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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
Aset 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 empty 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
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.