Review Chapter
 What you Should Learn
 REALLY – WHAT YOU SHOULD
HAVE ALREADY LEARNED
 If not, then you might be in too high of a
course level – decide soon!!!
Henry David Thoreau - author
“It affords me no satisfaction to
commence to spring an arch
before I have got a solid
foundation.”
Objective
Understand the
structure of algebra
including language and
symbols.
Objective
Understand the
structure of algebra
including language and
symbols.
Definiton
Expression – a
collection of constants,
variables, and
arithmetic symbols
Definition
Inequality – two expression
separated by <, <, >, >,
 -2>-3
4 < 5
4 < 4
Definition
Equation – two expression set
equal to each other
 4x + 2 = 3x - 5
Def: evaluate
When we evaluate a numerical
expression, we determine the
value of the expression by
performing the indicated
operations.
Definition
Set is a collection of objects
Use capitol letters to represent
Element is one of the items of
the collection
Normally use lower case letters
to describe
Procedure to describe sets
Listing: Write the members of a set
within braces
Use commas between
Use … to mean so on and so forth
Use a sentence
Use a picture
Julia Ward Howe - Poet
“The strokes of the
pen need deliberation
as much as the sword
needs swiftness.”
Examples of Sets
{1, 2, 3}
{1, 2, 3, …, 9, 10}
{1, 2, 3, … } = N = Natural
numbers
Set Builder Notation
{x|description}
Example
{x|x is a living United States
President}
Def: Empty Set or Null set is the
set that contains no elements
 Symbolism
{}
Symbolism – element
“is an element of”

1 N
Def: Subset: A is a subset of B if
and only if ever element of A is
an element of B
 Symbolism
A B
Examples of subset
{1, 2}

{1, 2, 3}
{1, 2}  {1, 2}
{ }  {1, 2, 3, … }
Def: Union
symbolism: A
B
A union B is the set of all
elements of A or all elements of
B.
Example of Union of sets
A = {1, 2, 3}
B = {3, 4, 5}
A
B = {1, 2, 3, 4, 5}
Real Numbers
 Classify Real Numbers
– Naturals = N
– Wholes = W
– Integers = J
– Rationals = Q
– Irrationals = H
– Reals = R
Def: Sets of Numbers
Natural numbers
N = {1,2,3, … }
Whole numbers
W = {0,1,2,3, … }
Integers
 J = {… , -3, -2, -1, 0, 1, 2, 3, …}
Naturals
Wholes
Integers
Def: Rational number
Any number that can be
expressed in the form p/q where
p and q are integers and q is not
equal to 0.
Use Q to represent
Def (2): Rational number
Any number that can be
represented by a terminating or
repeating decimal expansion.
Examples of rational numbers
Examples: 1/5,
-2/3, 0.5,
0.33333…
Write repeating decimals with a
bar above
.12121212… =
.12
Def: Irrational Number

 H represents the set

A non-repeating infinite decimal
expansion
2
Def: Set of Real Numbers = R
R = the union of the set of
rational and irrational numbers
Q
H R
Def: Set of Real Numbers = R
R = the union of the set of
rational and irrational numbers
Q
H R
Def: Number line
A number line is a set of points
with each point associated with
a real number called the
coordinate of the point.
Def: origin
The point whose coordinate is 0
is the origin.
Definition of Opposite of
opposite
For any real number a, the
opposite of the opposite of a
number is
-(-a) = a

Definition: For All

Def: There exists
Bill Wheeler - artist
“Good writing is
clear thinking
made visible.”
Def: intuitive
absolute value
The absolute value of any real
number a is the distance
between a and 0 on the number
line
Def: algebraic absolute value
a  R
a  aif a  0
a if a  0
Calculator notes
TI-84 – APPS
ALG1PRT1
Useful overview
George Patton
“Accept challenges, so
that you may feel the
exhilaration of
victory.”
Properties of Real Numbers
 Closure
 Commutative
 Associative
 Distributive
 Identities
 Inverses
Commutative for Addition
a + b = b + a
 2+3=3+2
Commutative for
Multiplication
ab = ba
2 x 3 = 3 x 3
2 * 3 = 3 * 2
Associative for Addition
a + (b + c) = (a + b) + c
–2 + (3 + 4) = (2 + 3) + 4
Associative for Multiplication
(ab)c = a(bc)
(2 x 3) x 4 = 2 x (3 x 4)
Distributive
multiplication over addition
a(b + c) = ab + ac
2(3 + 4) = 2 x 3 + 2 x 4
X(Y + Z) = XY +XZ
Additive Identity
a + 0 = a
3 + 0 = 3
X + 0 = X
Multiplicative Identity
a x 1 = a
5 x 1 = 5
1 x 5 = 5
Y * 1 = Y
Additive Inverse
 a(1/a) = 1 where a not equal to 0
 3(1/3) = 1
George Simmel - Sociologist
“He is educated who
knows how to find out
what he doesn’t know.”
Order to Real Numbers
 Symbols for inequality
 Bounded Interval notation
 *** Definition of Absolute Value
 Absolute Value Properties
 Distance between points on # line
George Simmel - Sociologist
“He is educated who
knows how to find out
what he doesn’t know.”
The order of operations
 Perform within grouping symbols – work
innermost group first and then outward.
 Evaluate exponents and roots.
 Perform multiplication and division left to
right.
 Perform addition and subtraction left to
right.
Grouping Symbols
 Parentheses
 Brackets
 Braces
 Radical symbols
 Fraction symbols – fraction bar
 Absolute value
Algebraic Expression
 Any combination of numbers, variables,
grouping symbols, and operation symbols.
 To evaluate an algebraic expression, replace
each variable with a specific value and then
perform all indicated operations.
Evaluate Expression by
Calculator
 Plug in
 Use store feature
 Use Alpha key for formulas
 Table
 Program - evaluate
The Pythagorean Theorem
 In a right triangle, the sum of the square of
the legs is equal to the square of the
hypotenuse.
a b  c
2
2
2
Operations on Fractions
 Fundamental Property
 Add or Subtract
 Multiply
 Divide
Properties of Exponents
 Multiply
 Divide
 Opposite exponent
 Product to power
 Power to power
 Quotient to power
 Scientific Notation
COLLEGE ALGEBRA REVIEW
Integer Exponents
Integer Exponents
 For any real number b and any natural
number n, the nth power of b o if found by
multiplying b as a factor n times.
b  bbb
n
N times
b
Exponential Expression – an
expression that involves
exponents
 Base – the number being multiplied
 Exponent – the number of factors of the
base.
Exponential Expression – an
expression that involves
exponents
 Base – the number being multiplied
 Exponent – the number of factors of the
base.
Quotient Rule
m
a
mn

a
n
a
Integer Exponent
1
n
a  n
a
Zero as an exponent
a  1 a  0  R
0
Calculator Key
 Exponent Key
^
Sample problem
3
0
8x y
2 5
24 x y
5
y
 5
3x
more exponents
Power to a Power
a 
n
m
a
mn
Product to a Power
 ab 
r
a b
r r
Quotient to a Power
r
a a

 
r
b b
r
Sample problem
a b

a b 

4
2 3
2
3 5
2
b
 2
a
Scientific Notation
A number is in scientific
notation if it is written as a
product of a number between 1
and 10 times 10 to some power.
Calculator Key
EE
Mode - SCI
Sydney Harris:
“When I hear somebody
sigh,’Life is hard”, I am
always tempted to ask,
“Compared to what?”
Radicals
 Principal nth root
 Terminology
– Index
– Radicand
Properties of Radicals
 Product of radicals
 Quotient of Radicals
 Index is even or odd and radicand of any
Real number
Rational Exponents
 Definition
 Evaluation
 Evaluation with calculator
Operations on Radicals
 Add or subtract
 Multiply
 Divide
 **** Rationalize
Polynomials
 Multiply – FOIL
 Evaluate
 Product of polynomials
 Special Products
Sum and Difference
Squaring
Factoring
 Common Factor
 By Grouping
 Difference of Two Squares
 Perfect Square Trinomials
 General Trinomials
 Difference of Cubes
 Sum of Cubes
Rational Expressions
 Find Domain
 Simplify
 Multiply and Divide
 Add and Subtract
 Complex Fractions
Cartesian Plane
 Plot Points
 **** Distance Formula
 ** Midpoint Formula
 General Equation of Circle
Chapter Summary
 Text – Chapter Summary and Review– end
of chapter
 What You Should Learn – beginning of
each section
 Review Exercises – broken down by
sections
 Chapter Test – Good Practice
The END.
 Or The Beginning of possibly one of the
most challenging courses you will take that
will require the following:
–
–
–
–
–
Commitment
Time
Dedication
Perseverance
More Work than you Think if you want to be
successful!
Good Luck