Some
simplify to whole
numbers because the radicands
are perfect squares
a
2
=a
Other radicands are not perfect
squares, but can be simplified
ab
=
a
b
Adding and Subtracting
1. You must have the same
number or expression under the
to + or –
2. Add the coef. of like radicals
Multiplying Square Roots
1. Multiply coefficients
2. Multiply expressions under
the radicals and put under
one √
3. Simplify
Dividing Square Roots
1. Divide out common factors
to reduce
2. Simplify
3. Rationalize if denominator
still has
You cannot leave a radical in
the denominator
Two types:
1. Monomial in denom
2. Binomial in denom
Use a number between 1 and 10
times a power of 10.
There will be one digit then the
decimal point !
Dividing
Polynomials
1. Polynomial ÷ monomial
2. Polynomial ÷ binomial