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1.1 Fractions
•
•
Multiplying or dividing the numerator
(top) and the denominator (bottom) of a
fraction by the same number does not
change the value of a fraction.
Writing a fraction in lowest terms:
1. Factor the top and bottom completely
2. Divide the top and bottom by the greatest
common factor
1.1 Fractions
• Multiplying fractions:
a c
ac
 
b d
bd
• Dividing fractions:
a c
a d
ad
   
b d
b c
bc
1.1 Fractions
• Adding fractions with the same denominator:
a c
ac
 
b b
b
• Subtracting fractions with the same denominator:
a c
ac
 
b b
b
1.1 Fractions
•
To add or subtract fractions with different
denominators - get a common denominator.
• Using the least common denominator:
1. Factor both denominators completely
2. Multiply the largest number of repeats of each
prime factor together to get the LCD
3. Multiply the top and bottom of each fraction
by the number that produces the LCD in the
denominator
1.1 Fractions
• Adding fractions with different denominators:
a c
ad  bc
 
b d
bd
• Subtracting fractions with different denominators:
a c
ad bc
 
b d
bd
1.1 Fractions
• Try these:
12
(simplify)
16
7 3

?
9 14
9
3
 ?
10 5
1 5
 ?
9 9
5
2

?
7 21
5 1
 ?
9 4
1.2 Exponents and Order of
Operations
• Exponents:
4
3  3  3  3  3  81
• Note:
4
3  34
1.2 Exponents and Order of
Operations
•
PEMDAS (Please Excuse My Dear Aunt Sally)
1. Parenthesis
2. Exponentiation
3. Multiplication / Division
(evaluate left to right)
4. Addition / Subtraction
(evaluate left to right)
• Note: the fraction bar implies parenthesis
1.3 Geometry Review
• Acute angle –
0 < x < 90
• Right angle - 90
• Obtuse angle –
90 < x < 180
• Straight angle - 180
1.3 Geometry Review
• Complementary
angles – add up to 90
• Supplementary angles
– add up to 180
• Vertical angles – the
angles opposite each
other are congruent
1.3 Geometry Review
1 2
3 4
5 6
7 8
• When 2 parallel lines are cut by a transversal the
following congruent pairs of angles are formed:
– Corresponding angles:1 & 5, 2 & 6, 3
& 7, 4 & 8
– Alternate interior angles: 4 & 5, 3 & 6
– Alternate exterior angles: 1 & 8, 2 & 7
1.3 Geometry Review
1 2
3 4
5 6
7 8
• When 2 parallel lines are cut by a transversal the
following supplementary pairs of angles are
formed:
– Same side interior angles: 3 & 5, 4 & 6
– Same side exterior angles: 1 & 7, 2 & 8
1.3 Geometry Review
• Terminology:
– Corresponding angles – in the same relative
“quadrant” (upper right, lower left, etc.)
– Alternate – on opposite sides of the transversal
– Same side – on the same side of the transversal
– Interior – in between the 2 parallel lines
– Exterior – outside the 2 parallel lines
1.3 Geometry Review
1 2
3 4
5 6
7 8
• What type of angles are:
– 1 & 8
– 4 & 6
– 4 & 5
– 2 & 6
– 1 & 7
1.3 Geometry Review
• Triangles classified by number of congruent sides
Types of triangles
# sides congruent
scalene
0
isosceles
2
equilateral
3
1.3 Geometry Review
• Triangles classified by angles
Types of triangles
Angles
acute
All angles acute
obtuse
One obtuse angle
right
One right angle
equiangular
All angles congruent
1.3 Geometry Review
• In a triangle, the sum of the interior angle
measures is 180º
(mA + mB + mC = 180º)
A
C
B
1.3 Geometry Review
Figure
Area
Square
s2
Rectangle
lw
Parallelogram
bh
Triangle
½ (b  h)
Trapezoid
h(b1  b2 )
2
A   r C  2 r
Circle
1
2
1.4 Sets of Numbers and Absolute Value
•
Classifications of Numbers
Natural numbers
Whole numbers
Integers
Rational numbers – can be
p
expressed as q where p
and q are integers
Irrational numbers – not
rational
{1,2,3,…}
{0,1,2,3,…}
{…-2,-1,0,1,2,…}
-1.3, 2, 5.3147,
7
13
,
5 ,
23
5
47 , 
1.4 Sets of Numbers and Absolute Value
• The real number line:
-3 -2 -1
0
1
2
3
• Real numbers:
{xx is a rational or an irrational number}
1.4 Sets of Numbers and Absolute Value
•
Ordering of Real Numbers:
a < b  a is to the left of b on the number line
a > b  a is to the right of b on the number line
•
Additive inverse of a number x:
-x is a number that is the same distance from 0
but on the opposite side of 0 on the number line
1.4 Sets of Numbers and Absolute Value
•
•
Double negative rule:
-(-x) = x
Absolute Value of a number x: the distance
from 0 on the number line or alternatively
x 
x if x  0
 x if x  0
How is this possible if the absolute value of a
number is never negative?
1.5 Adding and Subtracting Real
Numbers
• Adding numbers on the number line (2 + 2):
-4 -3 -2 -1
0
1
2
3
2
2
4
1.5 Adding and Subtracting Real
Numbers
• Adding numbers on the number line (-2 + -2):
-4 -3 -2 -1
-2
-2
0
1
2
3
4
1.5 Adding and Subtracting Real
Numbers
• Adding numbers with the same sign:
Add the absolute values and use the sign of
both numbers
• Adding numbers with different signs:
Subtract the absolute values and use the
sign of the number with the larger absolute
value
1.5 Adding and Subtracting Real
Numbers
• Subtraction:
x  y  x  ( y )
• To subtract signed numbers:
Change the subtraction to adding the
number with the opposite sign
5  (7)  5  (7)  12
1.6 Multiplying and Dividing Real
Numbers
• Multiplication by zero:
x0  0
For any number x,
• Multiplying numbers with different signs:
For any positive numbers x and y,
x( y )  ( x) y  ( xy)
• Multiplying two negative numbers:
For any positive numbers x and y,
( x)(  y )  xy
1.6 Multiplying and Dividing Real
Numbers
• Reciprocal or multiplicative inverse:
If xy = 1, then x and y are reciprocals of
each other. (example: 2 and ½ )
• Division is the same as multiplying by the
reciprocal:
x
y
 x
1
y
1.6 Multiplying and Dividing Real
Numbers
• Division by zero:
x
For any number x,
0  undefined
• Dividing numbers with different signs:
For any positive numbers x and y,
x
y

x
y
 ( )
x
y
• Dividing two negative numbers:
For any positive numbers x and y,
x
y

x
y
1.7 Algebraic Expressions and
Properties of Real Numbers
• Algebraic expression 5 x 3  9 x 2  1 x  7
4
• x is a variable
5 is a coefficient
7 is a constant
• Evaluating an expression: substitute a value for
the variable and evaluate
example:the last expression when x=1
5(1) 3  9(1) 2  14 (1)  7  5  9  14  7  3 14
1.7 Algebraic Expressions and
Properties of Real Numbers
• Commutative property
(addition/multiplication)
• Associative property
(addition/multiplication)
ab  ba
ab  ba
(a  b)  c  a  (b  c)
(ab)c  a (bc)
1.7 Algebraic Expressions and
Properties of Real Numbers
• Identity property (addition/multiplication)
a0  a
a 1  a
• Inverse property (addition/multiplication)
1
a  (a)  0
a 1
a
• Distributive property
a (b  c)  ab  ac
(b  c) a  ba  ca