Algebra ( Period 5)

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 Honors ( Period 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 8 ( Period 1)

Chapter 3-2 Slope

Slope is the steepness of a line. The coefficient of the x term makes the slope steeper as it gets bigger.

4x is a steeper line than 3x, which is steeper than 2x
A fraction ( or decimal) coefficient makes the slope less than 45 degrees.

A 45 degree angle was a slope of 1 ( which actually is steep)

To make the line look like its going UP from left to right, the slope is POSITIVE
To make a line look like its going DOWN from left to right, the slope is NEGATIVE

There are several ways to think of slope to actually calculate it

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)

So Slope is the RATE OF CHANGE and  if it’s a CONSTANT rate of change, you have a LINEar relationship.

This slope, constant rate of change is the UNIT RATE in a real world problem.

TWO WAYS OF CALCULATING on a graph:

1  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

Math 8 ( period 1)

Constant Rate of Change 3-1

Quick Review of Proportions

A proportion = 2 equal ratios ( 2 equivalent fractions)

Solve using equivalent fractions or cross multiplication and then a one-step equation

(see if you can simplify the fractions before you multiply)

Solve the proportion for y

Equivalent Fraction approach:

Multiply both the top and the bottom by 7

y = 28

Cross Products Approach

you will get

3y = (21)(4)

Now divide each side by 3

DO THIS BEFORE multiplying on the right side? WHY?  Because a lot of times you will be able to cross cancel and keep the numbers smaller. IT will be easier for you! y = 28

Quick review of Coordinate Plane:
Formed by 2 perpendicular number lines that intersect at 0.

Origin: The point (0,0) where the number lines intersect

x axis: The horizontal line

y axis: the vertical line

ordered pair: a pair of numbers that finds a location on the coordinate plane with x first and y listed second

x coordinate: the first number in the ordered pair that tells you how far to go the right ( or left)

y coordinate:  The second number in the ordered pair that tells you how far to go up ( or down) 

Graph: Locating points on the coordinate plane to help see the patterns (if any) between the points.

You can express the points first in a table of ordered pairs before you graph

Here’s an example about the cost of photo CD’s where the ordered pair (x, y) represents (number of CD’s, cost of CDs in dollars)

The sets of pairs

(1,$3), ( 2, $6) (3, $9)

Make a table

# of CDs	Cost in $	(x, y)
1	3	( 1, 3)
2	6	(2, 6)
3	9	(3, 9)

Graph the table and you will see that the points appear to be on a straight diagonal line that slopes up from the left and to the right.

Is there a pattern?     Yes

Each point is one unit to the right and 3 units up from the previous point. What does that mean in terms of the info about the CD’s?

It means that every time this company makes another CD it costs them $3  more 
This is called LINEAR RELATIONSHIP

Notice that the word linear contains the root word LINE

In order to have a graph that is a line, there must be a constant rate of change between the change in both pieces of the data.

In the above example, the two data items are cost and number of CD’s As the cost goes up 3, the number of CD’s goes up 1… every single time—constantly!!

So it has a constant rate of change. You make this into a ratio where you always place the change in the y value ( the up or down data) over the change in the x-value ( the right or left data)

So the constant rate of change for the above examples is 3/1 or $ 3/ CD

Wait—this is the unit cost!! YES IT IS!!!

Your book shows another example where Marcus downloads 2 songs each minute on his phone

The Unit Rate for this is 2 songs/ minute

If we turn this into a graph, we would graph  the following points ( minutes, number of songs)

(0 min, 0 songs) ( 1 min, 2 songs) (2 min, 4 songs) ( 3 min 6 songs)  etc

The CONSTANT RATE OF CHANGE on the graph:

As the number of songs goes UP 2 each time, the minutes go to the RIGHT 1

Making that into a ratio up 2/ over 1 = 2/1

That’s familiar because it’s the UNIT RATE: 2 songs/ 1 minute

How can you easily tell whether it is a constant rate of change and therefore linear?  If you graph it… it will be a diagonal line.

From the TABLE, you see the difference  (change in each row)… If this is constant, you have a linear relationship.

In the Marcus example, you would have + 2, + 2, + 2, +2 etc  on the songs row and + 1, + 1, + 1, +1 etc on the minutes row.

YOU CAN HAVE A NEGATIVE  Constant Rate of Change

Your book gives the example of taking money out of the bank at a constant rate… or it could be a pool emptying of water at a constant rate.

Now the y value is going DOWN constantly  ( or is NEGATIVE) the graph will still be a gdiagonal, but it will look like it is going DOWN from left to right.

You can use tables and graphs to compare different ratios

Let’s say that in the previous example there is another company that has higher costs and so it costs $4 not $3  for each CD You can produce another table:

# of CDs	Cost in $	(x, y)
1	4	( 1, 4)
2	8	(2, 8)
3	12	(3, 12)

Now  graph of these ordered pairs and compare it to the other company’s graph. Again it is a diagonal line going up to the right But this line is STEEPER.. for every unit to the right, it increases 4 units up… not 3 units… making it steeper WHY?  Because the cost  is higher..

PROPORTIONAL LINEAR RELATIONSHIPS

Linear relationships with a constant rate of change are PROPORTIONAL if they go through the ORIGIN (0,0) 

There are 2 ways to show this…on the graph see if it goes through the origin.. plug and chug to see if ( 0,0) works. or compare several points placing the y value over the x value as a ration… those ratios need o be in the same to be proportion

REMEMBER: The definition of a proportion is 2 equal ratios

The book gives an example which we talked about in class
Look again at page 173

Fahrenheit and Celsius temperatures

It is a LINEAR RELATIONSHIP (a diagonal line) with a CONSTANT RATE OF CHANGE  (9/5) But it is not PROPORTIONAL because it does NOT go through the origin  and if you simplify a few ratios, they are NOT equivalent

Math 6A ( Periods 2 & 7)

 The Distributive Property 3.4  

Definition in words: To multiply a sum of difference by a number, multiply each number in the sum of difference by the number outside the parentheses. Then evaluate.

Distributive Property of Multiplication with Respect to Addition:

For any real  numbers a, b, and c,

a( b + c) = ab + ac

and

(b +c)a = ba +bc

Distributive Property of Multiplication with Respect to Subtraction:

For any real  numbers a, b, and c,

a( b - c) = ab - ac

and

(b - c)a = ba – bc

If you are given the problem 23 x 6 normally you would stack them and multiply but what if you consider

6(23) = 6(20+ 3) = 6 (20) + 6(3) 120

How about 8(26) = 8(20 + 6) = 8(20) + 8(6) = 160 + 48 = 208

But you could have changed it to 8 ( 25 + 1) = 8(25)  = 8(1) = 200 + 8 = 208

What should we do with 20(19)? well… 20(15 + 4) or 20(10 + 9) work but are they easy to multiply—especially in your head?

Why not try 20(20-1) = 20(20) – 20(1) = 400 – 20 = 380

20(39) = 20(40 – 1) = 20(40) – 20(1) = 800 – 20 = 780

25(39) has all sorts of possibilities

25(40 -1) = 25(40) – 25(1) = 1000 – 25 = 975

25( 36 + 3) = 25(36) + 25(3) = 900 + 75 = 975

We talked about the “trick” of multiplying any number by 25… if you forgot… come in and ask again. I would love to go over this. Hint: get out quarters and figure out how many dollars you have with all those quarters…

15(47) ?  again lots of possibilities to make the multiplication easier

15(50-3) = 15(50) – 15(3)

or change 15(47) = 47(15) { what property allows you to change the order?}

Then 47(10 + 5) = 47(10) + 47(5)

How do you multiply by 5 quickly?  Multiply by 10 and take half. Why does that work?

Now the book gives the following example:

3(7 + 2) = 3(7) + 3(2) = 21 + 6 = 27

My question, why would we even want to do that? It does not make it easier—in fact it is just too many steps. So you need to start to realize WHEN the DP (Distributive Property) helps you to make the multiplication easier!

8(53) could be an example when it makes sense because 8(50 + 3) = 8(50) + 8(3) and both of those are easy to multiply—even in your head.. and then add them! 424

What about mixed numbers? We were taught that if you had

 

You needed to change the mixed number to an improper fraction

But watch the Distributive Property in action:

We then reviewed some of the problems similar to our Apples & Bananas WS

4(n + 5) = 4n +4(5) = 4n + 20

9( 6 + x + 2) = 9(6) + 9x + 9(2) = 54 + 9x + 18

but then we combined like terms to get 9x + 72

Why not combine like terms inside the HUGS first?

9( 6 + x + 2) = 9(x + 8) = 9x + 72

Combining LIKE TERMS

Like terms are terms that have the same variable(s) raised to the exact same sxponent(s)

Constants are like terms

5x+ 19 + 2x + 2 becomes

7x + 21

Jose is x years old. His brother is 2 years older than Jose. Their aunt, Maria is three times as old as Felipe. Write and simplify an expression that represents Maria’s age

Jose x

Felipe x + 2

Maria 3( x + 2)

3(x + 2) = 3x + 6

Art Museum question:

	Museum	Exhibit
Child (under 5)	Free	Free
Student	$8	$x
Regular	$12	$4
Senior	$10	$3

A class of 30 students visit an art museum and a special exhibit. Use the DP to write and simplify an expression for the cost.

each student would be 8 + x so for 30 students you would have

30(8 + x) = 240 + 30x

We estimate reasonable values for x—and had a lively discussion about why we picked the price for the student’s exhibit fee.

Then we evaluated for the price of $2

30(8 + 2) = 30(10) = 300

and we also substituted in for 240 + 30x  = 240 + 30(2) = 260 + 60 = 300

We found that the values were the same $300

Extension:

Factoring Expressions:

When you factor an expression, you can factor out any common factor!

Factor 20 -12

Find the GCF of 20 and 12  … the book says by listing the factors… we use inverted division or the box method

12 = 2²∙3

20 = 2²∙5

So the GCF(20,12) = 4

Write each term of the expression as the product of the GCF and the remaining factor(s)

20 – 12 = 4(5) – 4(3)

=4(5 -3)

14x -98

What is the GCF?  Look only at the numbers this time

14 = 2∙7

98 = 2∙7∙7

The GCF( 14, 98) = 2∙7 = 14

so 14x -98 = 14 (x) – 14(7)

14(x -7)

Algebra Honors ( Period 4)

Inequalities Involving Absolute Value 5-5

The inequality │x│<  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 ( Period 5)

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

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 > x  so   13 > x which means x < 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

Algebra ( Period 5)

Inequalities Involving Absolute Value 5-5

The inequality │x│<  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 4)

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

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 > x  so   13 > x which means x < 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