Algebra Honors ( Period 6)

Chapter 3-3 Rate of Change and Slope

We’ve already looked at the slope (m) of lines—today we will connect slope to the RATE of the CHANGE of the linear function (the line). the rate of change for a line is a CONSTANT… it is the same value EVERYWHERE on the line

This change, also know as the slope, is found by  finding the rise over the run between ANY 2 points.  rise/run

The rise is the change in y and the run is the change in x.

In a real world example, the rate of change is the UNIT RATE

If you are buying video games that are all the same price on BLACK FRIDAY, two data points might be

# of computer           Total
games                          cost

4                                  $156
6                                  $234

The slope or rate of change  is the  change in y/ the change in x

(234- 156)/ 6-4

78/2

or $39/ video game

Again, as long as the function is linear, or one straight line, it has a constant rate of change, or slope between ANY TWO POINTS

The constant rate of change, or slope, is the rise over the run—or the change in y over the change in x

or
y₂ – y₁/ x₂-x₁

Slope = rise/run ( rise over run)
=change in the y values/ change in the x values =
Difference of the y values/ Difference of the x values

Mrs Sobieraj uses “Be y’s first!” Be wise first!  meaning always start with the y vales on top (in the numerator)

TWO WAYS OF CALCULATING on a graph:

       1) Pick 2 points and use the following formula
Difference of the 2 y –values/ Difference of the 2 x-values
The formal is restated with SUBSCRIPTS on the x’s and y’s below: (memorize this) y₂ – y₁/ x₂-x₁  The subscripts just differentiate between point one and point two. You get to decide which point is point one or two. I usually try to keep the difference positive, if I can—but often, one of them will be negative and the other will be positive.

EXAMPLE:   ( 3, 6)  and (2, 4)    y₂ – y₁/ x₂-x₁       6-4/3-2 = 2/1 = 2

    2)Count the slope on the GRAPH using rise over run.
From the point (2,4) count the steps UP ( vertically) to (3,6): I get 2 steps
Now count how many steps over to the right (horizontally): 1 step
Rise = 2 and Run = 1 or 2/1 = 2

HORIZONTAL LINES  have only a y intercept (unless it’s the line y = 0 and then that is the x-axis) The equation of a horizontal line is y = b where b is a constant. Notice that there is NO X in the equation. For example y = 4 is a horizontal line parallel to the x-axis where the y value is always 4 What is the x value? All real numbers! Your points could be ( (3, 4) or ( 0, 4) or ( -10, 4)
Notice y is always 4! The constant rate of change  or slope is 0

If you take any 2 points on a horizontal line the y values will always be the same so the change ( or difference) in the numerator = 0.

EXAMPLE  y = 4

Pick any two points Let’s us ( 3,4) and (-10, 4)

(4 - 4)/ (3 - ^-10) becomes ( 4-4)/ 3 + 10 = 0/13 = 0

VERTICAL LINES ( which are NOT functions)  have only an x intercept ( unless it is the line x = 0 and then it is the y-axis) The equation of a vertical line is x = a, where a is a constant. Notice that there is NO Y in this equation.

EXAMPLE: x = 4

This is a vertical line parallel to the y axis 4 steps to the right of it. Pick any two points on this line Let’s use ( 4, -1) and (4, 7)

This time the change in y is -1  - 7 = -8

and the change in x is 4 -4 = 0

BUT -8/0 is UNDEFINED

Make sure you write undefined for the slope!

Finding a Missing Coordinate if you know 3 out of 4 values and the Slope

Say you know the following:
(1,4) and (-5, y) and the slope is given as 1/3

Find the missing y value

Use the slope formula

Change in y/ change in x

(y – 4)/- 5 – 1  and you know that the slope is 1/3

That means

(y – 4)/- 5 – 1   = 1/3

(y – 4)/-6 = 1/3

Solve

3(y -4)= -6

3y – 12 = -6

 y = 2

Or you could have divide both sides by 3 FIRST

y - 4 = -2

y = 2

Algebra (Periods 1 & 4)

Chapter 3-3 Rate of Change and Slope

We’ve already looked at the slope (m) of lines—today we will connect slope to the RATE of the CHANGE of the linear function (the line). the rate of change for a line is a CONSTANT… it is the same value EVERYWHERE on the line

This change, also know as the slope, is found by  finding the rise over the run between ANY 2 points.  rise/run

The rise is the change in y and the run is the change in x.

In a real world example, the rate of change is the UNIT RATE

If you are buying video games that are all the same price on BLACK FRIDAY, two data points might be

# of computer           Total
games                          cost

4                                  $156
6                                  $234

The slope or rate of change  is the  change in y/ the change in x

(234- 156)/ 6-4

78/2

or $39/ video game

Again, as long as the function is linear, or one straight line, it has a constant rate of change, or slope between ANY TWO POINTS

The constant rate of change, or slope, is the rise over the run—or the change in y over the change in x

or
y₂ – y₁/ x₂-x₁

Slope = rise/run ( rise over run)
=change in the y values/ change in the x values =
Difference of the y values/ Difference of the x values

Mrs Sobieraj uses “Be y’s first!” Be wise first!  meaning always start with the y vales on top (in the numerator)

TWO WAYS OF CALCULATING on a graph:

       1) Pick 2 points and use the following formula
Difference of the 2 y –values/ Difference of the 2 x-values
The formal is restated with SUBSCRIPTS on the x’s and y’s below: (memorize this) y₂ – y₁/ x₂-x₁  The subscripts just differentiate between point one and point two. You get to decide which point is point one or two. I usually try to keep the difference positive, if I can—but often, one of them will be negative and the other will be positive.

EXAMPLE:   ( 3, 6)  and (2, 4)    y₂ – y₁/ x₂-x₁       6-4/3-2 = 2/1 = 2

    2)Count the slope on the GRAPH using rise over run.
From the point (2,4) count the steps UP ( vertically) to (3,6): I get 2 steps
Now count how many steps over to the right (horizontally): 1 step
Rise = 2 and Run = 1 or 2/1 = 2

HORIZONTAL LINES  have only a y intercept (unless it’s the line y = 0 and then that is the x-axis) The equation of a horizontal line is y = b where b is a constant. Notice that there is NO X in the equation. For example y = 4 is a horizontal line parallel to the x-axis where the y value is always 4 What is the x value? All real numbers! Your points could be ( (3, 4) or ( 0, 4) or ( -10, 4)
Notice y is always 4! The constant rate of change  or slope is 0

If you take any 2 points on a horizontal line the y values will always be the same so the change ( or difference) in the numerator = 0.

EXAMPLE  y = 4

Pick any two points Let’s us ( 3,4) and (-10, 4)

(4 - 4)/ (3 - ^-10) becomes ( 4-4)/ 3 + 10 = 0/13 = 0

VERTICAL LINES ( which are NOT functions)  have only an x intercept ( unless it is the line x = 0 and then it is the y-axis) The equation of a vertical line is x = a, where a is a constant. Notice that there is NO Y in this equation.

EXAMPLE: x = 4

This is a vertical line parallel to the y axis 4 steps to the right of it. Pick any two points on this line Let’s use ( 4, -1) and (4, 7)

This time the change in y is -1  - 7 = -8

and the change in x is 4 -4 = 0

BUT -8/0 is UNDEFINED

Make sure you write undefined for the slope!

Finding a Missing Coordinate if you know 3 out of 4 values and the Slope

Say you know the following:
(1,4) and (-5, y) and the slope is given as 1/3

Find the missing y value

Use the slope formula

Change in y/ change in x

(y – 4)/- 5 – 1  and you know that the slope is 1/3

That means

(y – 4)/- 5 – 1   = 1/3

(y – 4)/-6 = 1/3

Solve

3(y -4)= -6

3y – 12 = -6

 y = 2

Or you could have divide both sides by 3 FIRST

y - 4 = -2

y = 2

Algebra Honors ( Period 6)

Chapter 3-2 Solving Linear Equations by Graphing

We need LOTS of graph paper. YOU must graph on graph paper—using a ruler or a straight edge! Make sure to label your x and y axes!  Put arrows on them!

Linear function: A line in the format:

f(x) = x or y = x

This is called the PARENT GRAPH

This parent graph has a FAMILY of GRAPHS related to it that has similar characteristics but is in someway different 
The Slope is Different
The y intercept is Different

The ROOT of the function or line is the X – intercept of the graph  It is called a ZERO on the graphing calculators. To find the root, find the value of x that makes the equation true when the y value is 0.

Linear functions ( equations) have at most 1  ROOT ( solution) because once the line intercepts the  x-axis it cannot curve around and intercept it again

Again, the FUNCTION is the ENTIRE GRAPH … and is in the form f(x) = x   or y = x

The linear EQUATION related to the function is only concerned with one value, the x- intercept.  The y value would be ZERO on the x-axis so you set the function = 0 and solve!

Here are the synonyms:
x-intercept= the solution = the root = the zero

 If you graph the function
f(x) = 2x – 8, you will see the x intercept is ( 4, 0)

Therefore, the solution, the root, the zero of the function is 4.

set f(x)  or y = 0
2x – 8 = 0
2x = 8
x = 4

You can solve a linear function 2 ways:

Graphically- Graph the function and read the x-intercept
Algebraically- Set y or f(x) equal to 0 and solve for x

Notice you are solving multi-step equation but now the value means that you found the root of the function. How often will you be able to read the exact answer from a graph? NOT OFTEN!

Therefore we usually solve for the root Algebraically!

Example:
Find the root or zero of y = 20- .75x

set y = 0

0 = 20 - .75x

x = 26 2/3

Algebra Honors ( Period 6)

Chapter 3-1 Graphing Linear Equations

There are FOUR types of linear graphs and this chapter begins with an  OVERALL, BIG  picture

Positive Slope- slants up from left to right
Negative Slope- slants down from left to right
Horizontal line- stays flat from left to right ( constant function)
Vertical Line- stays straight up and down ( Not a function—why??)

Somethings to look for:
Domain
Range
End behavior
Intercepts
Extrema
Positive/Negative
Increasing/Decreasing
Symmetry

A Linear Equation is an equation that forms a line when it is graphed. Linear equations are often written in the form Ax + By = C
This is called standard form. In this equation C is called  a constant  Ax and By are variable terms.

A ≥ 0
A and B BOTH cannot be 0
A, B,and C are ALL integers with a GCF= 1

If you see a term such as xy attached to together it cannot be a linear equation. If the exponent on a variable is different than the understood 1,  it is not a linear equation

in 3x + 2y = 5

A = 3
B = 2
C = 5

In x = -7 ( Yes that is in Standard Form)

A = 1
B = 0
C = -7

Identify Linear Equations

Determine whether each equation is  a linear equation. Write the equation in Standard form
y = 4 – 3x   YES

To put this equation in standard form, we need to move the -3x term to the other side, using the  Addition Property of Equality and the Additive Inverse Property.  So that the x and y values are on the SAME side and the constant is always on the other side to the right of the equal sign.
3x + y = 4
A = 3
B = 1
C = 4

6x –xy = 4   NO

the term xy has two variables the equation cannot be written in AX + By = C . It is not a linear equation

(1/3)y = -1  Yes

It becomes y = -3

A= 0
B = 1
C = -3

A linear equation can be represented on a coordinate graph. The x- coordinate of the point at which the graph of the equation crosses the x-axis is called the x-intercept. The y- coordinate of the point at which the graph of the equation crosses the y-axis is called the y-intercept.

The graph of  linear equation has AT MOST one x- intercept and ONE y-intercept ( unless it is the equation x = 0, which is the y-axis or y = 0, which is the x-axis. In those two special cases every number is a y-intercept or an x-intercept, respectively)

Real World Example  Swimming Pool Page 157 in your textbook

A swimming pool is being drained at a rate of 720 gallons per hour. The table on Page 157 shows the function relating the volume of water in a pool and the time in hours that the pool has been draining.

Find the x- and y- intercepts on the graph of the function.

Looking at the table we see that the x intercept is 14 ( that is when y is 0)
and the y-intercept is 10,080 ( that is the value of y, when x = 0)

Describe what the intercepts mean in this situation: This should remind you of our unit at the beginning of the year!

The x intercept 14 means that after 14 hours the pool is completed drained because it has a volume of 0 gallons!

The y- intercept of 10,080 means that the pool contained 10,080 gallons of water at time 0 ( or before it started to drain)

Graph by Using Intercepts
Graph 2x + 4y = 16 using just the x-intercept and y-intercept

2x + 4(0) = 16   replace y with 0 (or as taught in class cover over the y value and solve)

2x = 16 so x = 8 ( when y = 0) ( 8,0)

This means the graph intersects the x-axis at (8,0)

Now

2(0) + 4y = 16  replace x with 0 ( or as taught in class- cover over the x value and solve)

4y = 16
y = 4  ( when x = 0)  ( 0, 4)

This means the graph intersect the x-axis at (0, 4)

Plot these two point and draw a line through them

Notice that this has both an x- intercept and  y-intercept

Some lines have only an x- intercept and NO y-intercept  or vice versa

y = b is a horizontal line that has only a y- intercept (unless b=0)

The graph of x = a is a vertical line that has only an x- intercept (unless a = 0)

Lines that are neither vertical or horizontal cannot have more than one x- and/or y-intercept.

Graphing Using an XY Table

Another way to graph is choosing random x values , plugging those into the equation to find the corresponding y values, and graphing those points you found.

Although 2 points determine a line, it is always best to find 3 points so that you are sure you did not make a mistake on either of the first two points.

If the coefficient of x is a fraction, select a value that is  multiple of the denominator so hopefully you won’t end up with fractions to graph! 

Algebra (Periods 1 & 4)

Chapter 3-1 Graphing Linear Equations

There are FOUR types of linear graphs and this chapter begins with an  OVERALL, BIG  picture

Positive Slope- slants up from left to right
Negative Slope- slants down from left to right
Horizontal line- stays flat from left to right ( constant function)
Vertical Line- stays straight up and down ( Not a function—why??)

Somethings to look for:
Domain
Range
End behavior
Intercepts
Extrema
Positive/Negative
Increasing/Decreasing
Symmetry

A Linear Equation is an equation that forms a line when it is graphed. Linear equations are often written in the form Ax + By = C
This is called standard form. In this equation C is called  a constant  Ax and By are variable terms.

A ≥ 0
A and B BOTH cannot be 0
A, B,and C are ALL integers with a GCF= 1

If you see a term such as xy attached to together it cannot be a linear equation. If the exponent on a variable is different than the understood 1,  it is not a linear equation

in 3x + 2y = 5

A = 3
B = 2
C = 5

In x = -7 ( Yes that is in Standard Form)

A = 1
B = 0
C = -7

Identify Linear Equations

Determine whether each equation is  a linear equation. Write the equation in Standard form
y = 4 – 3x   YES

To put this equation in standard form, we need to move the -3x term to the other side, using the  Addition Property of Equality and the Additive Inverse Property.  So that the x and y values are on the SAME side and the constant is always on the other side to the right of the equal sign.
3x + y = 4
A = 3
B = 1
C = 4

6x –xy = 4   NO

the term xy has two variables the equation cannot be written in AX + By = C . It is not a linear equation

(1/3)y = -1  Yes

It becomes y = -3

A= 0
B = 1
C = -3

A linear equation can be represented on a coordinate graph. The x- coordinate of the point at which the graph of the equation crosses the x-axis is called the x-intercept. The y- coordinate of the point at which the graph of the equation crosses the y-axis is called the y-intercept.

The graph of  linear equation has AT MOST one x- intercept and ONE y-intercept ( unless it is the equation x = 0, which is the y-axis or y = 0, which is the x-axis. In those two special cases every number is a y-intercept or an x-intercept, respectively)

Real World Example  Swimming Pool Page 157 in your textbook

A swimming pool is being drained at a rate of 720 gallons per hour. The table on Page 157 shows the function relating the volume of water in a pool and the time in hours that the pool has been draining.

Find the x- and y- intercepts on the graph of the function.

Looking at the table we see that the x intercept is 14 ( that is when y is 0)
and the y-intercept is 10,080 ( that is the value of y, when x = 0)

Describe what the intercepts mean in this situation: This should remind you of our unit at the beginning of the year!

The x intercept 14 means that after 14 hours the pool is completed drained because it has a volume of 0 gallons!

The y- intercept of 10,080 means that the pool contained 10,080 gallons of water at time 0 ( or before it started to drain)

Graph by Using Intercepts
Graph 2x + 4y = 16 using just the x-intercept and y-intercept

2x + 4(0) = 16   replace y with 0 (or as taught in class cover over the y value and solve)

2x = 16 so x = 8 ( when y = 0) ( 8,0)

This means the graph intersects the x-axis at (8,0)

Now

2(0) + 4y = 16  replace x with 0 ( or as taught in class- cover over the x value and solve)

4y = 16
y = 4  ( when x = 0)  ( 0, 4)

This means the graph intersect the x-axis at (0, 4)

Plot these two point and draw a line through them

Notice that this has both an x- intercept and  y-intercept

Some lines have only an x- intercept and NO y-intercept  or vice versa

y = b is a horizontal line that has only a y- intercept (unless b=0)

The graph of x = a is a vertical line that has only an x- intercept (unless a = 0)

Lines that are neither vertical or horizontal cannot have more than one x- and/or y-intercept.

Graphing Using an XY Table

Another way to graph is choosing random x values , plugging those into the equation to find the corresponding y values, and graphing those points you found.

Although 2 points determine a line, it is always best to find 3 points so that you are sure you did not make a mistake on either of the first two points.

If the coefficient of x is a fraction, select a value that is  multiple of the denominator so hopefully you won’t end up with fractions to graph! 

Math 6A ( Periods 2 & 5)

Dividing Integers 11-5

Essential Question: Is the quotient of two integers positive, negative or zero? How can you tell?

The rules for multiplying integers apply to dividing integers.

Look at your Fact Families

(2)(-3) = (-6)

so we know

-6 ÷ 2 = -3  and   -6 ÷ -3 = 2

The same life story applies as told about multiplying integers!  Hmmm… what was that story?

-15 ÷  3  ( integers with different signs) = - 5

12 ÷ 4 ( integers with same signs) = 3

12 ÷ (-3) ( integers with different signs) = -4

-12 ÷ (-4) ( integers with same signs) = +3

0 ÷  (-15) = 0

IMPORTANT:
What’s the general rule about dividing two integers with the same sign?
When dividing integers with the same sign, divide their absolute value and make the quotient positive.

What’s the general rule about dividing two integers with different sign?
When dividing integers with different sign, divide their absolute value and make the quotient negative.

-49/7 = -7

21/-3 = -7

Evaluating an Expression

Evaluate 10 - x² ÷ y  when x = 8 and y = -4
Substitute each value using {{HUGS}} 

10 – (8)² ÷ (-4)

10 -64÷ (-4)

10 –(-16)

10 + 16 = 26

Real Life Application

You measure the height of the tide using the support beams of a pier. Your measurements are  59 inches at 2PM and 8 inches at 8PM

What is the mean hourly change in the height?
How many hours are between  2 PM and 8 PM ?   6 hours
Use a model to help you solve this problem

The mean change in the height of the tide is -8.5 inches per hour

What is the mean of these integers:
3, -10, -2, 13, 11
First- add all the positives 3 + 13 + 11 = 27
Second add all the negative -10 + -2 = -12
Add them together +15
To get the mean divide by the number of integers you started with ( which was 5)
15/5 = 3
-8 -14 ÷ 2 + 5
Make sure you follow the Order of Operations ! So
-8 -7 + 5 
-15 + 5 
-10
Any number divided by ZERO is UNDEFINED  (You must know that term!)
4/0 = undefined  
You cannot divide by ZERO
However, notice 0/4 = 0
4 + 3÷0   since 3÷0  is undefined  --> the entire expression  is UNDEFINED! 

Math 6A (Periods 2 & 5)

Multiplying Integers 11-4


3 ⋅ -2 = -6
Its really repeated addition
or
-2 + -2 + -2 which we learned a few sections ago was equal to -6.

The product of a positive integer and a negative integer is a negative integer.

The product of ZERO and any integer is ALWAYS ZERO!!
a⋅0 = 0

Math imitates life...and Karma(?)
What was the story I told in class... it applies to
Multiplication & Division ...
+ ⋅ + = +
- ⋅ + = -
+ ⋅ - = -
- ⋅ - = +



The product of -1 and any integer equals the opposite of that integer.
(-1)(a) = -a

The product of two negative integers is a positive integer

For a product with NO ZERO factors:
-->if the number of NEGATIVE factors is odd, the product is negative
-->if the number of NEGATIVE factors is even, then the product is positive
Be a sign counter-- count your negatives!
Every integer and its opposite have equal squares!!

Remember-- if its all multiplication use the Associative & Commutative Properties of Multiplication to make your work EASIER!!

Using Exponents

Evaluate (-2)²
Write (-2)² as repeated multiplication

(-2)² = (-2)(-2) = 4

Evaluate
-5²
Notice that the exponent is ONLY touching the 5 so this is really
-(5∙5) = -25

Evaluate
(-4)³  =(-4)(-4)(-4) = 16(-4) = -64

Algebra:
Evaluate the expression when a = -2, b = 3, and c = -8
Always substitute your actual number in with {{Hugs}} that is  (  )
Example 1 ab would be ( -2)(3) = -6
Example 2 │a²c│= │(-2)²(-8)│ = │ (4)(-8)│=│-32│= 32
Example 3 -ab³ –ac  = - (-2)(3)² – (-2)(-8) = -(-2)(27) –(16) = -(-54) – 16 = +54 – 16 = 38
Did you get that one?
Look at each step to see what was done along the way!

Number Sense:
Find the next two numbers in the patterns:
7, -28, 112, -448, …
First what is happening to each?
It looks like each consecutive integer is 4 times the opposite of the previous number.
So the next two integers would be  1792,  -7168

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="">

Math 6A ( Periods 2 & 5)

The Coordinate Plane 6-5

The Coordinate Plane is formed by the intersection of a horizontal number line ( x-axis) and a vertical number line ( y-axis)
They intersect at the origin (0,0) and the ORIGIN separates the Coordinate Plane into FOUR regions called QUADRANTS


A PAIR of numbers whose ORDER is important is called an   ordered pair
(ordered, pair) (x, y)

(2,3) is not the same as (3,2)

The two perpendicular lines are called axes.
The x-axis deals with the 1st number of the ordered pair and the y-axis deals with the 2nd number of the ordered pair.
The AXES meet at a point called the Origin (0,0)
The plane is called the coordinate plane

There are 4 quadrants, You MUST use Roman Numerals to name them!!
Quadrant I ---> both the x and y coordinates are positive
(x,y) (+,+)

Quadrant II --> the x coordinate is negative but the y is positive
(-x,y) (-,+)

Quadrant III -->. both the x and y coordinates are negative
(-x,-y) (-,-)

Quadrant IV --> the x coordinate is positive but the y coordinate is negative
(x,-y) (+,-)

Reflect a point in the x-axis
Use the same x-coordinate and take then opposite of the y- coordinate

Reflect a point in the y-axis
Use the same y-coordinate and take then opposite of the x- coordinate

Reflect a point in both axes:
1) x followed by y
First follow the steps above so for (2,1) reflect in x followed by y means
(2,1)--> (2, -1), then --> (-2,-1)  What do you notice? Both coordinates are now the opposites of what they started as!

Algebra Honors ( Period 6)

Solving Multi- Step Inequalities 5-3

Again you will use your equation skills

1.      Do Distributive Property first (if necessary) do it carefully

2.      Combine like terms on each side of the “WALL”

3.      “JUMP” the variables to one side of the wall—that is get all the variables on one side of the inequality by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term, constants second by adding or subtracting

4.      Do it just like a 2 step or 1 step equation but remember;  FLIP THE SYMBOL if you multiply or divide by a NEGATIVE to balance

5.      If you end up with the variable on the right side, SWITCH the SIDES and the SYMBOL

REMINDERS:

Set builder notation

Get familiar with the following notation

{x│ x≥ 5} is read “ x SUCH THAT c is greater than or equal to 5”

Checking your solutions is an important set. Many students skip this step! Checking the solutions is especially important with inequalities because the direction of the inequality sign is often changed when writing solutions in set builder notation.