Download Math 105 Fraction Notes Name______________ Page one

Math 105
Giovanini
Fraction Notes
Name______________ Page one
REDUCING FRACTIONS
1.
Cancel the biggest common factor to both numerator and denominator
2.
Fractions must always be reduced.
ADDING OR SUBTRACTING OF LIKE FRACTIONS
1.
Add or subtract numerators, keep the denominator the same.
2.
Check for reducing.
ADDING OR SUBTRACTING OF UNLIKE FRACTIONS
1.
Find the LCD, write all denominators using LCD as denominator.
2.
Follow above rules for like fractions.
3.
Check for reducing.
MULTIPLYING FRACTIONS
1.
Check for any canceling or related factors on both top and bottom.
2.
Cancel, then take
top x top
bottom x bottom
DIVIDING FRACTIONS
1.
Invert (flipflop) the divisor, (the fraction or number after the divide sign.)
(Reciprocal),
2.
If the fractions are mixed numbers, you must rewrite the fractions as improper
fractions before you invert and multiply.
3.
Cancel related factors if you can.
4.
Multiply the resulting fraction, same steps as above.
5.
Division may be written horizontally or vertically.
SIMPLIFLYING COMPLEX FRACTIONS
Complex fractions are fractions in which numerator or denominator or both contain
fractions.
Method One
1.
Perform add or subtract in denominator and/or numerator. May first have to find
LCD.
2.
Do indicated division from this result.
3.
Check for reducing.
Method Two
1.
Multiply numerator and denominator of the complex fraction by the LCD OF ALL
THE FRACTIONS APPEARING IN THE COMPLEX FRACTION.
2.
Then simplify these results.
3.
Check for reducing.
When a fraction is negative the minus sign can go with the numerator, denominator, or out in front of the
fraction.
Page 2
x
x
x

 
y
y
y
Addition of Signed Numbers
1.
If the signs are the same ADD, Keep the sign.
2.
If the signs are different SUBTRACT, answer takes the sign of the larger number.
Subtraction of Signed Numbers
1.
2.
Change the subtract sign to an add, then change the following sign to its opposite.
Then follow the rules for addition of signed numbers.
Multiplication and Division of Two Signed Numbers
1.
2.
If the signs are the same, answer is POSITIVE.
If the signs are different, answer is NEGATIVE.
When multiplying an EVEN amount of negative numbers, the answer is POSITIVE.
When multiplying an ODD amount of negative numbers, the answer is NEGATIVE.
The Sets of Real Numbers
The real numbers are any number on the number line. They are made up of the rational numbers and the
irrational numbers.
The real numbers represent all positive and negative numbers, all fractions and all terminating, repeating and
non-repeating decimals.
The rational numbers are any number that can be written as a fraction or terminating or repeating decimal.
A terminating decimal is a decimal that has a set number of terms.
A repeating decimal is a decimal that has one or more digits that repeat in a pattern forever.
We would put a bar over the group of numbers that are repeated.
In order to find the equivalent terminating decimal divide the denominator into the numerator, add the decimal,
and some zeros.
The irrational numbers are any number that can’t be written as a fraction or terminating or repeating decimal.
These are the non-repeating decimals. They never form a pattern and never terminate.
The following are examples of irrational numbers.
2 , 3 , 5 , 7 and π
The Sets of Real Numbers
Page 3
The Real Numbers are any number on the number line.
The Rational numbers are any number that can be written as a fraction or terminating or repeating
decimal.
The irrational numbers are any number that can’t be written as a fraction or terminating or repeating
decimal.
Integers
... ,  4 ,3 ,  2,  1 , 0 , 1 , 2 , 3 , 4, ...
Whole Numbers
 0, 1 , 2 , 3 , 4,...
Natural Numbers
1, 2 ,
3 , 4, ...
Real numbers
/
\
Rational numbers
Irrational numbers
/
Integers
/
Whole numbers
/
Natural numbers
Sometimes a number needs to be written in a more simplified form to decide which set of numbers it
belongs to.
12
16  4
6
2
Given the set of numbers
5


  14 , 4π , 25 , 0 , ,  10 , 11 , 7 , .65 , .975 
2


1.
2.
3.
4.
5.
6.
Which of the above numbers are real numbers?
Which of the above numbers are rational numbers?
Which of the above numbers are irrational numbers?
Which of the above numbers are integers?
Which of the above numbers are whole numbers?
Which of the above numbers are natural numbers?
Properties of Real Numbers
a.
ASSOCIATIVE PROPERTY
Associative property of addition
2  5  8  2  5  8
b.
Page 4
a  b  c  a  b  c
25  75  90  25  75  90
Associative property of multiplication
5  20  11  5  20  11
a  b  c  a  b  c
4  25  65  4  25  65
COMMUNITIVE PROPERTY
Commutative property of addition
59  95
a  bb  a
15  85  85  15
Commutative property of multiplication
6  7  7 6
a  bb  a
2  14  14  2
ab  c  ab  ac
65x  2y  30x  12y
c.
DISTRIBUTIVE PROPERTY
27x  3  14x  6
 3a9b  4   27ab  12a
d.
ADDITIVE IDENTITY
and
5 05
055
a  0  a and
 9  0   9 and
0  a a
0  9  9
e.
ADDITIVE INVERSE
2   2  0 and
2  2  0
a   a  0 and
10   10  0 and
a  a  0
 10  10  0
f.
MULTIPLICATIVE IDENTIY
and
6  1 6
1 6  6
a  1 a
2  1 2
1 a a
1 2  2
g.
MULTIPLICATIVE INVERSE
7
h.
1
1
7
and
1
71
7
1
1
a
1
5 1
5
a 
and
and
and
and
MULTIPLICATION PROPERTY OF ZERO a  0  0 and
and
and
11  0  0
0  11  0
300
1
 a 1
a
1
51
5
0  a 0
030
Notes on Empty set, Order of Operations, and Absolute Value
Page 5
Empty set or null set is a set that has no elements. The symbol for empty set is  .
This is another symbol for empty set  , but use  .
Variables are letters that can take on different values depending on the problem.
Multiplication can be shown with a dot  , or parentheses around one or both of the numbers. Write
the dot between the two numbers so it doesn’t look like the decimal. Please don’t use the times sign,
as it looks like the letter x.
Order of Operations
1.
2.
3.
4.
5.
6.
Copy the problem.
Substitution, if there are any variables in the problem substitute the given values into the
problem and simplify.
Parentheses, Brackets, the Division Bar, the Absolute Value Bars, all act as grouping symbols.
Work inside the innermost one first and simplify.
Simplify all roots and radicals in order from left to right.
Multiply and divide in order from left to right
Add or Subtract in order from left to right.
PBMDAS PLEASE BURY MY DEAD AUNT SALLY
PARENTHESES, POWERS& ROOTS, BRACKETS, MULTIPLICATION,
DIVISION, ADDITION, AND SUBTRACTION
PEMDAS
PLEASE EXCUSE MY DEAD AUNT SALLY
PARENTHESES, BRACKETS, EXPONENTS & ROOTS, MULTIPLICATION,
DIVISION, ADDITION, AND SUBTRACTION
ABSOLUTE VALUE
Absolute value is the distance between zero and the given number on the number line.
Since absolute value is a distance it is never negative. However there may be a minus sign out in front
of the absolute value bars. If there is, it is understood to be a negative one in front of the absolute
value bars. Then the final answer would be negative.
Absolute value has two vertical bars as the symbol.
The absolute value of zero is zero, since we haven’t moved either left or right on the number line.
These bars act as one of the grouping symbols. Simplify inside the bars first, before taking the absolute
value of what is inside the bars.
Math105
Giovanini
Inequality Notes
Name________________
Page 6
An inequality is a statement that shows two quantities are not equal.
The symbols for inequality are  ,  ,  ,  .
This is the less than sign. 
This is the less than or equal to sign. 
This is the greater than sign. 
This is the greater than or equal to sign. 
Set Builder notation is a method of writing a set, using a special notation of set braces and a straight line.
 x has some property For example x x  3 
Some books use parentheses and brackets while some books use open or closed dots to determine if the
endpoints are included in the graph of an inequality. Some books also use interval notation as a shortcut way of
writing the solution set.
An interval is a section or piece of a number line.
Interval notation is a special notation that uses symbols to describe the interval or section of the number line.
When writing a solution set in interval notation, parentheses are always used next to the infinity symbol.
  ,  
Parentheses or open dots indicate the endpoints are not included in the graph. Also
unshaded portions of a number line indicate points not included in graph.
Brackets or closed dots indicate the endpoints are included in the graph. Also shaded portions of a number
line are used to indicate points included in the graph.
Describe in words how to graph each of following inequalities, make sure to tell which endpoints to use.
Also write the solution set in interval notation.
x
x  9 
x
x  5
Line with an arrow going to the right, parentheses at –9.
Sol. Set   9 ,  
Line with an arrow going to the left, bracket at 5.
Sol. Set    , 5
x
x  7 

Line with an arrow going to the right, bracket at –7.
Sol. Set
 7 ,  
x
x  3
Line with an arrow going to the left, parentheses at 3.
Sol. Set    , 3 
Page 7
Describe in words how to graph each of following inequalities, make sure to tell which endpoints to use.
Also write the solution set in interval notation.
x
5  x  4
x
2  x  7
Parentheses at –5 and bracket at 4, shade between the two numbers
Sol. Set   5, 4 
Bracket at 2 and parentheses at 7, shade between the two numbers
Sol. Set
x
1  x  11
x
2  x  9
 2 ,7 
Parentheses at 1, parentheses at 11, shade between the two numbers
Sol. Set  1,11 
Bracket at –2 and bracket at 9, shade between the two numbers
Sol. Set   2 , 9 
Describe in words how to graph each of following inequalities, make sure to tell which endpoints to use.
Also write the solution set in interval notation.
x
x  7 
x
x  4
x
3  x  9
x
 6  x  11
x
8  x  2