Transcript MATH 010

MATH 010
JIM DAWSON
1.1
INTRODUCTION TO INTEGERS
This section is an introduction to:
•
Positive Integers
•
Negative Integers
•
Opposites
•
Additive Inverse
•
Absolute Value
1.2 ADDING AND
SUBTRACTING INTEGERS

If the signs are the same : ADD the absolute
values and place the common sign in the
answer.
6+7=13
- 13+(-5)= -18

If the signs are different : SUBTRACT the
absolute value of the smaller number from
the absolute value of the larger number.
Place the sign of the larger number using
absolute value in the answer.
14+(-6)= 8
-21+10= -11
1.3 MULTIPICATION AND
DIVISION OF INTEGERS
1.
Determine the sign of the answer first:
Count the negative signs: even number of
negative signs – the answer is positive
An odd number of negative signsthe answer is negative
2.Multiply or Divide using the absolute values
-5 x (-6)=30
7 x (-4)= -28
1.4 REVIEW OF FRACTIONS
AND DECIMALS WITH SIGNS

The rules for sign are the same for
fractions and decimals as they were for
integers.
CONVERTING BETWEEN
FRACTIONS, DECIMALS,
AND PERCENTS
 Change
a percent to a decimal.
Move the decimal point TWO
places from right to left.
 Change a decimal to percent.
Move the decimal point TWO
places from left to right.
FRACTION TO A PERCENT
 Change
the fraction to a
decimal(numerator divided by
denominator) and move the
decimal point TWO places from
left to right.
CHANGE A PERCENT WITH A
FRACTION TO A FRACTION
Drop the % and multiply by 1
over 100.
•EXPONENTIAL NOTATION
AND SOLVING EXPONENTS
If the base is negative and does not have
parentheses around it the sign of the answer
is ALWAYS negative.
 If the base is negative and has parentheses
around it; look at the exponent to find the
sign of the answer
 Even numbered exponent: positive answer
 Odd numbered exponent: negative answer

ORDER OF OPERATIONS
AGREEMENT
1.
2.
3.
4.
Priority #1-GROUPING SYMBOLS
Priority #2- EXPONENTS
Priority #3- MULTIPLY AND DIVIDE
AS THEY OCCUR FROM LEFT TO
RIGHT
Priority #4- ADD AND SUBTRACT AS
THEY OCCUR FROM LEFT TO RIGHT
TRANSLATE AND SIMPLIFY
The translation must be done first and then
simplify using the rules learned previously
in the chapter.
 The answer must be in descending order.

2.1 EVALUATING VARIABLE
EXPRESSIONS

COMBINING LIKE TERMS
 Add or subtract the terms with the same
variable part
 Place the answer in descending order
 2a+3b-4a+7b=2a-4a+3b+7b
 -2a+10b
2.2 SIMPLIFYING VARIABLE
EXPRESSIONS

Combining like terms
 Combine the terms with the same
variable part or the constants
 -3a+7+5a-9=-3a+5a and7-9
 2a-2
MULTIPLYING VARIABLE
TERMS

Multiply the number parts and bring the
variable into the answer.
 -3x(5)=-3(5)=-15x
 7(-4a)=-7(4)=-28a
 (-2b)(-6)=-2(-6)=12b
MULTIPLYING VARIABLE
TERMS

Multiply the number parts and bring the
variable into the answer.
 -6(-4a)=-6(-4)= 24a
 (-5x)(-3)=(-5)(-3)=15x
APPLYING THE
DISTRIBUTIVE PROPERTY
The Distributive Property is used to remove
parentheses.
 If the terms inside the parentheses are
different, multiply the term on the outside
by every term on the inside.
 Place the answer in descending order.
 3(2x-4)=3(2x) and 3(-4)=6x-12


If you cannot combine like terms inside the
parentheses, multiply the outside term by
each term inside the parentheses.
 -3(4x+2)=-3(4x) and –3(2)
 -12x-6
 Place the answer in descending order
SIMPLIFYING A GENERAL
VARIABLE EXPRESSION

Use the Distributive Property to remove
parentheses and brackets
 Combine like terms when possible
 Place the answer in descending order
 Be careful with sign!
2.3 TRANSLATING VERBAL
EXPRESSIONS
Memorize the expressions on p.67
 Rules for Parentheses
 Use parentheses to infer multiplication
when needed
 Use parentheses to separate two
processes together not separated by a
number or a variable


Use parentheses to separate a more than
or less than phrase with a number and
letter next to the phrase from any other
phrase in the expression
EXAMPLES OF
TRANSLATING VERBAL
EXPRESSIONS
7 ADDED TO 3 LESS THAN A NUMBER
 7+(n-3)
 4 TIMES THE DIFFERENCE BETWEEN
A NUMBER AND 4
 4(n-4)
 THE SUM OF 2 AND THE PRODUCT OF
A NUMBER AND 9

EXAMPLES OF
TRANSLATING
2+9x
 6 TIMES THE TOTAL OF A NUMBER
AND 8
 6(n+8)
 5 INCREASED BY THE DIFFERENCE
BETWEEN 10 TIMES a AND THREE
 5+(10a-3)

TRANSLATE AND SIMPLIFY
Translate the verbal expression FIRST and
then simplify using the rules that were
applied earlier in the chapter.
 The answer must be in descending order if
the expression was able to be simplified.

DEFINING THE UNKNOWNS
In order to define an unknown quantity,
assign a variable to that quantity, and then
attempt to express other unknown quantities
in terms of the same variable.
 These are equations of one variable;
therefore, the same variable must be used
when defining any unknowns.

3.1 SOLVING EQUATIONS OF
THE FORM x+a=b
The Addition Property of Equations
 The goal is to solve for the unknown
 VARIABLE = CONSTANT
 Find the number that is on the same side of
the equation as the variable and use the
opposite process on both sides of the
equation to solve the unknown quantity.

SOLVING EQUATIONS USING
THE ADDITION PROPERTY
x+a=b

X+4=12
 Find the number that is on the same side of the
equation as x and use the opposite process to
remove the number from the x side . The
number to be removed is 4. It`s opposite is –4.
The unknown can be solved by the following;
x+4-4=12-4;x=8
 You must do the same thing on both sides of the
equation; -4 on both sides
SOLVE AN EQUATION OF
THE FORM ax=b

Use the Multiplication Property of Equations to
solve the unknown
 Find the number that is on the same side of the
equation as the unknown and multiply both
sides by the reciprocal and the sign that is with
the number.
 Apply the Division Principle as a shortcut with
integers and decimals when possible.
THE BASIC PERCENT
EQUATION

Percent x Base = Amount
 P x B =A
 20% of what number is 30?
 Translate and solve the verbal
expression. Change the percent to a
decimal or fraction.
 0.20 x n = 30; Solve for n; 30 divided
0.20 = n; n = 150
MORE EXAMPLES OF THE
BASIC PERCENT EQUATION




70 is what percent of 80?
Translate and solve. Change the answer to a
percent.
 70 = n x 80; 70 divided by 80 = 0.875 which is
87.5%; n = 87.5%
What is 40% of 80?
Translate and solve. Change the percent to a
decimal.
 40% = 0.40;n = 0.40 x 80;n =32
3.2 GENERAL EQUATION
PART 1

Solve an equation of the form ax+b=c
 The goal is to write the equation as
variable=constant.
 5x+6=26 ; solve for x by applying the
Addition Property of equations to +6
 5x+6-6=26-6
 5x=20; divide both sides by 5; x=4
 Check by replacing x with 4
3.3 GENERAL EQUATION
PART 2
To solve an equation of the form
ax+b=cx+d
 Apply the Addition Property of Equations
twice and then the Multiplication Property
of Equations to solve the unknown.
 7a-5=2a-20; subtract 2a from both sides ;
5a-5=-20; add 5 to both sides; 5a=-15;
divide both sides by 5; a=-3 and check.

3.4 TRANSLATE AND SOLVE
Use the translation rules from chapter 2 and
solve the equations of one variable.
 Consecutive Integer Formulas
 Consecutive Integers:n,n+1,n+2
 Consecutive Even Integers; n,n+2,n+4
 Consecutive Odd Integers; n,n+2,n+4

SUM OF TWO NUMBERS
WORD PROBLEMS
Define the unknowns first.
 Smaller number is x; Larger number is the
sum minus x (the smaller number).
 Translate and solve for the smaller number
first and then the larger number. Each
problem must have two answers and add to
equal the original sum.

ADDITION AND
SUBTRACTION OF
POLYNOMIALS
Monomial- a polynomial of one term.
 Binomial- a polynomial of two terms.
 Trinomial- a polynomial of three terms.
 Quadnomial- a polynomial of four terms
 Descending Order- the exponents of the
variable decrease from left to right in the
answer.

4.1 ADDING AND
SUBTRACTING
POLYNOMIALS


Addition of polynomials
 Combine the like terms inside both sets of
parentheses(same sign-ADD; different signsSUBTRACT).
Subtraction of polynomials
 Multiply each term in the second polynomial by
–1(there is a minus sign in front of the
parenthese) then combine the like terms in both
polynomials.
4.2 MULTIPLYING
MONOMIALS
Multiply the coefficients and add the like
variable exponents.
 Simplifying powers of monomials
 Distribute the outside exponent to each
exponent in the monomial. Simplify the
coefficient completely in the answer. This
is the only time exponents are actually
multiplied.

4.3 MULTIPLICATION OF
POLYNOMIALS


Monomial times a polynomial.
 Multiply the monomial by applying the
distributive property to each term inside the
parentheses( the polynomial)
Multiplying two polynomials.
 Apply the distributive property by multiplying
each term in the first polynomial by each term
in the second polynomial and then combine the
like terms. Place the answer un descending
order.
TO MULTIPLY TWO
BINOMIALS



Use the FOIL method to multiply two binomials.
This is the simple application of the distributive
property in an ordered method.
F0IL METHOD;F- first terms are to be
multiplied;O- outside terms are multiplied; Iinside terms are multiplied;L-last terms are
multiplied. Combine the middle two terms if
possible.
MULTIPLYING BINOMIALS
WITH SPECIAL PRODUCTS
The Sum and Difference of two terms.
 Do FOIL; the middle two terms will
cancel; the answer will be a binomial
with a minus sign between the terms.
 The Square of a binomial.
 Do FOIL; the middle two terms will be
the same so add them; the answer will be
a trinomial.

4.4 NEGATIVE EXPONENTS

Division of monomials.
 To divide two monomials with the same
base, subtract the smaller exponent from
the larger exponent.
 Zero as an exponent.
 If zero is the dominant exponent the
answer is always 1.
RULES FOR SIMPLIFYING
NEGATIVE EXPONENTS

The negative exponent must be made positive by
moving it to the opposite place in the fraction.
This may be done first in the problem, but
especially in the answer.
 If there is a like base in the numerator and
denominator and both exponents are negative
they must be switched and made positive; then
use division rules to simplify.
MIORE RULES FOR
NEGATIVE EXPONENTS
If the bases are the same and one of the
exponents is negative and one is positive,
move the negative exponent to the positive
exponent and ADD the exponents together.
 When multiplying negative exponents,
combine the like base`s exponents together
using sign rules for addition and
subtraction. Make neg. exponents positive.

SCIENTIFIC NOTATION

In scientific notation, a number is expressed
as the product of two factors, the first
number must be a number between one and
ten(use of a decimal point may be needed),
and the other number a power of ten.
 To find the exponent in a number greater
than one, count the place values after the
first number.
MORE ON SCIENTIFIC
NOTATION

To write a decimal in scientific notation.
 Place a decimal point after the first
number in the decimal.
 To write the power of ten, count the place
values from the decimal point to the first
number in the decimal, this is the
exponent.
4.5 DIVISION OF
POLYNOMIALS

TO divide a polynomial by a monomial.
 Divide each term of the
polynomial(numerator) by the
monomial.Simplify the expression.
TO DIVIDE POLYNOMIALS

The process for dividing polynomials is
similar to the one for dividing whole
numbers. The use of long division is the
method.
 Steps: Divide the like variable terms and
place the answer in the quotient. Multiply
the quotient by each term on the outside
of the problem.
STEPS FOR DIVISION
Step 3 is to subtract the products( change
the sign of the second term and combine the
like terms).
 The process starts over; divide, multiply,
and subtract.
 If there is a remainder, write it as a
fraction.

5.1 GREATEST COMMON
FACTOR
Find the GCF of the coefficients which is
the largest number the numbers are divisible
by evenly.
 Find the GCF of the variable parts by
choosing the variable part with the smallest
exponent, but the variables must be in
common.

FACTORING BY GCF
Find the GCF of each term in the
polynomial and write it outside a
parentheses.
 Divide each term in the polynomial by the
GCF and write it inside the parentheses.
This is factoring by GCF.

FACTOR BY GROUPING
The polynomial must be a quadnomial(four
terms).
 Steps for factoring by grouping:
 Group the first two terms and the second
two terms with parentheses. The sign in
front of the third term is not inside the
parentheses.

STEPS FOR GROUPING
Find the GCF of each set of terms and
factor it out.
 To write the answer; write the common
binomial factor once and combine the
GCF`s into one binomial and check the
sign of this binomial to make sure it is
right.

5.2 FACTOR BY EASY
METHOD
 Factor
out a GCF first, if possible.
 Find the signs of the binomials and
place the correct variable in each
binomial.
EASY METHOD
 Find
the factors of the last term
whose sum or difference equals the
middle term. Write the correct
factors in the binomials.
5.3 TRIAL FACTORING
 Factor
out a GCF first, if possible.
 If the first term is 2 or greater,
factor by trial factors.
 Find the signs of the binomials
using the same rules as easy
method.
TRIAL FACTORING
 Find
the factors of the first and last
terms and place them in a chart.
 Do outer and inner FOIL with the
factors. The answer must match the
middle term. Write the factors in
the correct binomials and check.
5.4 SPECIAL FACTORING
 To
factor the difference of two
squares.
The problem must be a binomial
with a negative sign.
The signs of the binomials will
be (+) and (-).
THE DIFFERENCE OF TWO
SQUARES
 Find
the perfect squares of both
terms and set the up as the
difference of two squares.
 Write the terms twice, once in each
binomial.
PERFECT- SQUARE
TRINOMIALS
 This
method may be used as a
shortcut to trial factoring.
 Criteria:
Must be a trinomial with a (+)
sign in front of the last term.
CRITERIA FOR SPECIAL
FACTORING
The first and last terms must
have perfect squares.
Multiply the perfect squares
together twice and add them. The
answer must match the middle
term or factor by another
method.

5.5 SOLVING EQUATIONS BY
FACTORING
 The
Principle of Zero Products
states that if the product of two
factors is zero, then at least one of
the factors must be zero.
 If a x b = 0, then a =0 or b =0.
QUADRATIC EQUATION
 A Quadratic
Equation is in
standard form when the
polynomial is in descending order
AND equal to zero.
 Factor and solve. Each problem
will have two answers.
6.1 TO SIMPLIFY A
RATIONAL EXPRESSION
 Factor
the numerator and
denominator.
 Divide by the common factors.
 Be careful with the sign of the
simplified answer.
TO MULTIPLY RATIONAL
EXPRESSIONS
 Factor ALL numerators
and
denominators.
 Divide by the common factors.
 Multiply the numerators.
 Multiply the denominators.
TO DIVIDE RATIONAL
EXPRESSIONS
 Change
division to multiplication
and invert the second fraction.
 Follow the steps for multiplication
to simplify the problem.
 Be careful with the sign of the
answer.(Multiplying and dividing)
6.2 FINDING THE LCM
 Factor
the denominators first.
 To find the LCM:
 What is the greatest number of
times a term(monomial) occurs or
a set of terms(binomial) occurs?
ADDITION AND
SUBTRACTION
 Factor
the denominators.
 Find the LCM of the denominators
and place them under one fraction
bar.
 Place the fractions in higher
terms.(Divide and Multiply)
ADDITION AND
SUBTRACTION STEPS
 Combine
like terms in the
numerator.
 Simplify the answer(factor and
cancel).
6.4 COMPLEX FRACTIONS
 Find
the LCM of the denominators of
the fractions in the numerator and
denominator.
 Multiply the LCM by every term in the
numerator and denominator.
 Simplify the answer(factor and cancel).
6.5SOLVING EQUATIONS
WITH FRACTIONS
 Find
the LCM of the denominators.
 Multiply the LCM by every term in
the problem.
 Solve and check( if any
denominators equal zero the
answer is NO SOLUTION).
6.6 RATIO AND PROPORTION
 Cross
multiply and solve for the
unknown.
 In the word problems, make sure
that the rates are set up with like
units on top and like units on the
bottom.
6.7 LITERAL EQUATIONS
 A literal
equation is an equation
that has more than one variable.
 Use the Addition and
Multiplication Properties to help
solve for one of the variables.
10.1 RADICAL EXPRESSIONS
 A square
root of a number x is a
number whose square is x.
 The square root of 16 is 4.
 The number 4 is considered the
perfect square of 16.
PRODUCT PROPERTY
 If
the number under the radical
does not have a perfect square,
apply the Product Property of
Square Roots.
 Find the first number that has a
perfect square that goes into the
number evenly.
625
= 25=
360 = 3610
Find the square root of 36 which
is 6 and leave 10 under the
radical sign.
MORE ON THE PRODUCT
PROPERTY
 Write
the perfect square on the
outside of the radical and the
number that does not have a
perfect square on the inside the
radical.
TO SIMPLIFY VARIABLE
RADICAL EXPRESSIONS
 If
there is a variable under the
radical, find the perfect square by
dividing the exponent by 2 and
write the answer.
10.2 ADDITION AND
SUBTRACTION OF
RADICALS
Apply the Product Property to
each term. If the terms under the
radical are the same, combine the
like terms outside the radical.

10.3 MULTIPLICATION AND
DIVISION OF RADICALS
 Multiply
the terms under the
radicals and place the answer under
one radical.
 Apply the Product Property.
MULTIPLYING RADICALS
 If
there are parentheses then apply
the Distributive Property or FOIL
and then combine like terms if
possible.
 Conjugates are the sum and
difference of two terms.
DIVISION OF RADICALS
 Rewrite
the radical expression as
the quotient of the square roots.
 Apply the Product to each term.
 Simplify and write the answer.
 The answer cannot have radical in
the denominator.
DIVISION OF RADICALS
 If
the denominator has a radical,
the process is called rationalizing
the denominator.
 Multiply the denominator by both
the numerator and denominator.
 Simplify what is left.
7.1 GRAPHING
 To
graph the ordered pair (2,3), the
2 is plotted along the x-axis and 3
is plotted on the y-axis.
 The origin is 0 on the graph.
 2 is the x-coordinate and 3 is the ycoordinate.
RELATIONS AND
FUNCTIONS
 A relation
is any set of ordered
pairs.
 The domain is the set of first
coordinates.
 The range is the set of second
coordinates.
FUNCTIONS
 A function
is a relation in which no
two ordered pairs that have the
same first coordinate have different
second coordinates.
GRAPHING EQUATIONS OF
THE FORM Y=MX+B
 In
this section we learn to graph
equations three different ways.
 Define a set of three values for x
and solve the equation, graph the
three sets of ordered pairs and
connect them with a straight line.
9.2 ADDITION AND
MULTIPLICATION
PROPERTIES
 To
solve an inequality, apply the
same properties that were used to
solve equations of one variable.