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Exam
Review
Christian Martin
Algebra (6th period)
Ms.Hardtke
5/17/10
Addition Property of Equality
if the same number is added to both sides
of an equation, the two sides remain
equal.
 Ex. If
X=Y, then X+Z=Y+Z

Multiplication Property of
Equality


Is when both sides of
an equal equation is
multiplied and the
equation remains
equal
Ex.
Reflexive Property of Equality

The reflexive property of equality says that
anything is equal to itself
For example
A=A or, 145=145

Symmetric Property of Equality
Symmetric property of equality is two
variables that are different but have the
same number/amount (equal symmetry)
 Ex. If a=b, then b=a

Transitive Property of Equality
When numbers or variables are all equal
 Ex. If a=b and b=c, then c=a
 If A=3.3 and 3.3=B, then A=B

Associative Property of Addition
The sum of a set of numbers is the same
no matter how the numbers are grouped.
For example:
 2+(3+Y)= (2+3)+Y= 5+Y

Associative Property of
Multiplication
The product of a set of numbers is the
same no matter how the numbers are
grouped. For example
 5(4a)=(5x4)a=20a
 by using the associative property we
switch the 5 and 4

Commutative Property of
Addition
The sum of a group of numbers is the
same regardless of the order in which the
numbers are arranged, for example:
 3+7=7+3
 X+2 = 2+X

Commutative Property of
Multiplication
The product of a group of numbers is the
same regardless of the order in which the
numbers are arranged, for example:
 3xA=Ax3
 Xy=yx

Distributive Property
Is distributing something as you separate or
break it into parts.
Ex.



A(b+c)= Ab + Ac
2x(3y+8)=6xy + 16x
Property of Opposites/Inverse
Property of Addition
A number added to its opposite integer will
always equal zero.
 Ex.
 A+(-A)=0
 3+(-3)=0 or (-3)+3=0

Property Of Reciprocals/Inverse
Property of Multiplication


For two ratios, if a/b =
c/d, then b/a = d/c
a(1/a) = 1


A number times its
reciprocal, always
equals one
A Reciprocal is its
reverse and opposite
(the signs switch from
+ to — or vice versa)
Identity Property of Addition
Identity property of addition states that
the sum of zero and any number or
variable is the number or variable.
 4+0=4, -11+0=-1, y+0=y

Identity Property of
Multiplication
Identity property of multiplication states
that the product of 1 and any number or
variable is the number or variable itself
 4×1=4, -11×1=-11, y×1= y

Multiplicative Property of Zero
The product of any number and zero is
zero
 Ax0=0

Closure Property of Addition
Closure property of addition states that the
sum of any two real numbers equals
another real number.
2+5=7

Closure Property of
Multiplication
Closure property of multiplication states
that the product of any two real numbers
equals another real number.
Ex.
 4 × 7 = 28

Product of Powers Property
Is that when you multiply powers having
the same base, add the exponents.
Ex.
 am × an = am+n.

Power of a Product Property
This property states that the power of a
product can be obtained by finding the
powers of each factor and multiplying
them.
Ex.
(ab)m = am × bm

Power of a Power Property
the power of a power can be found by
multiplying the exponents.
Ex.
 (am)n = amn

Quotient of Powers Property

This property states that to divide powers
having the same base, subtract the
exponents.
Power of a Quotient Property

This property states that the power of a
quotient can be obtained by finding the
powers of numerator and denominator and
dividing them.
Zero Power Property
If a variable has an exponent of zero, then
it must equal one
 30=1

Negative Power Property
When a fraction or a number has negative
exponents, you must change it to its
reciprocal in order to turn the negative
exponent into a positive exponent
 2-2 = 1/2 2 = 1/4

Zero Product Property
When both variables equal zero, then one
or the other must equal zero
 Ex.

if ab=0, then either a=0 or b=0
if XY = 0, then X = 0 or Y = 0 or both X and Y
are 0.
Product of Roots Property

The product is the same as the product of
square roots
Quotient of Roots Property

the quotient is the same as the quotient of
the square roots
Mini Quiz
What Property of Equality states that
anything is equal to itself?
A) Reflexive B) Symmetric C) Multiplication
D) None
A) Reflexive [remove for answer]
1.
Solving 1st power inequalities in
one variable


Here is a table explaining inequalities
Symbols
Words
Example
>
greater than
x+3>2
<
less than
7x < 28
≥
greater than or
equal to
5≥x–1
≤
less than or equal
to
2y + 1 ≤ 7
Also when dividing by a negative, the inequality sign
must be switched in order for the statement to be
true
Inequalities cont.
A conjunction is true only if both the
statements in it are true
 A conjunction is a mathematical operator
that returns an output of true if and only
if all of its operands are true.
 Ex. -2<x and x<3

-3
-2
-4
0
1
2
3
Ineq. Cont.



A disjunction is statement which connects
two other statements using the word or.
To solve a disjunctions of two open
sentences, you find the variables for which at
least one of the sentences is true. The graph
consists of all points that are in the graph
Ex. -2<x or x<3
-3
-2
-1
0
1
2
3
Mini Quiz
Solve the inequality:
x-1>4

Answer: x>5
Linear Equations
Solving equations in two variables
 Graphing points
 Standard/General Form
 Slope- Intercept Form
 Point-Slope Form
 Slopes

Solving equations in two
variables

the solutions for an equation in two
variables are ordered pairs in the form
(a, b).
Ex.
 x = y + 1 is true when x = 3 and y = 2. So, the
ordered pair
 (3, 2)
 is a solution to the equation.
Graph Points
A Graph Point contain an X and a Y
 The X-number line crosses horizontally
and the Y-line vertically.

Y line graph
X line graph
Standard/ General Form
Standard/ General form
 Ax + By = C
 The terms A, B, and C are integers (could
be either positive or negative numbers or
fractions)

Slope-Intercept Form

where m is the slope of the line and b is the y-intercept,
which is the y-coordinate of the point where the line
crosses the y axis. This can be seen by letting x = 0,
which immediately gives y = b. Vertical lines, having
undefined slope, cannot be represented by this form.

The graph of this equation is a straight line.
The slope of the line is m.
The line crosses the y-axis at b.
The point where the line crosses the y-axis is called the y-intercept.
The x, y coordinates for the y-intercept are (0, b).




Point-Slope Form

where m is the slope of the line and (x1,y1) is any
point on the line. The point-slope and slopeintercept forms are easily interchangeable. The
point-slope form expresses the fact that the
difference in the y coordinate between two
points on a line (that is, y − y1) is relative to the
difference in the x coordinate (that is, x − x1).
The proportionality constant is m (the slope of
the line).
Slopes
To find a slope use this formula:
 m= X2 - X1

Y2-Y1
Slope of a Straight Line



One of the most important properties of a
straight line is in how it angles away from the
horizontal. This concept is reflected in something
called the "slope" of the line.
To find the slope, you will need two points from
the line and you will get these points by doing
the point-slope formula.
Then use the formula from the previous slide.
Mini Quiz

What is the equation for the Point-Slope
Form?
Answer:
Linear Systems
Substitution Method
 Addition/ Substitution Method (Elimination)

Substitution Method
Ex. X+Y=2
Y= 2x-1 substitutes for the Y in the
equation above making it
X+(2x-1)=2
3x-1=2
3x=3
X=1 then plug the x-coordinate into the
equation to find the y-coordinate
Y=2(1)-1
Y=2-1 = 1 thus the answer will be (1,1)

Addition/Subtraction Method
1. Algebraically manipulate both equations so that all the
variables are on one side of the equal sign and in the same
order. (Line the equations up, one on top of the other.)
2. If needed, multiply one of the equations by a constant so
that there is one variable in each equation that has the same
coefficient.
3. Subtract one equation from the other.
4. Solve the resulting equation for the one variable.
5. Using the value found in the step 4, substitute it into either
equation and solve for the remaining variable.
6. Substitute the values for both variables into the equation not
used in step 5 to check.
Factoring
Greatest Common Factor (GCF)
 Difference of Squares
 Sum and Difference of Cubes
 PST
 Reverse Foil
 Grouping 2x2

GCF

Ex.
3x3 + 27x2 + 9x

To factor out the GCF in an expression like the one above, first find
the GCF of all of the expression's terms.

3(1, 3)
 27(1, 3, 9, 27)
 9(1, 3, 9)


Next, write the GCF on the left of a set of parentheses:
3x(
)
Next, divide each term from the original expression
(3x3+27x2+9x ) by the GCF (3x), then write it in the
parenthesis
=3x(x2 + 9x + 3)
Difference of Squares
When the sum of two numbers multiplies
their difference then the product is the
difference of their squares. Symmetrically,
the difference of two squares can
be factored:
 x² − 25 = (x + 5)(x − 5)
 x² is the square of x. 25 is the square of 5.

Sum & Difference of Cubes

A polynomial in the form a3 + b3 is called
a sum of cubes. A polynomial in the
form a3 − b3 is called a difference of
cubes.
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
PST
A perfect square trinomial is:
(ax)2 + 2abx + b2
 Recognizing it can save a lot of time
 Ex.
x2 + 2x + 1 = 0
(x + 1)2 = 0
X+1=0 x= -1

Reverse Foil
3x2 + 10x + 8
 =(_ + _)(_+_)
 =(3x + _)(_+_)
 use trial and error to fill in the blank spots
 Answer: (3x+4)(3x+2)

Grouping 2x2
Consider the expression
3x2 + 6x - 4x - 8
 Factor this by grouping two at a time:
(3x2 + 6x) - (4x + 8)
 Pull out the GCF of each:
3x (x + 2) - 4 (x + 2)
 Pull out the GCF again:
(3x - 4) (x + 2)

Mini Quiz

Which factoring method most likely
requires trial and error?
Answer: Reverse FOIL
Rational Expressions
Reducing
 Multiplying and Dividing
 Add and Subtract

Reducing



When you reduce a fraction such as 4/6 to 2/3
you do so by noticing that there is a factor
common to both the numerator and the
denominator (a factor of 2 in this example),
which can divide out of both the numerator and
the denominator.
Use exactly the
same procedure to
reduce rational expressions.
Multiplying and Dividing







Multiplication
Both the numerators and the denominators
multiply together
Common factors may be cancelled before
multiplying
Division
For Division multiply by the reciprocal of the
divisor
Invert the second fraction, then proceed with
multiplication
Don’t cancel the factors before it is written as a
multiplication
EX. Mult.
• First factor all the expressions:
(I also put the denominators in
parentheses because then it is easier to
see them as distinct factors)
• Now cancel common factors—any
factor on the top can cancel with any
factor on the bottom:
Now just multiply what’s left.
You usually do not have to multiply out
the factors, just leave them as shown.
Addition and Subtraction
First you must find the LCD
 The LCD is built up of all the factors of the
individual denominators, each factor
included the most number of times it
appears in an individual denominator.
 The product of all the denominators is
always a common denominator, but not
necessarily the LCD (the final answer may
have to be reduced).

Add and Sub. Cont.



Given equation:
Factor both denominators:
Assemble the LCD:
Note that the LCD
contains both
denominators
Add and Sub. Cont.

Build up the fractions
so that they both
have the LCD for a
denominator:

Now that they are over the
same denominator,
you can add the numerators:
simplify:

Quadratic Equations
Solve by Factoring
 Solve by taking square root of each side
 Solve by completing the Square
 Quadratic Formula
 Discriminant

Solve by Factoring






Here is a quadratic equation in standard form
x2 - 2x - 24=0
(x-6)(x+4)=0
x-6=0 or x+4=0
X=6 or X=-4
For more examples of how to solve a quadratic
equation by factoring, using the square root
property, completing the square, and a better
explanation of this problem go to
www.mathtv.com
Quadratic Formula
• Ex. y =x2+2x+1
-2
X=
22 – 4 (1)(1)
2(1)
-2
4-4
2
-2
=
0
2
=
2
2
Discriminant



The discriminant is what you need to solve a
quadratic equation
A quadratic equation with real coefficients can
have either one or two distinct real roots, or two
distinct complex roots. In this case the
discriminant determines the number and nature
of the roots. There are three cases:
If the discriminant is positive, then there are two
distinct roots, both of which are real numbers
Mini Quiz

What is the Quadratic Formula?
Answer:
Functions


The parentheses in f(x) ("f of x") do not mean
multiplication. They are part of what is
called functional notation. f is the name of the
function. And whatever appears within the
parentheses is called the argument of the
function. It is upon the argument that the
function called f will "operate."
Thus, the function f has been defined as follows:
f(x) = x² + 1.
Functions cont.

A Function is a correspondence between
two sets, the domain and the range, that
assigns to each member of the domain
exactly one member of the range. Each
member of the range must be assigned to
at least one member of the domain.


Funct. Cont.
Linear functions are have
no exponents in them such
as 2x2. Wherever the
variables appear, they are
not raised to a power.
The a term is the slope of
the line and controls its
'steepness'. A positive
value has the slope going
up to the right. A negative
slope goes down to the
right. The b term is the y
intercept - the point where
the line crosses the y axis.
Adjust the sliders above to
vary the values of a and b,
and note the effects they
have on the graph.




y = ax + b
Where: x,y are the
coordinates of any
point on the line
A is the slope of the
line
B is the intercept
(where the line
crosses the y-axis)
Funct. Cont.
A quadratic function is one of the form f(x)
= ax2 + bx + c, where a, b, and c are
numbers with a not equal to zero.
 The graph of a quadratic function is a
curve called a parabola. Parabolas may
open upward or downward and vary in
"width" or "steepness", but they all have
the same basic "U" shape.

Simplifying expressions with
exponents
Product of powers:
23 x 34= 63+4 which can be simplified to 67
 Power of a product:
(ab)4 = a4xb4 which simplifies to a4b4
 Power of a Power:
(a2)2 = a2(2)= a4

Simplifying expressions w/
exponents cont.

Quotient of Powers:
a4/b2= a4-2 =a2
Word Problems
A foot ball game has attendance of 1200. Adult
tickets cost $5 and student tickets cost $2. The
total tickets receipts were $3660. The visiting
team is entitled to half of the adult ticket sales.
How much money should be given to the visiting
team?
5a + 2(1200 - a) = 3660
a = 420
5a + 2400 - 2a = 3660
5a/2
3a + 2400 = 3660
=5(420)/2
3a = 1260
=2100/2
=$1050

Word Problems cont.

At Bison High School, there are 16 students in
English Club, 16 students in Science club and
20 students in Math Club. Of these students,
there are 5 students in both the English and
Science Clubs; 6 students in both the Science
and Math clubs; and 8 in both the English an d
Math clubs. If only 2 students are in all 3 clubs,
how many students are in at least one of the
clubs?

35
Line Best Fit or Regression Line


A Regression line is a line draw through and
scatter-plot of two variables. The line is chosen
so that it comes as close to the points as
possible.
When asked to draw a linear regression line or
best-fit line, you are most certainly being asked
to draw a line through data point on a scatter
plot. In order to solve these problems you must
use a calculator’s scatter-plot.
Mini Quiz

What is the difference between line best-fit
and a regression line?
Answer: there is no difference