Math 6A ( Periods 2 & 5)

 Subtracting Integers 11-3

The life story about someone who was so negative-- you wanted to take a little negativity away but since you can't do that you add a little positiveness-- works in math as well!!
Instead of subtracting ... "ADD THE OPPOSITE!"
We will prove it in class with our little red and yellow tiles... If you need to review, make your own out of red and yellow paper-- or whatever colors you want!!

In life-- to take away a little negative-- add some positive

To take away an integer... add its opposite.

Rule from our textbook
for all integers a and b

a - b = a + (the opposite of b) or
a - b = a + (-b)

Instead of subtracting.. "ADD THE OPPOSITE"
make sure you do the "check, check" ... you need to have two check marks... one changing the subtraction to addition and the other changing the sign of the 2nd number to its opposite.
NEVER TOUCH the sign of the first number

5 - ^- 2=
5 + ⁺ 2= 7

-2 - 5 =
- 2 + ^- 5 =
before I give you the answer look... we are looking at
-2 + ^-5
We are adding two negatives.. so we are back to the rules from Section 11-2...
when adding the same sign just add the number and use their sign so
- 2 + ^- 5 = -7

But what about -2 - ^-5 ?
adding the opposite, we get
-2 + ⁺ 5 .
Now, the signs are different so the rule from Section 11-2 is
ask yourself... "Who wins?" and "By how much?"

Okay, here the positive wins so I know the answer will be +
and by how much means.. to take the difference
5-2 = 3
so -2 + ⁺ 5 = +3
Do you need to put the + sign? No, but I like to in the beginning to show that I checked WHO WON!!

What about
-120 - ^-48?
add the opposite
-120 + ⁺ 48 follows the
Different signs rule... so
ask yourself
Who wins? answer: the negative.. so I know the answer will be negative..
And "By How Much?" take the difference 120-48 = 72
so
-120 + ⁺ 48 = -72
or

-120 - 48
-120 + (-48) = -168

NEVER EVER CHANGE THE FIRST NUMBER'S SIGN!!

WALK THE LINE, the number line-- that is!!
Remember... Attitude is such a little thing... but it makes a BIG difference!!
Always start with a positive attitude!!
When you walk the line, Which way are you always facing when you start???

Always attempt to get everything into addition so we can follow the rules of Section 11-2 Adding Integers.
1) SAME SIGN rule---> just add the numbers and use their sign
-4 + -5 = -9
2) DIFFERENT SIGNS rule
ask yourself those 2 important questions
a) Who wins? (answer is either negative or positive)
b) By how much? (Take the difference)

-15 + 7 -15 - 4 - 7
What do you see?
I see one zero pair... so cross them out!
left with -15 -15 -14 -7
WOW-- they are all negative so just add them and use their sign ( rule from Adding Integers!)

Some good strategies to follow:
1. Look at the problem
2. Make sure to write the problem down correctly
3. Change to an addition problem by  "ADD the OPPOSITE"
4. Look for ZERO PAIRS ( cross them out)
5. Add like signs ( add all the positives together; add all the negatives together)
6. Take the difference
7. Work across from left to right!

Algebra Honors ( Period 6)

Inequalities Involving Absolute Value 5-5

The inequality │x│< 3 means that the distance between x and 0 is less than 3

Graph:

so x > -3 and x <  3

The set builder notation or solution set is {x │ -3 < x < 3}

When solving absolute value inequalities there are two cases to consider:

Case 1 The expression inside the absolute value symbols is nonnegative

Case 2  The expression inside the absolute value symbol is negative

The solution is the intersection of these two cases.

When the absolute value inequality is less than… I think of “less thAND” … it is the intersection of two parts. It is the “YO”  we talked about in class

│m + 2 │ < 11

Rewrite  │m + 2 │ < 11       for both the above cases

m + 2 < 11                   and            -(m+2)  < 11

m  < ; 9                   this one  becomes m + 2 >; -11

m < ; 9     and     m >  -13

or think m + 2 < 11   AND  -11 < m + 2

The solution set is { m │ -13 < m <; 9}

│y -1│ <  -2    WAIT… this can NEVER be TRUE. There is no solution The solution set is the empty set.  {  }  or

When the absolute value inequality is a greater than… I think of  “greatOR”  it is the union
of two parts..  It is the “DORKY DANCER we talked about in class… going one way and then the other.
│x│ > 3 means that the distance between x and 0 is greater than 3…

Graph:

so x <  -3 OR  x >; 3 The solution set is { x │ x < -3 or x >  3}

We must consider two cases always
Case 1 The expression inside the absolute value symbols is nonnegative
Case 2 The expression inside the absolute value symbols is negative

Solve      │3n + 6│≥ 12

Case 1
3n + 6 is non negative

3n + 6 ≥ 12              n ≥ 2        That was easy

Case 2
3n + 6 is  negative

The book sets it up  -(3n + 6) ≥ 12

That would mean    3n + 6 ≤ -12

n  ≤ -6     so

 n  ≥2   OR   n  ≤ -6

The solutions set is { n │ n  ≥2   OR   n  ≤ -6}

Graph   

Math 6A ( Periods 2 & 5)

Adding Integers 11-2

Rules: The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.

So- if the two numbers have the same sign, use their sign and just add the numbers.

-15 + -13 = - 28

-10 + -4 = -14

Rules: The sum of a positive integer and a negative integer is :

POSITIVE… IF the positive number has a greater absolute value

NEGATIVE… IF the negative number has a greater absolute value

ZERO… IF both numbers have the same absolute value

Think of a game between two teams- The POSITIVE TEAM and The NEGATIVE TEAM.

30 + -16 … ask yourself the all important question…
“WHO WINS?
in this case the positive and then ask
“BY HOW MUCH?”
take the difference 14

14 + - 52…
“WHO WINS?”
the negative… “BY HOW MUCH?”
38
so the answer is -38


(-2 + 3) + - 6 you can work this 2 ways

(-2 + 3) + - 6 = 1 + -6 = -5 or
using all the properties that work for whole numbers
Commutative and Associative properties of addition
can change expression to (-2 + -6) + 3 or -8 + 3 = -5 you still arrive at the same solution.

You want to use these properties when you are adding more than 2 integers.
Additive Inverse Property ( Zero Pairs)
Sum of an integer and its opposite ( its additive inverse) is ZERO
First look for zero pairs—you can cross them out right away!!
3 + (-3) = 0
-9 + 9 = 0

Then you can use C(+) to move the integers around to make it easier to add them together rather than adding them in the original order. In addition, you can use A(+) to group your positive and negative numbers in ways that make it easier to add as well.

One surefire way is to add all the positives up… and then add all the negatives up.
At this point ask yourself that all important question… WHO WINS? …
use the winner’s sign..
and then ask yourself..
BY HOW MUCH?

example:

-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13

Taking a good scan of the numbers, do you see any zero pairs?
YES—so cross them out and you are left with
-4 + 5 + (-4) + 13
add your positives 5 + 13 = 18
add your negatives and use their sign – 4 + -4 = -8

Okay, Who wins? the positive
By how much? 10
so
-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13 = 10

if a= 4, b = -5 and c = -8
whats
a + b
When you substitute in, use HUGS   (  ) parentheses around what you are substituting in for the variables.
(4) + )  Look at the different signs and use the rules above
Find the number with the greatest absolute value-- then take the difference
Use the sign of the number with the greater abs value.

What about  -b + c
Read -b as "the opposite of b"  and you won't go wrong!
-(-5) + -8
wait that's
5 + (-8) = -3
The opposite of the opposite of 5 =5
WOW!
Notice

 │a + b + c │ =  │4 + (-5) + (-8) │=  │-9 │ = 9

That is different from

 │a │ +│b │+│c │ =  │4 │+  │-5 │+ │-8 │  = 4 + 5 + 8 = 17

Math 6A ( Periods 2 & 5)

Integers and Absolute Value 11-1

The following numbers are integers:

… -3, -2, -1, 0, 1, 2, 3,…

The absolute value of an integer is the distance between the number and zero on a number line. The absolute value of a number a is written │a│

Find the absolute value of 2 and graph it on the number line

Find the absolute value of -3 and graph it on a number line

Remember plotting a point involves actually putting a “closed dot” on the number line at the point you are graphing! The numbers go BELOW the number line—not above it

Are absolute values always positive?  What about ZERO?

When you write the notation for the absolute value, it means “take the absolute value of the number inside the symbols”

A number line can be used to compare and order integers.
Numbers to the left are LESS THAN numbers to the right.
Numbers to the right are MORE THAN numbers to the left.

Compare 1 and  │-4│     One way is to graph them on the number line.

The freezing point is the temperature at which a liquid becomes a solid.

Substance	Freezing Point ⁰C
Butter	35
Airplane Fuel	-53
Honey	-3
Mercury	-39
Candle Wax	55

Which substance has the lowest freezing point?

Is the freezing point of mercury or butter closer to the freezing point of water,  0⁰C?

The freezing point of water is  so you can use absolute values to compare
Mercury:  │-39│= 39                       Butter: │35│= 35

Because 35 is less than 39, the freezing point of butter is closer to the freezing point of water.

Good thinking: Determine whether the statement is true of false

A. If x < 0 then  │x│ = -x     ( THIS IS TRUE!... why?)

B. The absolute value of every integer is positive. (THIS IS FALSE… why?)

Algebra Honors ( Period 6)

Solving Compound  Inequalities 5-4

Solving  Compound  Inequalities 5-4

Inequalities containing AND….  When considered together two inequalities such as

h  ≥  52  AND  h  ≤  72 form a compound inequality.

A compound inequality contains AND is only true if BOTH inequalities are true.

Its graph is where the graphs of the two inequalities overlap. This is called the intersection of the two graphs.

For instance,

x ≥ 3 is graphed

x< 7 is graphed

x ≥ 3  and  x < 7
3 ≤ x <7 o:p="">

The statement   3 ≤ x < 7 can be read as “x is greater than or equal to 3 and less than 7”  or “x is between 3 and 7 including 3” or “3 is less than or equal to x which is less than 7.”

Solve  -2  ≤ x – 3 < 4 Then graph the solution set.

First express   -2  ≤ x – 3 < 4  using AND

-2 ≤ x -3      AND        x -3 < 4   Write the inequalities separately

Solve both of them!

1 ≤ x   and x < 7

so

the solution set is

{ x│1≤ x < 7}

Now graph the solution set

Inequalities containing OR….  A compound inequality containing or is TRUE if AT LEAST one of the inequalities is TRUE. Its graph is the UNION of the graph of the two inequalities 

The human ear can only detect sounds between the frequencies 20 Hertz and 20,000 Hertz. Write and graph a compound inequalities that describes the frequency of sounds humans cannot hear.

Let f = the frequency

f ≤ 20  OR f ≥ 20,000

Graph:

Notice that the graphs do NOT intersect. Humans cannot hear sounds at a frequency less than 20 Hertz or greater than 20,000 Hertz. The compound inequality is  { f│ f ≤ 20 or   f ≥ 20,000}

Intersections and Unions:

The graphs of compound inequalities containing AND will be an intersection. The graphs of compound inequalities containing OR will be a union.

The Triangle Inequality Theorem states that the sum of the measures of any two sides of  a triangle is greater that the measure of the third side!

Write and solve three inequalities to express the relationships among the measure of the sides of the triangle show here

           

x + 9 < 4      x < -5 (wait a minute—it can’t be negative—or even 0)

x + 4 < 9    x < 5

4 + 9 13

Since x > 5 AND x < 13 the solution set includes the whole numbers… 6, 7, 8, 9, 10, 11, 12

The compound inequality is   5< x <13 o:p="">