Transcript Reviewing Cant Hurt

Reviewing Can’t
Hurt
By: James Christian
and Jack Gryniewicz
Solving 1st Power Equations in
One Variable
Example problem: 5(4+n) = 2(9+2n)
 5(4+n) = 2(9=2n) Distribute the 5 and the 2
 20+5n = 18+4n Subtract 4n from both sides
 20+n = 18 Subtract the 20 from both sides to get n alone
 n = -2
Solving 1st Power Equations in
One Variables (Cont’)
Fractional Coefficients- Example:
x 1 1
 
2 2 2
1x  1
x 1
1st: get rid of denominator by multiplying by 2
2nd: divide by 1 on both sides
YOUR ANSWER IS 1
Solving 1st Power Equations in
One Variables (Cont’)
Variables in the
Denominator-Example:
2 x 4

3 x 9
1.
2.
3.
9(2  x)  (3  x)(4)
Multiply by the LCD on
both sides
9(3-x)
Distribute
Solve for x
18  9x  12  4x
6
x
5
Solving 1st Power Equations in
One Variables (Cont’)

Special Cases
1.
Variables cancels and you get the same number on both
sides of the equal sign
ALL REALS
x  5  5  x
2.
55
Variables cancel but the numbers on either side of the equal
sign are not the same
x  5  x  2
5  2

Properties
 Addition Property of Equality

If the same number is added to both sides of an equation, the
two sides remain equal. That is, if x = y, then x + z = y + z
 Multiplication Property of Equality

If a = b then a·c = b·c
 Reflexive Property of Equality

If something is equal to its identical twin a=a
 Transitive Property of Equality

If a = b, c = b so a = c
 Symmetric Property of Equality

If something flipped sides of the equal sign

So if a = b, then b = a
Properties
 Distributive Property

The sum of two numbers times a third number is equal
to the sum of each addend times the third number. For
example 4 * (6 + 3) = 4*6 + 4*3
 Inverse Property of Addition

The sum of a number and its additive inverse is always
zero. (x + (-x) = 0)
 Closure Property of Addition

Sum (or difference) of 2 real numbers equals a real
number
Properties
 Commutative Property of Addition

When two numbers are added, the sum is the same regardless of
the order of the addends. For example 4 + 2 = 2 + 4
 Associative Property of Addition

When three or more numbers are added, the sum is the same
regardless of the grouping of the addends. For example (2 + 3) + 4
= 2 + (3 + 4)
 Additive Identity Property

The sum of any number and zero is the original number. For
example 5 + 0 = 5
Properties
 Commutative Property of Multiplication

When two numbers are multiplied together, the product is the same
regardless of the order of the multiplicands. For example 4 * 2 = 2 *
4
 Associative Property of Multiplication

When three or more numbers are multiplied, the product is the same
regardless of the grouping of the factors. For example (2 * 3) * 4 = 2
* (3 * 4)
 Identity Property of Multiplication

When three or more numbers are multiplied, the product is the same
regardless of the grouping of the factors. For example (2 * 3) * 4 = 2
* (3 * 4)
Properties
 Multiplicative Identity Property

The product of any number and one is that number. For example 5 *
1 = 5.
 Multiplicative Inverse Property

any rational number times its reciprocal equals 1.
 Multiplicative Property of Zero

The product of 0 and any number results in 0.That is, for any real
number a, a × 0 = 0.
 Closure Property of Multiplication

For any two whole numbers a and b, their product a*b is also a whole
number.
Example: 10*9 = 90
Properties
 Product of Powers Property

This property states that to multiply powers having the
same base, add the exponents.
 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.
 Power of a Power Property

This property states that the power of a power can be
found by multiplying the exponents.
Properties
 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 the power is zero then the number will turn into 1.
Properties
 Negative Power Property


A negative exponent just means that the base is on the
wrong side of the fraction line, so you need to flip the
base to the other side.
The Zero Product Property

simply states that if ab = 0, then either a = 0 or b = 0
(or both). A product of factors is zero if and only if one
or more of the factors is zero.
Properties
 Product of Roots Property

If you multiply two roots together you get the product.
5  9  45
 Quotient of Roots Property

The square root of a quotient is equal to the quotient
of the square root of the numerator and the square root
of the denominator.
a
a / b = ---b
Properties
 Root of a Power Property

The squared and the square root cancel each other out.
a2  a
 Power of a Root Property

The square root and the square cancel each other out.
2
5 5
 Power of a Root Property
Quiz Time!!! Lets see what you
have learned.

A.
Addition Property (of Equality)

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
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

B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
L.
M.
N.
O.
P.
Q.
R.
S.
T.
U.
V.
W.
X.
Y.
Z.
AA.
BB.
CC.
Multiplication Property (of Equality)
Reflexive Property (of Equality)
Symmetric Property (of Equality)
Transitive Property (of Equality)
Associative Property of Addition
Associative Property of Multiplication
Commutative Property of Addition
Commutative Property of Multiplication
Distributive Property
Prop of Opposites or Inverse Property of Addition
Prop of Reciprocals or Inverse Property of Multiplication
Identity Property of Addition
Identity Property of Multiplication
Multiplicative Property of Zero
Closure Property of Addition
Closure Property of Multiplication
Product of Powers Property
Power of a Product Property
Power of a Power Property
Quotient of Powers Property
Power of a Quotient Property
Zero Power Property
Negative Power Property
Zero Product Property
Product of Roots Property
Quotient of Roots Property
Root of a Power Property
Power of a Root Property
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
x9 ● x3 = _________
= ________
If x3 = y9, then y9 = _________
9(x – y) = ____
(x9)3 = ____
(xy)3 = ____
If x3 = y9 and y9 = z12, then _________
=_________
(– 9xy)0 ____
x3 ● ________ = x3
12.
x3 ● y9 = y9 ● ____
ANSWERS
1. R
2. V
3. D
4. J
5. T
6. S
7. E
8. X
9. W
10. N
11. L
12. I
x3
 _______  1
y
Inequality Rules
1.
2.
3.
When the sign has the equal to option the
graph will have a closed circle and when the
sign is only greater than or less than the
graph has an open circle.
Disjunction is an “or” statement when both
statements can be true. Satisfies one or both
statements.
A conjunction is when both statements are
true and the statement has an “and” in it.
Special Cases of Inequalities
 Watch for no solution- conjunction does
not combine, but rather looks like a
disjunction when graphed.
 Watch for when every number worksdisjunction looks like a conjunction when
graphed.
 One arrow- disjunction when one arrow
proves the other to be right as well.
Solving Inequalities in One
Variable
 Don’t forget the Multiplication
Property of Inequality! If you
multiply or divide by a negative,
the sign must be reversed.
Solution Set:
{x: x > -2}
-5x 10

x -2
-2
SIGN REVERSAL!!!
Try Solving These Inequalities
on Your Own
 Disjunction- 5x > 25 or 3x < 9
 Conjunction- -3x < 33 and x < 2
Linear Equations in Two
Variables
y1  y 2
x1  x 2
(“m” stands for the slope)
 Slope:


m
If you are given the points (3, –2) (9, 2)

Your slope would be:
 Standard Form:
4 2

6 3
Ax + By = C

A, B, C are integers (positive or negative whole numbers)
No fractions nor decimals in standard form.

Traditionally the "Ax" term is positive.

 Point-Slope Form:

y – y1 = m(x – x1)
For this one, they give you a point (x1, y1) and a
slope m, and have you plug it into this formula: y
– y1 = m(x – x1)
Graphing
 Given the equations:


y=x+1
y=2x
 What would your intersecting point
be????

The point (1,2) is where the two lines
intersect.
Linear Systems
How many points do these
lines have in common?
Think about:
2x + y = 5 or y = - 2x + 5
How to Find the Common
Points in Linear Systems

1.
 3x  y  1
Methods
7x  2y  4
Substitution- goal
is to get one
variable equal to y  3x  1
an equation and 7x  2(3x  1)  4
substitute that
expression into the 7x  6x  2  4
other equation for
x  2  4
that variable.
x  2  Are we done ?
How to Find the Common
Points in Linear Systems
 Method 2Estimate the SOLUTION
of a SYSTEM on a graph.
Where do they intersect?
 y  2x  5


3
 y  2 x  2
5
THE
SOLUTION IS :
fx =
gx =4 -2x+5

3
2
x-2
3
2
(2,1)
1
2
4
How to Find the Common
Point of Linear Systems
 Method 3- Elimination or
Addition/Subtraction
-Goal is to combine equations in order
to cancel one variable
Steps to solving:
1. Find the LCM of one of the two
variables.
2. Multiply each individual equation by
the necessary factor to cancel.
3. Add the two equations if they have
opposite signs, in not then subtract.
4. Solve for other variable.
5. Back substitute into other equation
to find other variable
 5x  3y  5

3x  2y  16
-3y and +2y could be
turned into -6y and
+6y
10x – 6y = 10
9x + 6y = -48
ADD!!!
19x = -38
x = -2
NOW BACK
SUBSTITUTE
A System Can Be . . .
 If lines are parallel and answer
is null set

4
2
This is inconsistent
-2
4
 If two lines cross at one point
 This is Consistent
 When same line is used twice
 This is Dependent
-4
2
-2
4
-4
2
-2
Factoring
 The 7 methods of Factoring:







GCF
Difference of Squares
Sum & Difference of Cubes
PST
Reverse Foil
Grouping 2x2
Grouping 3x1
Factoring Continued…..
 Factoring with GCF:

When factoring GCF is the first thing you look for if all
of the terms have a multiple in common you divide
that out of each.
 Factoring with Difference of Squares:

WHEN THE SUM of two numbers multiplies their
difference -

(a + b)(a − b)
--- then the product is the difference of their squares:
 (a
+ b) (a − b) = a² − b²
Factoring Continued…..
 Factoring with Sum & Difference of Cubes
 You can use the difference of squares rule to factor
binomials that can be written in the form a2 – b2.
Sometimes the terms of a binomial have common
factors. If so, the GCF should always be factored out
first.
 Formulas: a2 - b2 (a + b) (a - b) or (a - b) (a - b)
 Factoring with PST (Perfect Square Trinomial) :

If the square root of “a” and “c” can be found and if twice
their product is equal to middle term, then the trinomial can
be factor out as Perfect Square Trinomial (PST).
Factoring Continued…..
 Factoring with Reverse Foil:
 ax² + bx + c
 The difference between this trinomial and the one
discussed above, is there is a number other than 1 in
front of the x squared. This means, that not only do
you need to find factors of c, but also a.
 Factoring by Grouping
 When factoring by grouping, you must first identify
patterns of common factors.
Need more help???
 Inspiration file for more Factoring Help
Rational Expressions
 Simplifying by factoring and canceling

Ex: x2 + 9x + 18
x2 + 4x - 12
1st – factor (x + 6)(x + 3)
(x + 6)(x – 2)
2nd – cancel (x + 3)
(x – 2)
Rational Expressions (cont)
 Addition and Subtraction
– factor
 2nd – multiply by missing factor on top and
bottom of each equation
 3rd – simplify
 4th – look for more possible cancelation
factors
 Example problemhttp://www.youtube.com/watch?v=omv7Di2o8
-Y&feature=channel
 1st
Rational Expressions (cont)
 Multiplying
– look for
possible factoring
 2nd – then cancel
if possible from
any top to any
bottom
 3rd – multiply
across
 1st
Dividing
1st – take
reciprocal of
2nd fraction
 2nd – just multiply
the ration
expressions
Quadratic Equations in One
Variable
 First you must set equal to zero.
 Then you factor.

Use Zero Product Property to finish the
problem.
Quadratic Equations in One
Variable
 X2= 36

Take square root of both sides
 X=6
(final answer)
Quadratic Equations in One
Variable
 Completing the Square
 EX:

x2+6x-6=0
Move the -6 to the other side
 X2+6x

=6 (leave space to complete square)
Take half of six and square it to complete square
 x2+6x+9=6+9

The trinomial is a PST
 (X+3)2=15

Take the square root of both sides
 X+3=√15

Subtract three from both sides
 X=-3+-√15
Quadratic Equations in One
Variable
2 - 4ac
• QUADRATIC
-b±
b
x
=
FORMULA:
2a
• The b, a, and c
are coefficients
• Plug the
numbers from
you equation in
for these letters
Quadratic Equations in One Variable
•What does the discriminant tell you?
•The “zeros” of a function are the x-intercepts on it’s graph.
Use the discriminant to predict how many x-intercepts each
parabola will have and where the vertex is located.
1. y = 2x2 – x - 6
1–4(2)(-6)=49  2 rational zeros
opens up/vertex below x-axis/2 x-intercepts
2. f(x) = 2x2 – x + 6
1–4(2)(6)=-47  no real zeros
3. y = -2x2– 9x + 6
81–4(-2)(6)=129 2 irrational zeros
opens up/vertex above x-axis/No x-intercepts
opens down/vertex above x-axis/2 x-intercepts
4. f(x) = x2 – 6x + 9
36–4(1)(9)=0  one rational zero
opens up/vertex ON the x-axis/1 x-intercept
Functions
 What does f(x)= mean???

f (x)= means the same thing as y=
 What is the domain and range???


Domain: The set of numbers x for which f(x) is defined
Range: The set of all the numbers f(x) for x in the
domain of “ f ”
Parabolas
 Vertex: -b/2a
 x intercepts: given that y is zero for all x
intercepts plug 0 into where all of the y’s
would be.
 y intercepts: given that x is zero for all y
intercepts plug 0 where all of the x’s
would be.
Parabolas Continued….
 Equation: y = x2 – 6x + 5




Vertex: (3, -4)
y intercept: (0,5)
x intercepts: (5,0) and
(1,0)
Given these points your
graph would be:
Simplifying Expressions with
Exponents
 X5 x X8=x13 (just add exponents together)
 (3 x 4)2= (12)2= 144 (PEMDAS)
 (22)4 = 28 = 256
 X4/x3=x4-3 = x (just subtract)
 (9/3)2 = (3)2 = 9
 (5x+5y+7m)0 = 0
 X-2 = 1/X2
Simplifying Expressions with
Radicals
 √5 x √7 = √35
 √1/√2 = √1/√2 x √2/√2 = √2/2 (rationalizing
the denominator)
 √x2 = x (the root and power cancel each other out)

2√4
=4
Word Problems!!!!     
 A collection of 33 coins, consisting of nickels, dimes, and
quarters, has a value of $3.30. If there are three times as
many nickels as quarters, and one-half as many dimes as
nickels, how many coins of each kind are there?



number of quarters: q
number of nickels: 3q
number of dimes: (½)(3q) = (3/2)q
There is a total of 33 coins, so:
q + 3q + (3/2)q = 33
4q + (3/2)q = 33
8q + 3q = 66
11q = 66
q=6
Word Problems!!!!     
 A triangle has a perimeter of 50. If 2 of its sides are equal
and the third side is 5 more than the equal sides, what is
the length of the third side?


Step 1: Assign variables:
Let x = length of the equal side
Sketch the figure
x
x
x +5
Continued….
Last problem continued…
 Plug in the values from the question and from the sketch:
50 = x + x + x+ 5
 Combine like terms: 50 = 3x + 5
 Isolate variable x:
3x = 50 – 5
3x = 45
x =15
 The length of third side = 15 + 5 =20

Answer: The length of third side is 20
Word Problems!!!!     
 Polar bears are extremely good swimmers and can travel
as long as 10 hours without resting. If a polar bear is
swimming with an average speed of 2.6 m/s, how far will
it have traveled after 10.0 hours?

speed, v = 2.6 m/s
time, t = 10.0 h = 10.0 h × 3600 s/h = 3.6 × 104 s
Unknown: distance, d = ?

d =2 .6 m / s × (3.6 × 104 s)


d
= 9.4 × 104 m = 94 km
Word Problems!!!!     
 Florence Griffith-Joyner set the women’s world record for
running 200.0 m in 1988. At the 1988 Summer Olympics
in Seoul, South Korea, she completed the distance in
21.34 s. What was Florence Griffith-Joyner’s average
speed?
 Substitute distance and time values into the speed
equation, and solve.

v = d / t = 200.0 m / 21.34 s

v = 9.372 m/s
Line of Best Fit or Regression
Line
 A line of best fit (or "trend"
line) is a straight line that
best represents the data on
a scatter plot.
This line may pass through
some of the points, none of
the points, or all of the
points.
 Given this set of data:
For graphing on calculator view this
Total
Fat
(g)
Total
Calori
es
Hamburger
9
260
Cheeseburger
13
320
Quarter Pounder
21
420
Quarter Pounder with
Cheese
30
530
Big Mac
31
560
Arch Sandwich Special
31
550
Arch Special with
Bacon
34
590
Crispy Chicken
25
500
Fish Fillet
28
560
Grilled Chicken
20
440
Grilled Chicken Light
5
300
Sandwich
Line of Best Fit or Regression
Line Continued….
 Your best fit line would look like this:
650
600
550
500
y
450
400
350
 You use the line of
best fit when your
data is scattered.
300
250
200
150
100
50
THE END!!!!!