Math 6H Period 3, 6 & 7

Solving Equations & Inequalities 2-4 & 2-5

Inequalities Continued

We know about > greater and as well as < less than
so now we look at
≥ which means " greater than or equal to" and
≤ which means " less than or equal to"

This time the boundary point ( or endpoint) is included in the solution set.
The good news is that we still solve these inequalities the same way in which we solved equations-- using the properties of equality.
w/4 ≥ 3
we multiply both sides by 4/1
(4/1)(w/4) ≥ 3(4/1) by the X prop =
1w ≥ 12
w ≥ 12 by the ID(x)

Which means that any number greater than 12 is part of the solution AND 12 is also part of that solution

Take the following:
b - 3 ≤ 150 ( we need to add 3 to both sides of the equation)
+3 = +3 using the + prop =
b + 0 ≤ 153
or b ≤ 153 using the ID (+)

Problem Solving: Using Mathematical Expression 2-6

Seventeen less than a number is fifty six

I suggest lining up and placing the "equal sign" right under the word is
then complete the right side = 56
after that take your time translating the left
Start with a "let statement."
A "let statement" tells your reader what variable you are going to use to represent the number in your equation.
So in this case Let b = the number
it becomes
b- 17 = 56
Now solve as we have been practicing for a couple of weeks.
b - 17 = 56
+17 = +17 using the +prop=
b + 0 = 73
b = 73 by the ID(+)

How could we check?
A FORMAL CHECK involves three steps:
1) Re write the equation ( from the original source)
2) substitute your solution or... "plug it in, plug it in...."
3) DO the MATH!! actually do the math to check!!

so to check the above
b - 17 = 56
substitute 73 and put a "?" above the equal sign...

73 - 17 ?=? 56
Now really do the math!! Use a side bar to DO the MATH!!
That is, what is 73- 17? it is 56
so 56 = 56


Practice some of the class exercised on Page 50, Just practice setting up the equations from the verbal sentences.

Algebra Period 4

The Multiplication Property of Inequality 4-3
Solving Inequalities  with multiplication or division:
Again, you will use your equation skills, but this time with the Multiplicative Inverse Property.

ONE MAJOR DIFFERENCE FROM EQUATIONS;

When you multiply or divide by a NEGATIVE to BALANCE, you must SWITCH the inequality SYMBOL! 

(REMEMBER --->Does not apply to adding or subtracting negatives.)


EXAMPLE: -3y > 9

You need to divide both sides by NEGATIVE 3 so the symbol will switch from > to < in the solution y < -3 is the answer

If you want to understand why:

3 < 10

Now multiply both sides by -1 (mult prop of equality)

You get -3 < -10, but THAT'S NOT TRUE!!!

You have to SWITCH THE SYMBOL to make the answer true: -3 > -10


REMEMBER: when you MULTIPLY or DIVIDE by a NEGATIVE, the symbol SWITCHES



Using the Properties Together 4-4


Same as equations except make sure you switch the symbol if you multiply or divide by a negative!

Always finish with the variable on the left-- makes it so much easier to graph

Check with whatever solution is easiest in the solution set!

Somethings to remember with two step inequalities:

Before you start, you may want to clear fractions or decimals, but if you don't mind using them, just get started with the checklist below.
If you want to clear them, you should do that right after you distribute (between steps 1 and 2 below)

1. Do distributive property first (if necessary)

2 Combine like terms on each side of the wall (equal sign)

3. Jump the variables to one side of the wall (get all the variables on one side of the equation) by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term)

4. Add or subtract

5. Multiply or divide

6. Make sure the variable is on the LEFT side when finished.

Algebra Period 4

Inequalities & Their Graphs 4-1
Introduction to INEQUALITIES and graphing them:

Writing an inequality

Graphing an inequality - open dot is < or >

Closed dot mean less than or EQUAL or greater than or EQUAL
(think of the = sign as a crayon that you can use to COLOR IN THE DOT!)

Different from equations: Inequalities have many solutions (most of the time an infinite number!)

Example: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)

n ≥ 3 means still means that every real number greater than 3 is a solution, but now 3 is also a solution.
The endpoint ( in this case 3) is called the boundary point.


GRAPHING INEQUALITIES:

First, graphing an equation's solution is easy

1) Let's say you solved an inequality and you discovered that y = 5, you would just put a dot on 5 on the number line

2) But now you have the y ≥ 5

You still put the dot but now also darken in an arrow going to the right

showing all those numbers are also solutions

3) Finally, you find in another example that y > 5

You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!


TRANSLATING WORDS:

Some key words to know
AT LEAST means greater than or equal

AT MOST means less than or equal


I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!

I want at most 15 minutes of homework means that I can have 15 minutes,
but I'm hoping for even less!



The Addition Property of Inequalities 4-2

Solving Inequalities with adding or subtracting

Simply use the Additive Inverse Property as if you were balancing an equation!

The only difference is that now you have more than one possible answer.

Example: 5y + 4 > 29

You would -4 from each side, then divide by 5 on each side and get:
y > 5

Your answer is infinite! Any real number bigger than 5 will work!
It is IMPOSSIBLE to check every value in an infinite solution set. However, once you have determined the endpoint-- or boundary point, you can verify by checking a representative or sample number from the supposed solution set to verify that your inequality has been solved correctly.

Math 6H ( Periods 3, 6, & 7)

Solving Equations & Inequalities 2-4 & 2-5

GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation


Here are the steps and justifications (reasons)

1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)

2. What is the operation the variable is doing with that number in its way?

3. Get rid of that number by using the opposite (inverse) operation

*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?

that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:

" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality

5. You should now have the variable all alone (isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of x = ____and box this in.

Pre Algebra Period 1

Simplifying Variable Expressions 2-3

Review of Algebraic terminology:

In the expression, 3y + 5

3 is the coefficient (number attached to variable - remember, "co" means to go along with)

y is the variable

5 is the constant (number not attached to variable)

terms are separated by ADDITION ONLY!



COMBINING LIKE TERMS:

1. Same variable (or no variable)

2. Same power
You can combine by addition or subtraction LIKE TERMS.

You cannot combine UNLIKE TERMS.


EX:
3a + 4a = 7a
 BUT

3a + 4b = 3a + 4b

3a + 4a² = 3a + 4a²


YOU SHOULD ALWAYS COMBINE LIKE TERMS BEFORE YOU EVALUATE!

IT'S MUCH SIMPLER!

-25a + 5a - (-10a) when a = -14

First combine like terms: -10a

Then plug in for a = -14:
-10(-14) = 140

VARIABLES AND EQUATIONS: Section 2-4

Is a given number a solution to an equation?

Substitute and evaluate to see.

Use the set signs { } and a ? over the equal sign as we did in class

9k = 10 - k for k = 1


9(1) = 10 - (1)

9 = 9 checks



Numeric Equations are either True or False

Equations with variables are OPEN (don't know if they are True or False)

Algebra Period 4

Chapter 3-7 Formulas


Some formulas you should already know (if you don't, it's time to memorize them!)

d = rt

p = 2l + 2w
in a rectangle

A = lw or bh

THIS LESSON IS NOT ON MEMORIZING, BUT WHAT YOU CAN DO AFTER YOU MEMORIZE THESE FORMULAS.

FORMULAS = special equations that have KNOWN relationships


Example: d = rt

Sometimes, you want to solve for distance, but other times you need the time or the rate.

You can manipulate the variables by balancing the equation until you solve for the wanted variable.

In the formula, d = rt, to solve for rate, divide both sides by t

To solve for time, divide both sides by r

Don't think of this as 3 different equations! 

Just learn the main one and use that to solve for what you need!


Example:
Solve for l:

P = 2l + 2w

Subtact 2w for both sides:
P - 2w = 2l

Divide by 2 on each side:
(1/2)(P - 2w) = l.


When the variable is in the denominator, it's usually easiest to use cross products:

Solve for x:


(3z)/x = 4/y
     
Cross products:
3yz = 4x



Divide both sides by 4:

x = 3yz/4
 
If you have a fraction, you can simply use the multiplicative inverse:


Solve for b:

3ab = 7c
4           


Either use cross products or multiply each side by 4/3a:

b = 7c (4/3a) = 28c/3a


Why can't you use cross products for the following?

Solve for f:

d = 2a/c + 2b/f
 

Math 6H Period 3, 6 & 7

Solving Equations & Inequalities 2-4 & 2-5
Any value of the variable that makes the equation or inequality a true sentence is called a solution. We solve an equation or inequality when we find ALL its solutions.

Remember when you solved the following:
? + 4 = 12.
Your teacher would ask you what number replaced the ?
We are going to learn how to solve equations and inequalities using Algebraic terms-- we will solve using inverse operations and we will justify our steps.
x - 4 = 12
add 4 TO BOTH SIDES of the equation using
the addition property of equality ( + prop =)
so
x - 4 = 12
+ 4 = + 4
x + 0 = 16 but the Identity Property of addition ( ID+) allows us to write
x = 16

Similarly
15n = 60
can be read as "fifteen times n equals sixty"
What undoes multiplication? What is the inverse operation of multiplication?
DIVISION
so we will divide both sides by 15 to isolate the variable.
15n = 60
15 = 15
using the Division Property of Equality ÷prop +
Then we get
1n = 4
but the Identity Property of Multiplication ( ID X) lets us write
n = 4
See the Class Notes Solving One-Step Equation ( Yellow Sheet) for more examples. Make sure to glue that into your spiral notebook.



GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation

Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite ( inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?
that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:
" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality

5. You should now have the variable all alone ( isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of x = ____and box this in.

Pre Algebra Period 1

EQUATIONS & INEQUALITIES: Properties 2-1




WHAT ARE PROPERTIES? (Why are they your friends?)

You can count on properties. They always work.
There are 0 COUNTEREXAMPLES!

COUNTEREXAMPLE = an example that shows that something does not work
(counters what you have said)
Because you can count on them, you can use them to JUSTIFY what you do.

JUSTIFY = a reason for doing what you did


PROPERTIES ARE EXCEPTIONS TO AUNT SALLY:

Commutative (order) Property
a + b = b + a

ab = ba

3 + 5 = 5 + 3

3 (5) = 5 (3)

(you can HEAR the change in order!)

Aunt Sally says that you always need to go left to right, but Commutative says not necessary if you have all multiplication or all addition.



Associative (groupings) Property
a + b + c = a + (b + c)

abc = a(bc)

(3 + 2) + 8 = 3 + (2 + 8)

(Why would you want to? Sometimes it's easier!)

[57 x 5] (2) = (57) [ 5 (2) ]

(you can't hear this property! but you can SEE it!)

Aunt Sally says you must always do parentheses first, but Associative says that you can actually take the parentheses away, put parentheses in, or change where the parentheses are if you have all multiplication or all addition.


These properties give you a choice when it's all multiplication OR all addition

There are no counterexamples for these two operations.


BUT THEY DO NOT WORK FOR SUBTRACTION OR DIVISION
(lots of counterexamples!
10 - 2 does not equal 2 - 10

15 ÷ 5 does not equal 5 ÷ 15)


SO WHY SHOULD YOU CARE????

Because it makes the math easier sometimes!

Which would you rather multiply:
(2)(543)(5) OR (2)(5)(543) ???

Commutative allows you to choose!


ANOTHER EXAMPLE: [(543)(5)](2)

Aunt Sally would say you must do the 543 by the 5 first since it's in [ ]
But our friend the Associative Property allows us to simply move the [ ]

[(543)(5)](2) = (543)[(5)(2)]
which is so much easier to multiply in your head!!!



TWO MORE FRIENDS: THE IDENTITY PROPERTIES OF

ADDITION AND MULTIPLICATION
F
For addition, we know that adding zero to anything will not change the IDENTITY of what you started with: a + 0 = a
(what you started with)
0 is known as the ADDITIVE IDENTITY.


For multiplication, we know that multiplying 1 by anything will not change the IDENTITY of what you started with:
(1)(a) = a
(what you started with)
1 is known as the MULTIPLICATIVE IDENTITY.

Sometimes 1 is "incognito" (disguised!)

We use this concept all the time to get EQUIVALENT FRACTIONS.

Say we have 3/4 but we want the denominator to be 12

We multiply both the numerator and the denominator by 3 and get 9/12

We actually used the MULTIPLICATIVE IDENTITY of 1, but it was disguised as 3/3

ANYTHING OVER ITSELF = 1
(except zero because dividing by zero is UNDEFINED!)

a + b - c = 1
a + b - c


We also use this property to SIMPLIFY fractions.
We simplify all the parts on the top and the bottom that equal 1 (your parents would say that we are reducing the fraction)


6abc/2a  =  3bc  
2a/2a = 1 and that's why we can divide the fraction by it.
                    

10a/5 =     2a                                                                                              
AGAIN, WE LOVE THESE PROPERTIES BECAUSE THEY MAKE OUR LIFE EASIER!
AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!!

Algebra Period 4

ABSOLUTE VALUE EQUATIONS:Section 3-8

Review of SOLVING equations:

1. Clear both sides of any decimals or fractions

2. Distribute if necessary (or divide both sides if compatible number)

3. Collect like terms on the SAME SIDE of the equation

4. Use the additive inverse to move any variables on BOTH SIDES of the equation

5. Use the addition property of equality and
then the multiplication property of equality to isolate the variable.


Generally, you solve these the same way you solve regular equations.

Make sure you balance equally on both sides!


Follow the steps of a 2 step equation.

1. Add the opposite (you can subtract as well)

2. Multiply by the reciprocal (you can divide as well)


THE DIFFERENCE?
YOU HAVE 2 POSSIBLE ANSWERS! (+ and -)


EXAMPLE: 2 l x l + 1 = 15

2 l x l + 1 - 1 = 15 - 1

2 l x l = 14

1/2 ( 2 l x l ) = 1/2 (14)

l x l = 7

x = {-7, 7}



Remember: If you get that the absolute value is negative, the answer is the NULL SET-- just like before!!

EXAMPLE: 2 l x l + 16 = 15

2 l x l + 1 - 1 = 15 - 16

2 l x l = -1

1/2 ( 2 l x l ) = 1/2 (-1)

l x l = -1/2

NOT POSSIBLE! So the answer is the null set

Algebra Period 4

Clearing An Equation of Fractions or Decimals 3-6

You can do equations with fractions or decimals as they are, but many 
students find it easier to "get rid of" them!


How? Just use the multiplicative property of equality! (your old friend!)


DECIMALS:
16.3 - 7.2y = -8.18

Multiply by the power of 10 needed to clear ALL decimals!

In the problem above, you would need to multiply by 100 to make -8.18 an integer

100 (16.3 - 7.2y) = 100 (-8.18)

USE THE DISTRIBUTIVE PROPERTY ON THE LEFT SIDE OF THE EQUATION

1630 - 720y = -818

SUBTRACT 1630 FROM EACH SIDE (or add -1630)

1630 - 1630 -720y = -818 - 1630

-720y = -2448

DIVIDE EACH SIDE BY -720

-720y/-720 = -2448/-720

y = 3.4

ALWAYS CHECK YOUR ANSWER!!!!!


Why do students get this wrong?
1)
1) They forget to multiply the integers by the power of 10 because it doesn't have any decimal places to "get rid of".
Remember: to stay in balance, you must distribute to EVERY term.
2) They distribute only the power of 10 that each term needs. 

For example, if the problem has .3y = 2.85, you will need to multiply by 100 EQUALLY on both sides to stay in balance, but students will end up with 3y = 285.

THIS IS WRONG!!!
YOU HAVE MULTIPLIED BY 10 ON LEFT SIDE AND 100 ON RIGHT SIDE!
YOU ARE OUT OF BALANCE!!!



FRACTIONS:
Are you "Fraction Phobic"????

Then you're going to love this!!! :)


2/3y + 1/2y = 5/6 + 2y

Instead of finding a common denominator and using fractions, we're going to...

MULTIPLY BY THE LCM OF ALL THE DENOMINATORS

In the problem above, the LCM of 3, 2, and 6 is 6

6 (2/3y + 1/2y) = 6 (5/6 + 2y)

USE THE DISTRIBUTIVE PROPERTY ON BOTH SIDES OF THE EQUATION

6(2/3y) + 6(1/2y) = 6(5/6) + 6(2y)

4y + 3y = 5 + 12y

COMBINE LIKE TERMS ON THE LEFT SIDE OF THE EQUATION

7y = 5 + 12y

SUBTRACT 12y FROM BOTH SIDES TO GET ALL VARIABLES ON ONE SIDE

7y - 12y = 5 + 12y - 12y

-5y = 5

DIVIDE BOTH SIDES BY -5

-5y/-5 = 5/-5

y = -1

ALWAYS CHECK YOUR ANSWER!!!!!


Why do students get this wrong?
1)They forget to multiply the integers by the LCM because it doesn't have any denominator to "get rid of".
Remember: to stay in balance, you must distribute to EVERY term.

2) They distribute only the number that each denominator needs. Again, you're out of balance.

3) They multiply by the wrong LCM...you'll end up still having a denominator.
You have not achieved your objective under this method! Your objective is to get rid of every denominator!



Couldn't I just solve these problems keeping the decimals and fractions?

Yes, you could, but I want you to learn this method because you might like it!


Also, you will to know how to get rid of the denominators in Chapter 10 when you have very complicated denominators with variables.

Math 6H Period 3, 6 & 7

Writing Inequalities 2-3

2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7

2 < 7 reads 2 is the less than 7
7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.

Notice the mathematical sentence ( inequality)
Two is less than seven or 2 <7
is different from the mathematical phrase ( expression)
Two less than seven. 7-2

The point of the number line that is paired with a number is called the graph of that number.
Check out the graph on page 39 of our textbook. When you graph numbers on the number line, make sure to place a dot DIRECTLY ON the number line at that particular number's location. Again, check out our textbook for examples!!

Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.

A number n is between 6 and 12 would be 6 < n < 12 or 12 > n > 6


Notice the subtle differences
in the sentence
Six is greater than a number t
and the phrase
six greater than a number t

Six is greater than a number t becomes 6 > t
while
six greater than a number t becomes t + 6

What about the following inequality:
A number p is greater than a number q
is p > q

The value in cents of d dimes is less than the value in cents of n nickles.

If you need to-- set up your T-charts ( refer to your class notes) one for dimes and the other for nickles.

10d represents the number of dimes and 5n represents the number of nickles

so 10d < 5n

Algebra Period 4

EQUATIONS 3-3 continued TWO STEPS WITH DISTRIBUTIVE PROPERTY

Usually, you want to do DISTRIBUTE FIRST!
UNLESS THE FACTOR OUTSIDE THE ( ) CAN BE DIVIDED
OUT OF BOTH SIDES PERFECTLY!!!!

EXAMPLE: 
5y - 2(2y + 8) = 16
5y - 4y - 16 = 16 [distribute]
y - 16 = 16 [collect like terms]
y = 32 [solve by adding 16 to both sides]

EXAMPLE:
-3(4 + 3x) = -9
[Don't distribute! Divide by -3.
4 + 3x = 3
The -3 goes into both sides perfectly!)
3x = -1 [Subtract 4 from both sides]
x = -1/3 [Divide both sides by 3]

Chapter 3-5:
TWO STEPS WITH VARIABLES ON BOTH SIDES OF EQUATIONS
simplify each side of the equation first

Then use the ADDITIVE INVERSE PROPERTY to move variable to the other side

Usually, we try to move the smaller coefficient to the larger because sometimes that avoids negative coefficients

But that is not always the case, and you may move whatever side you choose.
EXAMPLE:
3y - 10 - y = -10y + 12
2y - 10 = -10y + 12
+10y +10y
12y - 10 = 12
+ 10 +10
12y = 22
12 12
y = 11/6

Math 6H Period 3, 6 & 7

Writing Mathematical Equations 2-2

The process of writing equations really is just writing two equal expressions and joining them by an equals sign. The words "is" "equals" or "equal to" all indicate that two phrases NAME the same number.
The equals sign is the VERB in a mathematical sentence-- without it you have a mathematical expression!!
Eight increased by a number x is equal to thirty-seven.
In translating this mathematical sentence, I always start but placing the equals sign directly under the words "is equal"
so my first step would be

Eight increased by a number x is equal to thirty-seven.
---------------------------------> = <------------------
Then I would translate each mathematical phrase separately.
Yes, thirty-seven is a mathematical phrase!!
8 + x = 37

Ten is two less than a number n
10 = n - 2

Twice a number w equals the sum of the number and four
2w = n + 4

Notice that when you are indicating multiplication the number (coefficient) ALWAYS is placed in front of the variable.

So three times a number b would be 3b
The only time you see the letter first-- is when you are looking for our ROOM-- which is P8... a mathematician did not label the room numbers!! :)

Sometimes we need to write an equation for a word sentence that involves measurements. MAKE SURE that each side of the equation uses the SAME UNIT of Measurement!!
For example. Write an equation for: The value of d dimes is $27.50
WE know that the value of d dimes is 10d cents( from our previous lesson) .. but $27.50 is in terms of dollars so we need to change it to cents . $27.50 is 2750 cents ... So our equation becomes 10d = 2750.

Pre Algebra Period 1

Multiplying & Dividing Integers 1-9

They have the SAME rules!!!!

Math Book Rules:
If you have 2 signs that are the SAME -
answer is POSITIVE
If you have 2 signs that are DIFFERENT -
answer is NEGATIVE

Good Guy/Bad Guy Rules:
GOOD thing happens to GOOD person= GOOD = POSITIVE

BAD thing happens to BAD person = GOOD = POSITIVE
(they got what they deserved!)

GOOD thing happens to a BAD person = BAD = NEGATIVE
(we hate when good things happen to people who don't deserve it!)

BAD thing happens to a GOOD person = BAD = NEGATIVE
(we hate when that happens because it's so UNFAIR!)

Finger Rules:
If your index finger represents the negative sign, then if you have 2 negatives, you have the index fingers of both your left hand and your right hand and they make a plus sign!
If you just have one negative, it just stays negative because you don't have another finger to cross it!
If you have more than 2 negatives, you just keep using your index fingers to determine the sign.
We did this in class. It's fun! But once you get it, you won't need to keep doing it! (unless you want to keep having fun!!!)

WHAT IF THERE IS MORE THAN 2 SIGNS?
Use Aunt Sally Rules and go left to right
or
BE A SIGN COUNTER:
an ODD number of NEGATIVES = NEGATIVE
an EVEN number of NEGATIVES = POSITIVE

EXAMPLE: (-2) (-5) (-3) = -30 (odd number of negatives)
(2) (-5) (-3) = +30 (even number of negatives)
(2) (5) (-3) = -30 (odd number of negatives)

Averages ( or the Mean): Adding up all the numbers and dividing by the number of numbers