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NUMBER SYSTEMS
The natural numbers are the numbers we count with:
1, 2
...
The whole numbers are the numbers we count with and zero:
0, 1, 2, 3, 4, 5, 6,
...
and zero:
...
-The positive integers are the natural numbers
...
-3, -4,
...
The fractions may be proper (less than one; Ex: k) or
improper (more than one; Ex: ft)
...
125 =
rational: Ex: 4 =
y
...
All integers are
The real numbers can be represented as points on the number
line
...
Ex:
0,
To,
v'3 -
Real numbers
...
25
2
3 etc
...
9, 0
...
Irrationals
The imaginary numbers are square roots of negative numbers
...
Ex: J=49 is imaginary and equal to iV49 or 7i
...
where a and bare
real and i = FI is imaginary
...
The Fundamental Theorem of Algebra says that every
Integers
~~
- - Whole Dwnbers
Natural numbers
)'t2+3
etc
...
I
SETS
A set Is ony collecllon-finite or infinite--
...
Ex: N
{L,2
...
} IS the !infinitel set of natural numbers
...
or a "is an element of" N
...
Beware: the set {O} is a set with one element, O
...
-Union of two sets: AU B is the set of all elements that are in
either set (or in both)
...
-Intersection of two sets: A n B is the set of all the elements
that are both in A and in B
...
Two sets with no elements
in common are disjoint; their intersection is the empty set
...
Ex: If we're talking about the set {l,2,3,4,5,6}, and
A = {1
...
6}
...
-Subset: A C C: A is a subset of C if all the elements of A are
also elements of C
...
A!""\ll
VENN DIAGRAMS
A Venn Diagram is a visual way to
represent the relationship between two
or more sets
...
Elements in an
overlapping section of nYQ sets belong
to
both
sets
(and
are
in
the
intersection)
...
@
A
1
4
B
2
3
6
AuB
Venn Diagram
A={1
...
6)
...
,G}
...
PROPERTIES OF ARITHMETIC OPERATIONS
Distributive property
(of additon over
multiplication)
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
...
Property
Multiplicotion (x or
...
(a
ldenlities
exist
o is a real number
...
Inverses
exist
Closure
~
ill
15
~
0
C
~
...
,
...
•
~
...
] ~ '"
...
!!!
g)
@(J)~Oaa
~;~~ ~
'C"'E,~ 0
g
:c
u
c
"0
<1>
~
...
a+ (-a)
=(-a)+a=O
Also, -(-a) = a
...
1 is a real number
...
If a i- 0, ~ is a real
number
...
a = b· a + c· a
Inequality « and»
Property
Equality (=)
Reflexive
a=a
Symmetric
If a = b, then b = a
...
then a
= c
...
a b is a real number
...
Example
<
less than
1 < 2 and 4 < 56
>
greater than
1 > 0 and 56 > 4
i
not equal to
0 i 3 and -1 i 1
~
less than or equal to
1 ~ 1 and 1 ~ 2
2:
greater than or equal to
1 2: 1 and 3 2: - 29
The shorp end aiways points toward the smoller number; the open
end toword the larger
...
Multiplicotion
and division
b, then
ac = be and
~ = ~ (if c i 0)
...
If a < band c > 0,
then ac < be
and %< ~
...
Mullipllcotion by zero: a
...
a = O
...
Sign
+ b) + c = a + (b + c) a· (b· c) = (a· b)
...
(b
inequality: ac > be
and ~ > ~
...
a = b, or a > b
...
will look iike a7 + b = c or like
IU"
b ("J" + d
...
c=9-(7-~) is a Iineor equation
j"
9 = 3 and 7(X + 4) = 2 and Vi = 5 ore not linear
But
in one vW'iable will always have (a) exactly
one real number solution, (b) rlO solutions, or (c) all real
numbers as solutions
...
Add 8x to both sides to get 5x + 8x - 18 = 77 or
13x-18=77
...
5
...
Stop if a = U
...
~~
1
...
-
3) + ~ = 9 - (~- ~)? Yes! Hooray
...
ll,
...
] ~\ 14f'
...
' :JII
Use the same procedure as for equalities, except flip the
inequality when mliltiplying or dividing by a negative
number
...
-The inequality may have no solution if it reduces to an
impossible statement
...
-The inequality may have all real numbers as solutions if it
reduces to a statement that is always true
...
Solutions given the reduced Inequality and the condition:
Multiply through by the LCM of the denominators
...
2
...
DETERMINING IF A UNIQUE
SOLUTION EXISTS
Use the distributive property and combine like terms on each
side
...
-
Multiply by 2 to get rid of fractions: 5x - 18 = 77 - 8x
...
Move variable terms and constant terms to different sides
...
6
...
~)
...
Combine like terms on the left side: ~x
Any linear equation can be simplified into the form ax = b for
some a and b
...
then x = ~ (exactly one solution)
...
If a = b = 0, then all
coefficient to begin with
...
Linear eCJlUlh'ons
Ex:
Distribute the right-side parentheses; ~x - 9 = 36 - 4x + ~
...
3
...
9 = 36 - 4 (x
-
~)
a>O
transformations, the new equation is false
...
-All real numbers are solutions to the original equation if,
after legal transformations, the new equation is an identity
...
r
...
a
a=O
b>O
none
a=O
b
a=O
b=O
none
x:<:;~
-The original equation has no solution if, after legal
none
all
all
none
all
ax
x<*
x>~
all
none
a:r:5b
x:$~
x~*
all
none
_
...
denoted Inl,
is its distance from O
...
Thus 131 = 3 and I-51 = 5
...
Formally,
l
...
If x < 0
x
-x
-The distance between a and b is the positive
value la - bl = Ib - aI- Ex: 15 - I = 3
...
PROPERTIES OF ABSOLUTE
VALUE
lal = Ibl means a = b or a = -b
...
If b ~ 0, then
1"1 = b means" = b or a = -b
...
1"1 > b means" < -b or " > b
...
1"1 > b means a could be anything
...
--Fee free 10 foclor au posiIiYe conslonts
...
Thus I-x - 11 = 4 is equivalent to
Ix + II = +4
...
)
-Use the Properities of Absolute Value to
unravel the absolute value expression
...
-Soive each one seporately
...
-Check specific solutions by plugging them in
If there are infinitely many solutions or no
solutions, check two numbers of large
magnitude
...
-Be especially careful if the equation contains
variables both inside and outside the
absolute value bars
...
---Ex
...
If2x ~ O
...
The first gives
x = 5: the second gives x = 1
...
-Ex
...
If ax ~ 0, then we can
rewrite
this
as
3x + 5 = 3x
or
3x + 5 = -3x
...
The second seems to give the
solution x = - ~
...
So -~ does not work
...
_
...
The first condition
forces the second
...
IlIf'~i6j:II~lal~\[tI&lij:[·I
...
Find all solutions to the
associated equalities
...
find all solutions
lhat the equation obtains by replacing the
absolute value with O
...
Determine the
solution intervals by testing a point in every
•
•
Unraveling tricks:
lal < b is true when b ~ 0 and -b < a < b
...
Ex: IIOI + II < 7x + 3 is eqUivalent to
7
...
Thus
the
equations
7x + 3 ~ 0,
-7x - 3 < lOx + I
...
SolVing the equations
...
The point
x = 0 is often good to test
...
Ex: IlOx + 11 < 7x + 3
...
-lOx-l=7x+3
...
The three points are
...
and - ~
...
At,
GRAPHING ON THE REAL
...
Solutions to one·
vorioble equations and (especiallyi inequalities
may be graphed on the real number line
...
Origin: A special point representing O
...
o
Open (ray or interval): Endpoints not
oa
I
Closed (ray or interval): Endpoints included
...
o
I
a
•
;
o
I
a
•
I
both inequalities
...
I
b
Ix - al < b; I
...
Plot the interval (open or closed)
a - b < x < ,,+ b (or" - b::; x ::; a + b
...
around which there is an open circle
...
I
a+b
:
I
b
•
I
a
Ix -
of the inequalities must be true
...
Equivalently, it is{x, x < -3} U {x : x > 5}
...
Shade the portions
that would be shaded by either one (or both)
if graphed independently
...
Ex: x > 5 OR
6 just means that x > 5
...
x ~
b
b
a
...
and b
...
o
a-b 0 a
Q
a
open circle if the endpoint is not included
...
a+b
a-b 0 a
x
I
included
...
Both
(or all) of the inequalities must true
...
Equivalently, it is {x: x > -3} n {x: x < 5}
...
At least one
Interval: A piece of the line; everything
may not be included
...
Ix - al = b means
that lhe distance between a and x Is b
...
b must be non-neg olive
...
Plot
two points: x = a + b, and x = 0 - b
...
Open circle around a
...
The endpoint may
or may not be included
...
which mayor
OTHER COMPOUNO
INEQUALITIES
oa
I ¢
represent negative numbers, and points to the
Ray: A half-line; everything to the left or the
I
x > '" Shaded open ray: everything to the
right of (and not including) a
...
convention, points to the left of the origin
right of the origin represent positive
numbers
...
GRAPHING ABSOLUTE
VALUE STATEMENTS
< b lopen IntervolJ
al
-4
0
2
I I I , I I I
Ix - al > b; Ix - al ~ b:
Ix -
The distance from a to x is more than (no less
than) b; or x is further than b away from a
...
a-b 0 a
I
I
I
b
b
Ix -
41 :0; 2 AND x
...
5
i
IF YOU CAN DETERMINE
ALL POTENTIAL
ENDPOINTS
...
Also test all endpoints to determine if
they're included
...
a+b
I
6
I [ I •
al ~ b
THE CARTESIAN
...
called axes
...
By convention, they are
numbered counterclock"wise starting with
the upper right (see the diagram at right)
...
Positive distances are
measured to the right; negative
...
y-axis: Usually, the vertical axis of the
coordinate plane
...
down
...
The first coordinate is
measured along the x-a
...
Ex: The point (I
...
r-~
Quadrant 1V
+
IV
+
I
y---+---+-
LINES IN THE CARTESIAN
PLANE
origin
...
the first
coordinate is called the abscissa; the
second, the ordinate
...
II
...
x-axis and the y-axis
...
If (a, b) and (c
...
-Horizontollines have slope O
...
-Lines that go "up left" and "down right"
(ending in II and IV) have negative slope
...
-The slopes of perpendicular lines are
negative reciprocals of each other: if two
lines of slope ffil and 7112 are perpendicular,
then mtm2 = -1 andm2 = -~
...
The relationship always can be
expressed as Ax + By = C for some rea]
numbers A, B, C
...
m<-1
mundefined
-1 <711 <0
A horizontal line at height b has equation y = b
...
where a line crosses the x-axis
...
at (a,O) is a
...
intercept
...
where a line crosses the y-axis
...
Stondard form: 711x - y = 711xo - Yo
...
b) is b
...
solve for b = Yo - 711xo to get the slope
intercept form
...
Equation:
Y-YI =m(x-xd = ~(x-xl)
...
Given slope 711 and x-intercept a:
Equation: x = !!; + a
...
Given a point on the line and Ihe equation of a
porallelline:
Find the slope of the parallel line (see Graphing
Linear Equations)
...
Use point-slope fonn
...
The slope of the original line is -;;!o
...
GRAPHING LINEAR EQUATIONS
A linear equation in two variables (soy x and yl
SLOPE-INTERCEPT FORM:
can be monipuloted-
terms and constonlterms are have been grouped
One of the easiest-to-graph forms of a linear
together-into the form Ax + By = C
...
of the equalion is a straight line thence the namel
...
-Using the slope to graph: Plot one point of
b is the y-intercept
...
If the slope is expressed as a ratio
line
...
keep plotting
points f' up and ±s over from the previous
POINT-SLOPE FORM:
point until you have enough to draw the
V - k = mix - h]
line
...
-finding intercepts: To find the v-intercept,
(h, k) is a point on the line
...
To find the x
intercept, set y = 0 and solve for x
...
less work: Find the x- and the y-intercepts
...
Slope: -~
...
x-intercept: ~
...
Plot those points
...
Done
...
Two
simultaneous linear equations in two variables will
have:
-Exactly one solution if their graphs
intersect-the most common scenario
...
-Infinitely many solutions if their graphs
coincide
...
J,=(O
~
(0
, ---co
T"""::=:;LO
Z~
...
...
:
...
~
...
!!l ~
c
:E
~~
...
The intersection of the graph gives the
simultaneous solutions
...
)
-Sometimes, the exact solution can be
determined from the graph; other times
the graph gives an estimate only
...
-If the lines intersect in exactly one point
(most cases), the intersection is the unique
solution to the system
...
S
j;
~ ~
8- j!!
~
~~ ~ul
8
o::j
~~~~~
'c"5~
~
...
...
Parallel lines have the same slope; if the
slope is not the same, the lines will
intersect
...
Plugging in 10 the second
equation gives 2
...
SolVing
for x gives x = 7
...
Check that (7,~) works
...
ax + by = c works well
...
Effectively, the two 1
equations to eliminate one of the variables
...
'
-If the coefficients on a variable in the two
y
3
equations are the same, subtract the
'2 2
y
equations
...
v = 3
equations
...
-If no simple combination is obvious,
simply pick a variable (say, x)
...
(say, y) in terms of the other (x): isolate y
If all went well, the sum or difference equation
on one side of the equation
...
fonn)
...
-Solve the resulting one-variable linear
-If by eliminating one variable, the other is
equation for x
...
If there are no
there are no solutions to the system
...
If all real numbers are solutions to
solutions; the two equations are
the sum (or difference) equation, then the
dependent
...
variables; there are infinitely many
-Check that the solution works by plugging it
solutions to the system
...
-Plug the solved-for variable into one of the
original equations to solve for the other
variable
...
{
x -4y
= 1
...
CRAMER'S RULE
The solution to the simultaneous equations
aX+by=e
{ ex + dy = f
if ad - be ,
...
I;I:a
is given by
o
...
E!
i'~I
...
Ii
l
There is a decent chance thol a system of Iineor
equalions has a unique solution only if there are
as mony equations as variables
...
(This is ollly
actually true if the equations are
"independent"-each
new equation
provides new information about the
relationship of the variables
...
-All of the above methods can, in theory, he
used to solve systems of more than two
linear equations
...
It's too hard to
visualize planes in space
...
-Adding or subtracting equations (or rather,
arrays of coefficients called matrices) is the
method that is used for large systems
...
In the notation a'l, a
...
The whole expression is "a to the nth
power," or the "nth power of a, or, simply, "a to
then
...
Exponentiation powers: (am)" = a~"
To raise a power to a power, multiply e
...
Ex: (-4)' = 16, whereas -(4') = -16
...
Ex:
(2xy)' = 4x'y', but (2+x +y)' '" 4 + x' +y'
...
a 2 is "u squared;" a3 is un cubed
...
Ex: 23
...
Zeroth power: aO = 1
To be consistent with all the other exponent
rules, we set aO = 1 unless a = O
...
n
=
...
This works well with all other
rules
...
2- 3 = ~ = 1
...
2-' = 23 +( -3) = 2° = 1
...
= -(a")
...
The expression yI(i reads
Radical Rule Summary
v
...
Sometimes, it is also referred to as the radical
...
It is usually dropped for square
va
I{?a=
vav'b=
SIMPLIFYING SQUARE ROOTS
Root of a power
A squore root expression is considered simplified if the radical has
no repeated factors
...
-Factor the radicand and move any factor that appears twice
outside of the square root sign
...
;xr; =
x
~;:;;;n
~~
*'
j
...
Use the the rule
= iji
...
If there are radicals in the denominator, combine them
into one radical expression Vd
...
Multiply the fraction by "a clever farm of 1:"
This
will leave a factor of d in the denominator and,
effectively, pull the radical up into the numerator
...
Simplify the radical in the numerator and reduce the
fraction if necessary
...
,I"j] = ~ =
= v'I5
$
...
t;t
-----
...
Factor the
radicand and use yI(;Tib = a v'b as for square roots
...
Use ~'!/O" = 'if
-Two unlike roots may be joined together
...
Ex:V'?~;:: 1~=Xl~
-When in doubt, use v'a"' = ( yJii)"' = a'li to convert to
fractional expqnents and work with them
...
-You may lose ± sign information
...
jl
...
fA' ~'fI j:':W ·'4~ I·'M" ~,,'
SIMPLIFYING HIGHER·POWEREO
RADICALS
and
~~
...
'i'"
~va
= \fa
Converting between notation
Ex: v'6ii = ~ = ~vT5 = 2115
...
Use
JX 2n + 1 = x"
...
(ab)" = a"b"
Root of a root
------'--
Product of roots
3
...
In such cases, we agree that the expression
always refers to the positive, or principal, root
...
;o
...
-When n is even and a
...
,/li
10
10
-If the simplified radical in the denominator is an nth
root if
...
-w
...
"Simplified" farm is nat necessarily simpler
...
subtracting, and multiplying real numbers and one
or several variables
...
- Expressions connected by + or - signs are
called terms
...
-The coeIficient of a term is the real number
(non-variable) part
...
Ex: 7y G and yxy5 are like terms
...
Like terms can be
added or subtracted into a single term
...
Ex: 2x' and
16:cy' z both have degree eight
...
-In a polynomial in one variable, the term with
the highest degree is called the Ieacing 1lllrm,
and its coefficient is the Ieacing coeIficient
...
-When subtracting a polynomial, it may be
easiest to flip all the ± signs and add it
16y 6
are both sixth-degree polynomials
...
1
...
Choose wisely
...
2
...
Use common sense: more, fewer, sum,
total, difference mean what you want them
to mean
...
Ex: "Half
of the flowers are blue" means that if there are
c flowers, then there are ~c blue flowers
...
Ex: "12% of the
flowers had withered" means that ~c
flowers were withered
...
Solve the equation(s) to find the desired
quantity
...
Check that the answer make sense
...
+ 5 and 4y 5 -
-Only like terms can be added or subtracted
By number of terms:
1 term: monomial
2 terms: binomial
3 terms: trinomial
instead
...
These ore often good variables candidates
...
Convert nnecessary:
Time:
1 min = 60s; 1 h = 60 min = 3,6005
Dis1ance: 1 ft = 12in; 1 yd = 3ft = 36i11
1mi = 1
...
280ft
Metric dis1ance:
1 m = 100em; 1 km = lOOOm
] in : : : : 2
...
28 ft; 1 mi ::::::: 1
...
-MNEMONIC: FOIl: Multiply the two First
terms, the two Outside terms, the two
Inside terms, and the two Last terms
...
I&lr"
!-;
The key is to muifiply every term by every term,
term by term
...
-Multiplying a monomial by any other
polynomial: Distribute and multiply each
5
...
...
-Multiplying two binomials: Multiply each term
of the first by each term of the second:
(a + b)(c + d) = (tc + be + ad + bd
...
:
(a + b)' = a' + 2ab + b'
(a - b)' = a' - 2ab + b'
(a + b)(a - b) = a' - b'
-After multiplying, simplify by combining
like terms
...
Ex: Supercar travels at 60 mi/b for 30 min and at
90 mi/h for the rest of its 45-mile trip
...
During this time,
Supercar travels 60mi/h x ~ Ii = 30mi
...
Supercar zips
through this part in ~~':;7!1 = ~ h
...
What is Supercor's overage speed for the trip?
T~~~8ti~~~::C~ = ~
=
67
...
This may seem low, but it's right: Supercar had
traveled at 60 mi/h and at 90 lUi/b, but only
one-fourth of the total journey time was at the
faster speed
...
How long will it take them
working together"
These problems are disguised rate problems
...
days, she works at a
rate of ~ house per day
...
Working for x days, they
have to complete one house:
j+~=1
...
2
days
...
5
days; a Sarah and a Justin need some length of
time in between