z
og
u
«
~
E
o.
<1>
'"
ill
"0
,..;
15
g
...
~
~
C
]
'"
o.
~
.~
.0
0
0>
~
...
•
No~ut
8~O:::l
g
N
~.!!!
g)
:c
@(J)~Oaa
u
~;~~
~
"0
c
'C"'E,~
0
<1>
~.>::
(;
SttS
c
84
Sto
c(
~
~x

9
+
4x
=
36  4
(x

~)
.
Combine
like
terms
on
the
left side:
~x

9 = 36  4
(x

~)
NUMBER
SYSTEMS
The
natural numbers are
the
numbers
we
count with:
1,
2.
3, 4, 5, 6,
, 27, 28,
..
The
whole numbers are the numbers
we
count with and zero:
0,
1,
2,
3,
4,
5,
6,
.
The
integers are the numbers
we
count with, their negatives.
and
zero:
...
,
3,
2,
1,
0,
I,
2,
3,
.
The
positive integers are the natural numbers.
The
negotfve integers are the "minus" natural numbers:
1, 2. 3, 4,
.
The
rcmonal numbers are
all
numbers expressible as
~
fractions. The fractions may be proper (less than one;
Ex:
k)
or
improper (more than one;
Ex:
ft).
Rational numbers can be
positive
(Ex:
5.125 =
¥)
or
negative
(Ex:

~).
All
integers
are
rational:
Ex:
4
=
y.
I
SETS
A
set
Is
ony collecllonfinite or infinite<lf things coiled
members
or
elements.
To
denote a
set,
we enclose
the
elements
in
braces.
Ex:
N
{L,2.3
....
}
IS
the
!infinitel
set of natural numbers. The
notation
(I
EN
means
that
a
Is
in
N.
or
a
"is
an
element
of"
N.
DEFINITIONS
Empty
set
or
null set:
0
or
(}:
The
set
without
any
elements.
Beware: the set
{O}
is a set with
one
element,
O.
It
is not the
same as the em
pty
set.
Union
of two sets:
AU
B
is
the
set
of
all
elements
that
are
in
either
set
(or
in both).
Ex:
If A =
(1,2,3)
and
B
=
(2,4,6),
thenAUB=
(l,2,3,4,6).
PROPERTIES
OF
ARITHMETIC
OPERATIONS
PROPERTIES
OF
REAL
NUMBERS
UNDER
ADDITION
AND
MULTIPLICATION
Real
numbers
satisfy
11
properties:
5 for
addition,
5
matching
ones for multiplication, and 1 that connects addition and
multiplication. Suppose a
l
b,
and c are real numbers.
Property
Addition
(+)
Multiplicotion
(x
or .)
Commutotive
a
+
b
=
b
+
a
a·b=b·a.
Assoclotive
(a
+
b)
+
c = a
+
(b
+
c)
a·
(b·
c)
=
(a·
b)
.
c
ldenlities
o
is a
real
number.
1
is
a real
number.
exist
a+O=O+a=a
al=la=a
o
is
the
additive
1 is
the
multiplkotive
identity.
identity.
Inverses
a
is a real
number.
If
a
i
0,
~
is a real
exist
a+
(a)
number.
=(a)+a=O
u·~=~·a=l
Also,
(a)
=
a.
Also,
t
=
a.
Closure
a
+
b
is
a real
number.
a
b
is
a real number.
LINEAR EQUATIONS
IN
ONE VARIABLE
A
linear equation
in
one
variable
is
an equation that, after simplifying
and
collecting
like
terms on each side.
will
look
iike
a7
+
b
=
c
or
like
IU"
b
("J"
+
d.
Each
Side can
Involve
xs
added
to
real numbers
and
muihplied
by
real numbers but not multiplied
by
other
rS
Ex:
1(~3)+.c=9(7~)
is
a Iineor equation
But
j"
9 = 3
and
7(X
+
4)
= 2
and
Vi
= 5 ore not linear
Linear
eCJlUlh'ons
in
one
vW'iable
will
always
have
(a)
exactly
one real number solution, (b)
rlO
solutions, or (c) all real
numbers
as
solutions.
FINDING
A
UNIQUE
SOLUTION
Ex:
J(t
3)
TT=9(x
~).
1.
Get
rid
of
fractions
outside
parentheses.
Multiply
through
by
the
LCM
of
the
denominators.
Ex:
Multiplyby4
toget3(t
3)
+4x=364(x
~).
2.
Simplify using
order
of
operations
(PEMDAS).
Use the distributive property and combine like terms on each
side. Remember to distribute minus signs.