Math 6A ( Periods 2 & 4)

 Graphing Inequalities  

You will need to look at the graphs in your textbook. .. page 397

Whenever we graph relations that are inequalities we must be aware of all the facts that can influence your work. You need to ask yourself, "What kind of numbers is the solution supposed to be?"
When you graphed inequalities such as
-3 < x < 2 where x was an integer we used a point on the number line for each integer that could be a solution to that inequality. To show every number in x < 2 we would use a number line and place an Open Dot at 2 indicating that 2 was NOT part of the solution and then draw a darkened ray away from 2 indicating 1, 0, -1, -2... were all part of the solution.
To show that this line has infinite solutions in that direction, you MUST place an arrow at the end of that darkened ray.

If the inequality was a " less than or equal to" " ≤" you would use a Closed Dot at 2 to indicate that 2 was part of the solution.


We graphed y ≥ x - 3
Notes will be added ...

WE will continue this lesson on Monday, April 15th....

We can now graph inequalities such as y ≥ x + 2
first you find the BOUNDARY LINE which is just y = x + 2 and you can use the 3 column table as we have done before or use a T chart as shown in class.

Remember you only need 2 points to determine a line---> but 3 points will help you make sure you have 3 correct points on the line!!

I am going to try to set up a T chart using "I" to separate the x and y
X I Y
-2 I 0
-1 I 1
0 I 2
1 I 3
2 I 4

(Note: it doesn't line up well here.. but hopefully you get the idea)

Plot those points on the graph and you have what appears to be a straight line. Since we are graphing y ≥ x + 2 we ARE including the line so we draw a solid line.

But.. what points are included?
Well, we know that (-2,0) works but we also see if we plug into our inequality that (-2,1) and (-2,2) work as well.

We need to shade the part above the line to indicate all those points are part of the solution as well.

Three set method for graphing an inequality

(1) Determine the boundary line. Draw it--
use a solid line if the boundary line is part of the graph (≤ or ≥)
use a dashed line if the boundary line is NOT part of the graph (< or >)

(2) Shaded either the part above the boundary line or the part below the boundary line.
If the inequality reads y > or y ≥ shade ABOVE the line.
If the inequality reads y < or y ≤ shade BELOW the line

(3) Always CHECK- choose a point you think works within the shaded region and see if it does work.. or use (0,0) and determine if it is part of the solution or not!!

Algebra Honors ( Periods 5 & 6)

 Percents 7.5
You have been doing this since at least 6th grade so this portion should really be review.
The word percent means hundredths or divided by 100

Some examples:
29 percent = 29% = 29/100 = 0.29
2.6 percent = 2.6 % = 2.6/100 = 26/1000 = 0.026
637 percent = 637% = 637/100 = 6 37/100 = 6.37
0.02 percent = 0.02% = 0.02/100 = 2/10000 = 0.0002
1/4 percent = 1/2% = 0.25 % = .25/100 = 25/10000 = 0.0025
33 1/3 percent = 33 1/3 % = 100/3% = 100/3 ÷ 100 = 1/3

Most calculators have a % key which will enable you to check your work!!
Remember in percent problems, the word, OF means Multiply and the word IS means EQUALS.

You can set up your problems either with an equation or with a proportion
Example 1:
15% of 180 is what number?
15/100 × 180 = x or
.15 × 180 = x

or set up a proportion
%/100 = 'is'/'of'
15/100 = x/180

In both cases x = 27

Example 2:
23 is 25% of what number?
23 = .25×n

or as a proportion
25/100 = 23/x

Again in both cases
x = 92

Example 3:

What Percent of 64 is 48?
x% × 64 = 48
or
x/100 × 64 = 48
64x/100 = 48

Set up as a proportion
x/100 = 48/64

x = 75
Remember to look at the question to make sure you have answered it. In this case you need to make sure you answer with 75%

When you solve with decimal coefficients, you can multiply both sides of the equation by a power of 10 (10, 100, 100 and so on) to get an equivalent equation with integral coefficients.

Example 4:
1.2x = 36 + 0.4x
Multiply both sides by 10 because the coefficients are tenths
12x = 360 + 4x
8x = 360
x = 45
{45}

Example 5:

94 = 0.15x + 0.08(1000 - x)
Multiply both sides by 100 because the coefficients are hundredths

9400 = 15x + 8(1000 - x)
9400 = 15x + 8000 - 8x
1400 = 7x
200 = x
{200}

Word Problem Example:
During a sale, a sporting goods store gave a 40% discount on sleeping bags. How much did Ross pay for a sleeping bag with an original price of $75?

Two different methods to find the same solution:
Method 1
Find 40% of $75? 0.40 × 75 = 30
Subtract the amount of discount from the original price
75-30 = $45
Ross paid $45 for the sleeping bag

Method 2
If the sleeping bag was discounted 40%, it then cost 100% -40% or 60% of its original price
Find 60% of 75 0.60 × 75 = 45
Ross paid $45 for the sleeping bag.

Math 6A ( Periods 2 & 4)

Graphs of Equations 11-9

An equation in two variables y = x + 2      produces an infinite number of ordered pairs
If we give x the value of 1, a corresponding value of y is determined
y = (1) + 2 = 3
The ordered pair is (1, 3)

If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)

What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -1
y = (-1) + 2 = 1 ( -1, 1)

I like to remember ordered pairs---> ( ordered, pairs)

We graphed the line on a mini graph stickie.
The line is graphed using a ruler and connecting all the points we plotted. Put arrows at each end (since a line continues with out end) and write the line's equation right above the line.


For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION


y = x + 2
in the future you will see it written as
f(x) = x + 2
so if x = 2
f(2) = (2) + 2 = 4
if x = 5
f(5) = (5) +4 = 9



We used a three column chart to compute our ordered pairs.
Please refer to the work sheet glued into your spiral notebook for the examples we completed from the class exercises found on Page 393 -- if you were absent, please come in one morning and I will review that chart with you.


The following equations create curves that are called PARABOLAS!! Notice the difference in these equations from our previous equations
y = x² +1
when we create your three column table using integers from -2 to 2
we notice
y = (-2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)
y = (-1)² +1 = 1 + 1 = 2 ordered pair (-1, 2)
y = (0)² +1 = 0 + 1 = 1 ordered pair (0, 1)
y = (1)² +1 = 1 + 1 = 2 ordered pair (1, 2)
y = (2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)

When you graph this... you get a "U" shaped graph.

Remember linear equations LINEar equations are lines!
and look like y = x + 2

PARABOLAS have the form y = x² or y = -x²

Let's try
y = 2 - x²
With our 3 column table
for values of x from -2 to 2
we find
y = 2 -(-2)² = 2 -(4) = -2 and the ordered pair is (-2,-2)
y = 2 -(-1)² = 2 - (1) = 1 and the ordered pair is ( -1, 1)
y = 2 -(0)² = 2 - 0 = 2 and the ordered pair is (0, 2)
y = 2 -(1)² = 2 -1 = 1 and the ordered pair is (1, 1)
y = 2 -(2)² = 2 - (4) = -2 and the ordered pair is (2, -2)

When you graph these ordered points you find you have an upside down U
hmmm... y = -x² results in a sad face parabola
and y = x² results in a happy face parabola!!

Algebra Honors (Periods 5 & 6)

Fractional Equations 7.4

The total resistance R of an electrical circuit with two resistors R₁ and R₂, that are connected in parallel is given by the formula
1/ R₁ +1/ R₂ = 1/R
What do you notice about the difference between this equation and those with fractional coefficients?

This formula is an example of a fractional equation.
An equation with a variable in the denominator of one or more terms is called a fractional equation. To solve a fractional equation, you can multiply BOTH sides of the equation by the LCD or you could use the method of solving a proportion when the equation consists of one fraction equal to another fraction.
3/x -1/4 = 1/12
The LCD of the fractions is 12x
Multiply BOTH sides of the equation by the LCD, 12x
Notice that x ≠ 0 because in this case 3/0 is undefined
12x(3/x – ¼) = (1/12)(12x)
36 -3x = x
36 = 4x
x = 9
{9}

(2-x)/(3-x) = 4/9
There are two different ways to solve this
First by finding the LCD, which is 9(3-x) Notice that x ≠3 Why?
9(3-x)[(2-x)/(3-x) = (4/9)[9(3-x)]
18-9x =12-4x
6 = 5x
6/5 = x
{6/5}
OR solve as proportion
(2-x)/(3-x) = 4/9

(2-x)(9) = (4)(3-x)
18- 9x = 12- 4x
we are at the same spot as with the first method and we arrive at the same solution
6 = 5x
6/5 = x
{6/5}

The following gets a little more complicated to do and to display here...
Solve
(2/b² - b) – 2/(b-1) = 1
Find the LCD by first factoring the denominators first
b² - b = b(b-1) so the LCD of the two fractions in this equation is in fact
b² - b BUT use it in factored form b(b-1) Notice: b ≠ 1 why?
(2/b² - b) – 2/(b-1) = 1
[b(b-1][ (2/b² - b) – 2/(b-1) ]= 1[b(b-1)]
which separates to
[b(b-1) (2/b² - b)] – [b(b-1)2/(b-1)] = 1
[b(b-1) (2/b(b - 1)] – [b(b-1)2/(b-1)] = 1
2 -2b=b(b-1)
or
2 – 2b = b² - b
solve for b now
0 = b² - b + 2b -2
0 = b² + b – 2
0 = (b-1)(b+2)
b = 1 and b = -2
Remember in this case b ≠ 1 because of the ORIGINAL EQUATION
the solution set is only
{-2)

Multiplying both sides of an equation by a variable expression sometimes results in an equation that has an extra root. You must check each root of the transformed equation to see if it satisfies the original equation.

Algebra Honors ( Periods 5 & 6)

Equations with Fractional Coefficients 7.3

Solving an equation with fractional coefficients can be easily accomplished by using the LCD of all the fractions in the equation. Clearing the equation of all fractions BEFORE attempting to solve the equation is probably the best way

Easy examples:
x/3 + x/7 = 10

The LCD of the fractions is 21
so multiply BOTH SIDES by 21

21(x/3 + x/7) = 10 (21)
7x + 3x = 210
10x = 210
x = 21
{21}

3a/5 – a/2 = 1/20

The LCD of the fractions is 20
20(3a/5 – a/2) = 1/20(20)
4(3a) -10a = 1
12a- 10a = 1
2a = 1
a = ½
{1/2}

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

The LCD of the fractions is 15
15[x/3- (x+2)/5] = 2(15)
5x –(3)(x+2) = 30
5x -3x-6 = 30
2x = 36
x = 18
{18}

2n + n/3 = n/4 + 5

The LCD is 12
12(2n + n/3) = (n/4 +5)(12)
24n + 4n = 3n + 60
28n = 3n + 60
25n = 60
n = 60/25 = 12/5
{12/5}


More complicated:
(1/4)(n + 2) – (1/6)(n – 2) = 3/2

The LCD is 12
12[(1/4)(n + 2) – (1/6)(n – 2)] = (3/2)(12)
3(n+2) – 2(n-2) = 18
3n + 6 -2n + 4 = 18
n = 8
{8}

Solving some word problems:
one eighth of a number is ten less than one third of the number. Find the number.
Let x = the number.
n/8 = n/3 – 10
LCD is 24
(24)(n/8)= (n/3-10)24
3n = 8n – 240
-5n = -240
5n = 240
n = 48
{48}

Math 6A ( Periods 2 & 4)

Graphs of Ordered Pairs 11-8

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

(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, 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) (+,-)

Math 6 High ( Period 3)

Discount & Sales Tax 7.7

Let’s go shopping!
A reduction in the price of an item is called a discount. Discounts are often calculated using percent of the original price.
To find the price of an item that is discounted:

1) find the amount of the discount

2) subtract the discount from the original price.

You buy a pair of hiking shoes. The original price of $65 is discounted by 35%. What is the sales price?

Solution:
Find the amount of the discount

Discount = 35% of $65

Discount = 65 ( .35)
=22.75

$22.75 is what you will SAVE!! It isn’t what you will pay to purchase them so

Remember

Sales price = Original price – discount
65- 22.75 = 42.25

$42.25

Which television’s sale price is less expensive? Let’s assume they are the same quality
Television A: Discount of 25% off the original price of $575
Television B: Discount of 30% off the original price of $585.

Television A:  Sales price = 575 – (0.25 · 575)
                        575 -143.75
                        431.75

Television B: Sales price = 585 –(0.30 · 585)
                        585- 175.5
                        409.50

Television B’s sales price of $409.50 is less expensive.

Finding Sales Tax

A sales tax is an amount added to the price of a purchased item. Sales tax is often calculated using a percent of the price.
To find the total price of an item including sales tax:

1) Find the amount of the sales tax

2) Add the sales tax to the price.

Example:
You buy an ipod shuffle. The price is $40.00 and the sales tax is 8.25% What is the total price you will pay?

Solution:
Find the amount of the sales tax.

Sales tax       = 8.25% of 40
                        =0.0825(40)
                        = 3.3

Which means the sales tax is $3.30

Add the sales tax to the price

Total price     = Purchase Price + Sales Tax
                        = $40 + $3.30 = $43.30

You will pay $43.30.

                       
Finding a Total Price

Shirts in a store are on sale for 30% off. The sales tax is 7.25%.  What is the total price of a shirt that has an original price of $29.95?

Solution: First find the discount and  the sale price.
Then find the sales tax and add that to the sale price to find the total price.
Careful—lots of steps here!

Finding the discount and the sales price.

Discount         = 30% of 29.95
                        = 0.3(29.95)
                        = 8.99
Sale Price     = Original Price – Discount
                        = 29.95 – 8.99 = 20.96

Sale Price is $20.96

That is the amount you calculate your sales tax upon so

Finding the Sales tax and the Total Price

Sales tax       = 7.25% of 20.96

                        = 0.0725(20.96)=1.52

Total Price     =  Sale Price + Sales Tax
                        = 20.96 + 1.52 = 22.48

The total price including sales tax is $22.48