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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, Order of Operations,
and Inequality
• Exponents:
4
3  3  3  3  3  81
• Note:
4
3  34
1.2 Exponents, Order of Operations,
and Inequality
•
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
1.2 Exponents, Order of Operations,
and Inequality
•
Symbols of equality / inequality
1. = is equal to
2.  is not equal to
3. < is less than
4.  is less than or equal to
5. > is greater than
6.  is greater than or equal to
1.3 Variables, Expressions, and Equations
• Variable – usually a letter such as x, y, or z, used
to represent an unknown number
• Evaluating expressions – replace the variable(s)
with the given value(s) and evaluate using
PEMDAS (order of operations)
1.3 Variables, Expressions, and Equations
•
Changing word phrases to expressions:
The sum of a number and 9
7 minus a number
Subtract 7 from a number
The product of 11 and a number
5 divided by a number
x+9
7-x
x–7
11x
5
x
The product of 2 and the sum of a 2(x + 8)
number and 8
1.3 Variables, Expressions, and Equations
•
Equation: statement that two algebraic
expressions are equal.
Expression
x–7
No equal sign
Can be evaluated or
simplified
Equation
x–7=3
Contains equal sign
Can be solved
1.4 Real Numbers and the Number Line
•
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 Real Numbers and the Number Line
• The real number line:
-3 -2 -1
0
1
2
3
• Real numbers:
{xx is a rational or an irrational number}
1.4 Real Numbers and the Number Line
•
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 Real Numbers and the Number Line
•
•
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 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 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
1.8 Simplifying Expressions Terms
• Term: product or quotient of numbers, variables,
and variables raised to powers
5
2
3
3z
x, 15 y ,  2 , xz
x y
2
• Coefficient: number before the variables
If none is present, the coefficient is 1
• Factors vs. terms:
In “5x +y”, 5x is a term.
In “5xy”, 5x is a factor.
1.8 Simplifying Expressions Terms
• When you read a sentence, it split up into words.
There is a space between each word.
• Likewise, an is split up into terms by the +/-/=
sign: 2
2
1
1
3
x 
2
x

6
y  3
• The only trick is that if the +/-/= sign is in
parenthesis, it doesn’t count:
2
3
x  3

1
2
x 
1
6
x  3
1.8 Simplifying Expressions
• Like Terms: terms with exactly the same variables
that have the same exponents
• Examples of like terms: 5 x and  12 x
3 x 2 y and 5x 2 y
• Examples of unlike terms
2 xy2 and  7 xy
x 2 y and 2x 2 z
1.8 Simplifying Expressions
• Combining Like Terms: the distributive
property allows you to combine like terms
• Examples of combining like terms:
5 x  (  12 x)  (5  12) x  7 x
3x y  5 x y  (3  5) x y  8 x y
2
2
2
2