Download 1.1-1.3

1.1 Introduction to Algebra
Arithmetic
vs.
Algebra
____________________________________ _________________________________
A variable is a ___________________________________________________________
An algebraic expression is a _______________________________________________
_______________________________________________________________________
ex.
Let x represent my yearly salary in dollars. If my boss makes three times
as much as me, we can say that her salary is ________.
If I make 50,000 dollars per year, then x = ___________,
and my boss then makes ___________________________ dollars.
We evaluate an expression by substituting a number for the variable as in the example
above.
ex.
Evaluate the algebraic expression
3(5 + 2 x − y ) − 1
for x = 2
x
and y = 6 .
Order of Operation
1. Perform all operations within parentheses or other grouping symbols.
2. Multiplication & Division in the order which they occur from left to right.
3. Addition & Subtraction in the order which they occur from left to right.
An equation is a __________________________________________________________
________________________________________________________________________
Solution(s) of an equation are _______________________________________________
________________________________________________________________________
ex.
Is 10 a solution to the equation 2 x − 8 = 7 + x ?
(Substitute 10 in for x everywhere and determine if both sides are equal.)
ex.
Is 15 a solution to the equation 2 x − 8 = 7 + x ?
Translating from English to Math (see p.4, table 1.1)
Five more than a number : __________________________________________________
Three less than a number : __________________________________________________
Nine decreased by twice a number : ___________________________________________
The quotient of a number and 5 is 4 : __________________________________________
Mathematical Models are formulas that describe relationships between real-world
variables.
1.2 Fractions
An improper fraction has a numerator that is___________ than its denominator.
ex.
A mixed number consists of the addition of an ________ and ________, but the addition
sign is invisible.
ex.
To rewrite an improper fraction into a mixed number, we _____________ the
denominator into the numerator and write in the form
quotient
ex.
14
=
5
remainder
denominator
ex.
228
=
7
To rewrite a mixed number as an improper fraction we multiply the ____________ and
the __________ and add the ___________, and place result over ___________.
ex.
1
2 =
4
A prime number is a natural number greater than 1, which only has ________ and ____
as factors.
ex.
A composite number is a natural number greater than 1, which is not a _____________.
Every composite number can be expressed as the product of ____________________.
ex.
6=
84 =
90 =
108 =
Fundamental Principle of Fractions:
a⋅c a
=
b⋅c b
Reducing a fraction to its lowest terms:
1.
Rewrite numerator and denominator as products of prime numbers.
2.
Divide out common factors.
ex.
8
=
10
32
=
48
Multiplying Fractions:
ex.
a c a⋅c
⋅ =
b d b⋅d
7 1
⋅ =
9 2
Dividing Fractions:
7
⋅2 =
9
a c a d
÷ = ⋅
b d b c
change to and
multiplication
ex.
2 1
6 ⋅2 =
3 4
7 3
÷ =
8 4
invert to get
reciprocal
10 9
÷ =
3 2
When Adding and Subtracting fractions they need to have the same denominators.
a b a+b
+ =
d d
d
and
a b a −b
− =
d d
d
If the denominators are different, first rewrite the fractions as equivalent fractions with the
least common denominator. (Find the LCD using prime factorization.)
ex.
2 10
+ =
9 9
1
1
−
=
24 45
3 1
+ =
4 6
1.3 The Real Numbers
A _______ is a collection of objects. The objects in a set are called the _____________.
ex.
{2, 4, 6, 8, ...} =
The Number Line
0
The sets that make up the real numbers
Natural numbers =
Whole numbers =
Integers =
Rational Numbers =
Irrational Numbers =
ex.
Natural number:
3
5, 0, - , 2π, -5, 0.45, 0.16 . Determine which is a
4
Rational number:
Whole number:
Irrational number:
Integer:
Real number:
Given the numbers 18, -3.5,
The Absolute Value of a number describes the ______________ from ____ on the
number line.
ex.
8=
0=
−5 =
The symbols < and > are called __________________.
a < b “a is ________ than b”
a ≤ b “a is _______________________ b”
a > b “a is ________ than b”
a ≥ b “a is _______________________ b”
ex.
8 2
−8 2
−8
−2
−8
2