Algebra Honors ( Period 6)

Chapter 4-4 Parallel and Perpendicular Lines 

Parallel Lines:

2 lines that are parallel to each other have the SAME SLOPE

Example:

y = 2x – 10

y = 2x + 4

Those two lines are  parallel because they both have a slope of 2

BE CAREFUL:
To be parallel they must have the SAME SLOPE BUT DIFFERENCE y-INTERCEPTS

If they have the same m and the same b then they are COLLINEAR
(or they COINCIDE) they are the same line and NOT parallel

Perpendicular Lines

2 lines that are perpendicular to each other have SLOPES that are:

Opposite signs

and

Reciprocals

y = 2x – 10 is perpendicular to y = (-1/2)x + 4

This is another twist to our Mystery Line Puzzles…

If you know that the mystery line is parallel  or perpendicular to another given line they you know the mystery line’s slope!!

EXAMPLE 1:

Your mystery line has a point of ( 2, -5) and is PARALLEL to the line y = 2x + 4

So you know that the mystery line’s slope
because it is the same as the given line  (m = 2)

Substitute the slope and the point given for the mystery line and solve for b
OR use Point Slope Form and solve easily!
This is why I like Point Slope: y + 5 = 2(x – 2)

y + 5 = 2x – 4

y = 2x – 9

If I used Slope Intercept form or y = mx + b

-5 = 2(2) + b

-5 = 4 + b

b = -9

Plug everything back in

y = 2x – 9

Same results—but I think that it takes longer ( students also tend to FORGET to put everything together)

EXAMPLE 2:

Your mystery line has a point of ( 2, -5) and is perpendicular to the line y = 2x + 4

So you know that the mystery line’s slope is the opposite sign and the reciprocal of the given line
(since the given line’s slope is 2, the mystery line’s slope must be –½ )

Substitute the slope and the point given on the mystery line and solve for b OR use Point Slope Form and solve easily!  

y + 5 = (-½)(x -2)

y + 5 =  -½x + 1

y =  -½x - 4

If I used Slope Intercept form or y = mx + b

-5 = (-½)(2) + b

-5 = -1 + b

-4 = b

Plug everything back

 y = -½x - 4

Algebra Honors (Period 6)

Chapter 4-2 Writing Equations in Slope-Intercept Form 

We usually use the slope- intercept form of the line as our  ”template”

We know that y = mx + b so we can substitute that in what we know (what the problem gives us as information) and solve for whatever we are missing

It helps to memorize this little rhyme (Mrs Sobieraj made it up!)

Oh mystery line,
What could you be?
If I could just find you,
 y = mx + b
First I’ll find m,
Then I’ll find b
Then I’ll put it all together
And I will see:
y = mx + b

The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:

1) Do I have the slope (m)? If not find it by using the slope formula or counting it if you have the graph—(carefully pick two sets of integer points)

2) Do I have the y- intercept (b) ? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.

3) Remember: Put it all together in ONE equation at the end!

There are FIVE general cases of mystery lines

First Case:
You are given the slope and the y intercept (that is the easiest case)
For example:  you are given m = 3/2 and b = - 7/5

Just plug in to the generic slope intercept equation

y = (3/2)x – 7/5

Second Case:
You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope ( make sure they are integers) If the y intercept is not an integer—then follow the FOURTH CASE (below) completely!

Put the information together in y = mx + b form

Third Case:
You are given a point and the slope and need to find the intercept ( b)
Example: ( 3, 1)is a point on the line and m = 2

Plug in the point and the slope and find b

That is, start with y = mx + b
You have a point ( 3, 1) plug it in to that equation:

1 = (2)(3) + b
1 = 6 + b

-5 = b or
b = -5

Now put it altogether with the given slope of m = 2 and the y intercept ( b) which you just found

y = 2x – 5

Fourth Case:
You are given a point and the y intercept and need to find the slope > Let’s use the point ( 3, 1) again but this time you are given b = 2

Again you can use y = mx + b . This time, however you are solving for m ( the slope)
1 = 3m + 2
-1 = 3m

-1/3 = m
m = -1/3

Again, NOW put it all together with the given intercept and the slope you just found
 y = (-1/3)x + 2

Fifth Case:
You are given 2 points on a line and need to find the slope and the y intercept
Example: ( 1, 3) and ( -2, -3) are 2 points on the line

You first need to find the slope using the formula
m = change in y/ change in x

m = (-3 -3)/(-2-1)  or  (3--3)/(1--2)  which really is (3+3)/(1+2) or
6/3 = 2

Now plug the slope in with one ( you get to pick—it will work with either) of the points and find the intercept, b

3 = 2(1) + b

3 = 2+b

b = 1

Finally put it all together

y = 2x +1

Chapter 4-3 Point Slope Form 

There is another way to solve for a line other than using slope-intercept form  Although most people use the slope- intercept form for all cases, the POINT-SLOPE Form  is actually easier—everything is built into the format!  You do not need to “put it altogether” at the end!

MY FAVORITE!!

It gives you exactly a point and the slope—just by looking at it!

You do not need to do anything BUT graph!!

Point- Slope Form of a line:

y-y₁ = m ( x- x₁)

Example: ( 3, 1) is a point on the line and m = 2

y – 1 = 2( x – 3)

What you have now is point-slope form of the line!

You can graph easily starting at (3, 1) and going up 2 and to the right 1

You can also simplify this and get the slope- intercept form of the line:

y - 1 = 2x – 6

y = 2x – 5

If you are trying to link the Slope-Intercept form to the Point-Slope form of the line:  The point slope version eliminates one step from using the slope intercept form

In the slope intercept form, you plug in the point and the slope and solve for b—
AND THEN rewrite the equation using the intercept that you found.

In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!

IF you don’t have the slope, you will need to compute it with the formula—just like you did with Slope Intercept FORM

The biggest difference is that any point is plugged into this form, while  the Slope- Intercept Form focuses on the ONE specific point on the y axis 

Algebra Honors ( period 6)

Chapter 3 -5 Arithmetic Sequences as Linear Functions

An arithmetic sequence is an ordered list of numbers ( called terms) where there is a common difference (d) between consecutive terms. the common difference can either be positive (increasing)  or negative ( decreasing)

Because an arithmetic sequence has a constant difference, it is a linear function. There is a formula using this common difference to find the equation of any arithmetic sequence:

a_n = a₁ + ( n-1)d

Notice that it is saying that any term in the sequence, a_n,  can be found by adding 1 less than the number of terms of the common difference to the first term, a_1.   Why 1 less than the number of the terms you want?

Take the following arithmetic sequence:

…17, 21, 25,29, 33….

If it’s the 2^nd term, there is only 1 common difference of 4 between the 2 terms. If it is the 3^rd term, there would be 2 common differences of 4 between the 3  terms, etc.

n is always positive because it represents the number of terms and that can’t be negative.

Where  a_n  represents any term in the sequence and a₁ represents the first term in a sequence, n represents the number of the terms in a sequence and d represents the common difference between consecutive terms in a sequence.

This is called an EXPLICIT FORMULA You can explicitly find any term number in the sequence as long as you know the 1^st term and the common difference. For example if the the common difference is 4 and the first term is 1 and you are trying to find the 27^th term

a_n = a₁ + ( n-1)d

a₂₇ = 1 + ( 27-1)4

a₂₇ = 1 + ( 26)4

a₂₇ = 1 + ( 104

a₂₇ = 105

You can also find the next term in the sequence if you know the
RECURSIVE FORUMULA

This is a Function Rule that tells you what the relationship between consecutive terms is. For example if the common difference is 4 and your last term is 101 the next term is 105—without knowing any of the other preceding terms. The terms in the sequence are shown as a list with 3 periods (called an ellipsis) at the end showing that it continues infinitely.

For example, the arithmetic sequences for the above example was

1, 5,9, 13, …

Graphing the terms of an arithmetic sequence shows that it is a linear function

a_n = 1 + ( n-1)4

Simplify

a_n = 1 + ( n-1)4

a_n = 1 + 4n -4

a_n = 4n -3

Now just substitute y for a_n   and x for n

y = 4x – 3

Notice that d is now the slope and also notice that the domain of the sequence is the natural numbers (the counting numbers) because you can’t have a negative term number!

Graph 3 points using 1, 2, 3 for the first 3 terms

(1, 1) ( 2,5) (3, 9)

Algebra Honors ( Period 6)

Chapter 3-4 Direct Variation

We’ve learned that the unit rate is the constant rate of change in a linear relationship and that it’s the slope of a line when it’s graphed. We’ve also learned that if a graph of an equation goes through the origin (0,0)  it’s proportional  and the ratio of any y value to it’s x value is a constant (which turns out to be the unit rate or constant rate of change or slope of the line)

When the linear relationship is proportional, we say it’s a DIRECT VARIATION. Now the constant rate of change, the slope, the unit rate, is called the CONSTANT OF VARIATION or the CONSTANT OF PROPORTIONALITY

This is not a new concept. IT IS  just NEW VOCAB!

We also say: y varies directly (constantly) with x.

The slope is now replaced by the letter k instead of m

Finding the equation of a line that is proportional

Find k (the slope) by counting the rise/run of the graph

Write the equation using the format  y = kx

Notice: if you always pick the origin as the point to count rise/run from—the slope (k) is always just y/x

In a word problem, if it says one amount VARIES DIRECTLY with another, you know that the origin is one of the points!!

You also know that the equation is y = kx
YOU just need to find k
and k is y/x of any point OTHER THAN THE ORIGIN

A babysitting example

The amount of money earned  VARIES DIRECTLY with the time worked.

THINK: the graph and equation go through (0,0)

THINK: Any other point will give you the slope, or constant of proportionality, or unit rate ( all the same thing) SO you only need one additional point.

We are given that she earns $30 for 4 hours. Find the equation.

Rise/Run = y/x
BECAUSE THEY SAID IT VARIED DIRECTLY!!

k = 30/4
Simplify
k = 7.5
So the equation is y = 7.5x

What does the 7.5 represent?
The unit rate of $7.50/ hour of babysitting!

A bicycling example 
The distance the cyclist bikes in miles VARIES DIRECTLY with the time in hours that he bikes.

THINK: The graph and equation go through the origin (0,0).THINK: Any other point will give you the slope, or constant of proportionality, or unit rate (all the same thing) SO you only need one additional point.

He bikes 3 miles in ¼ hour. Find the equation.

Rise/run = y/x
BECAUSE THEY SAID IT VARIES DIRECTLY

k = 3/¼  or 3/.25 Now the hardest part is doing this 3/.25

If you kept it as 3/¼  you could read this as 3 divided by ¼
THINK: instead of dividing, multiply by the reciprocal of ¼

or 3 (4/1) = 12 (Wait, wasn’t that much easier than dividing 3 by .25!!
k = 12

The equation is y = 12x

What does the 12 represent?

The unit rate of 12 miles/ hour – that’s the cyclist’s speed 12mph

  Determining whether a Table of Values is Direct Variation If you are given a table of values, you can determine if the relationship is direct variation by dividing 3 y’s by their x values and making sure that you get the SAME value. If you do, it is proportional, goes through the origin (0,0) and the slope of y/x is the unit rate ( which is now called the constant of variation)!

Example

Given 3 points (5, 20) , (6, 24), and (7, 28):

Divide each y/x

20/5 = 4

24/6 = 4

28/7 = 4

Since all the ratios simplify to the same value (4), it is a direct variation. The slope of 4 is the unit rate, which is the constant rate of change and is now also called the constant of variation.

Finding Additional Values for the Direct Variation once you have the EquationOnce you have the equation y = kx, you can find infinite additional values (points) that will work.
For example, in the first babysitting example, the equation is y = $7.50x, which we write as y = 7.5x  If she babysits for 20 hours, how much did she earn?
x = 20

so y = 7.5(20) = 150 so She earns $150.
If she earns $750, how many hours did she need to work?

Now y = 750  so  750 = 7.5x
It is a one-step equation and we get
x = 100 or 100 hours!

 Finding the Equation if you know 1 point and then Finding Additional Values

y varies directly with x. Write an equation for the direct variation. Then find each value

If y = 8 when x = 3, find y when x = 45

FIRST you need to find k

y = kx… In this case we have 8 = k(3) or 8 = 3k

Solve this 1 step equation—leaving it in fraction form!

8/3= k

so

y = (8/3)x

Now, find y when x = 45

y = (8/3)(45)

solve

y = 120

Applying direct variation to the Distance Formula d = rt

A jet’s distance varies directly as the hours it flies

If it traveled 3420 miles in 6 hours, how long will it take to fly 6500 miles?

k = 3420/6 = 570mph ( its speed)

6500 = 570t

t ≈11.4

about 11.4 hours

Algebra Honors ( Period 6)

Chapter 4-1 Graphing Equations in Slope-Intercept Form

The most used form of a linear equation: Slope Intercept form

You must restate the equation to get it into the following format:

y = mx + b

where        m= slope     and            b = y-intercept ( where the line crosses the y axis)

The m and b are called the PARAMETERS of the equation.

Solve the equation for “y” means isolate the y on one side of the equal signs

Example:  -3y = -2x  - 6

If you tried graphing randomly, (setting up a small table or t chart) you would problem select the x points of 0, 1, 2.  Only when x = 0 will give you an integer value for y. All the other y values are fractions/ decimals à which makes it difficult to graph accurately!
 

The slope- intercept form provides the explanation for this.

RESTATE -3y = -2x – 6 INTO SLOPE-INTERCEPT FORM:

 Divide BOTH sides by -3:

y = (2/3)(x) + 2

Look at the coefficient for x?
What  x values will give you integer answers for y?

They need to be multiples of 3.

Now look at the graph of  y = (2/3)(x) + 2
Notice that the +2 at the end is the y-intercept (without doing any work!) YAY!!!

Use the counting method for slope on your graph, you should have counted:
UP  2 and RIGHT 3. The slope therefore is 2/3
Look at the equation—it told you the slope was 2/3 without any work (YAY!!)

Graphing when the line is in Slope- Intercept Form

So if you have the slope- intercept form of the equation it is really easy to graph the line:

1) Graph the intercept on the y axis (That is the positive or negative constant at the end of your equation. Your HOME BASE)

2) Count the next point by using the slope of x coefficient as a fraction (so if you have an integer—place it “over” 1)

For the equation y = (2/3)(x) + 2

1) graph  a point at (0,2)

2) From (0,2) count up 2 and over to the right 3 to find the next coordinate ( 3, 4)

Remember slope is “ y over x” or “Rise over Run”
The numerator is the change in y  and the denominator is the change in x.

If it is positive you are counting up (positive) and to the right (positive) OR
you can count down (negative) and to the left (negative) because when multiplying 2 negatives become positive

If it is negative you are counting down (negative) and to the right (positive) OR
you can count up (positive) and to the left (negative) because then you would have a positive ∙ negative = negative

Example: m = -2/3 and b = -12
The line would be y = (-2/3)x - 12

Restate Standard Form to Slope- Intercept Form

Another example: 3x + 4y = 10 is the STANDARD FORM of a line
(Notice; x and y are on the same side of the equal sign, x is positive, there are NO fractions, and the constant is alone)
This equation is NOT easy to graph in this form because your y intercept will not be an integer ( In fact it will not be that easy to graph in slope intercept either—but we will get to that later)

Restate into slope intercept

Solve for y
First subtract 3x from both sides:

4y = -3x + 10

Now divide both sides by 4:

y = (-3/4)x + 10/4   or

y = (-3/4)x + 5/2

The slope is the coefficient of the x so

m = -3/4 (so you are sliding down at a little less than a 45 degree angle. Remember a slope of 1 or -1 is  45 degrees)

The y intercept is the constant b = 5/2 ( so the line crosses the y axis at 2½)

Notice that the ‘b’ is a fraction.
When this happens the slope- intercept form MAY NOT be the best form to graph the line. You MUST start at 2½ on the y axis and count down 3 and to the right 4. That is actually HARD to get accurate!

Neither 3 nor 4 is a factor of the constant 10 ( 3x + 4y = 10) so the intercepts will also be fractions.

We need the x term to end up with ½ so that when we add that to the b (which is 5/2) we will get an integer.
So try letting x be 2 because that will cross cancel with the -3/4 slope

y = (-3/4)(2) + 5/2 =  -3/2 + 5/2 = 1
so we just found a coordinate that has just integers ( 2, 1)
Graph that point. Now count the slope from THAT POINT ( instead of the y intercept)

HORIZONTAL LINES
You can think of these lines in Slope- Intercept form as y = mx + b IF your REMEMBER that the slope of a horizontal line = 0 àthe equation is y = (0)x + b
We have already learned that the equation of any horizontal line is y = a constant
Except for the horizontal line y = 0 ( which is the x axis) horizontal lines have no x intercept and therefore no intercepts ( no roots, no solutions, no zeros)

Finding the equation of a line by looking at its graph is easy if you can read the y intercept!

Simply plug in the y intercept as  b and then count the rise over run as the slope!

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 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)

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   

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="">

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.

Algebra Honors ( period 6)

Solving Inequalities  ( All 4 Op's) 5-1 &5-2

Solving Inequalities 5-2

Again you will use your equation skills but this time use the Multiplicative Inverse Property as you would if you were balancing an equation.

ONE MAJOR DIFFERENCE FROM EQUATIONS:

When you multiply or divide by a NEGATIVE coefficient (to balance) you must SWITCH the inequality SYMBOL (this does NOT apply to adding or subtracting negatives) You must rewrite the problem as you divide by a negative as shown below:

If you want to understand why:    3 < 10      you know that is true

Now multiply both sides by -1 ( multiplication property of equality lets you do that)
but you get -3 < -10 but THAT IS NOT TRUE

You have to SWITCH THE SYMBOL to make it true  -3 > -10

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

Doing 2 Steps with Inequality Signs:  same as equations except make sure you switch the symbols if you multiply or divide by a negative. (Rewrite that portion of the problem—as you multiply or divide)  Always finish with the variable on the left!

Check with whatever solution is the easiest in the solution set. If 0 fits—use it! 

NEVER USE THE BOUNDARY NUMBER   for instance if your solution was x ≤  4  You could check with any number less than 4—BUT NEVER USE 4!

With two steps—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 ( which is between Steps 1 and 2 below)

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

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

3   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

4     4.   Add or subtract

5      5.  Multiple or divide ( only FLIP THE SYMBOL if you multiply or divide by a NEGATIVE to balance)

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

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.

Algebra Honors ( Period 4)

Adding & Subtracting Polynomials 8-1

VOCABULARY

Monomial: a constant, a variable or product of variables and WHOLE number exponents – one term

Binomial: The SUM of TWO monomials

Trinomial: The SUM of THREE monomials

Polynomial: a monomial or the sum of monomials

Determine whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial or trinomial.

Example 1:  4y – 5xz 
  Yes 4y and -5xz is really the sum of 4y and -5xz. Degree: 2  binomial

Example 2:  -6.5
Yes, -6.5 is a real number. Degree: 0 monomial

Example 3: 7a^-3 + 9b   No it is not a polynomial because 7a^-3 the exponent is NOT a whole number exponent.

Example 4: 6x³ + 4x + x + 3 Yes  it actually simplifies to 6x³ + 5x  + 3 the Degree is 3 and it is a trinomial

Degree of monomial: The sum of the exponents of all the variables in the term

Degree of polynomial: The highest degree of any term in a polynomial

Although the terms of a polynomial can be written in any order, polynomials in one variable are usually written in standard form.

Standard form  (Descending form): Place variables in alphabetical order with the highest power  first

Leading coefficient: The coefficient of the term that has the highest degree.

Write each polynomial in standard form Identify the leading coefficient

Example 1: 3x² + 4x⁵ -7x
Find the degree of each term
The greatest degree is 5.  so    4x⁵ + 3x² -7x is its standard form , with a leading coefficient of 4

Example 2: 5y -9-2y⁴ -6y³ 
Find the degree of each term
The greatest degree is 4 so,  -2y⁴ -6y³ +5y -9   is its standard form, with a leading coefficient of -2. Notice that sign of the coefficient is kept!

Degree	Name
0	constant
1	linear
2	quadratic
3	cubic
4	quartic
5	quintic
6	6th degree

Add and Subtract Polynomials

Adding polynomials involves adding like terms. You can group by using a horizontal or vertical format.

Find each sum
Example 1; ( 2x² + 5x -7) + (3 – 4x² + 6x)
Horizontal method:
Group and combine like terms

[2x² + (– 4x² )] + [5x + 6x] + {-7 + 3]

– 2x²  + 11x -4

Example 2: (3y + y³ -5) + (4y²- 4y+ 2y³ +8)

Align like terms in columns and combine. Insert a placeholder to help align your terms, if necessary

 y³ + 0y² + 3y -5
+
2y³ + 4y² – 4y + 8

3y³ + 4y²  – y  + 3

You can subtract a polynomial by ADDING its additive inverse. To find the additive invers of a polynomial, write the opposite of each term. That is, distribute the negative to each of the terms.

-(3x² + 2x - 6) = -3x² - 2x + 6

Find each difference

Example 1 (3 – 2x + 2x²) – (4x -5 + 3x²)
Horizontal Method
Subtract 4x -5 + 3x² by ADDING its additive inverse so
( 3 – 2x + 2x²) – (4x -5 + 3x²) = ( 3 – 2x + 2x²) + (-4x +5  -3x²)

Group like terms
[2x²  + (– 3x²)] + [( -2x ) + (-4x)] + [3 + 5]
x² – 6x + 8

Example 2 (7p + 4p³ -8) – (3p² + 2 -9p)
Vertical Method

Align like terms in columns and subtract by adding the additive inverse

4p³ + 0p² + 7p – 8        ADD THE OPPOSITE   4p³ + 0p² + 7p – 8       
(-)     3p² - 9p + 2                                               (+)    - 3p² + 9p - 2
                                                                              4p³ -3p²+ 16p -10

Adding or subtracting integers results in an integers so the set of integers is closed under addition and subtraction. Similarly adding and subtracting polynomials results in a polynomial so the set of polynomials is closed under addition and subtraction.