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This is really not a clear question. I've done a lot with functions and functional notation, and f(x) cannot equal both -4 and -5 at the same time. Maybe you can restate the problem.

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What you are probably getting at here is a concept called a "Sample Space." It is the set of all possible outcomes, with no overlap and nothing left out.

Give a sample space for a real estate salesman who is selling one of three homes to a family. The choices would be { House #1, House $2, House #3, No Sale}. This would assume that the family is not going to buy TWO houses!

Is that what you had in mind?

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Two events are independent if the result of one does not affect the other.

One of the best examples I can think of is the event that "it is raining in Boston" is probably independent of the event "it is raining in Los Angeles". In other words the weather in Boston is independent of the weather in LA. However, the weather in San Francisco is NOT independent when compared to the weather in LA, since the cities are so close, that a storm system in California would likely affect the weather in both cities in the same way.

Another independent event would be drawing a card from a deck of cards, replacing the card, and then drawing the second card from the deck. These events, drawing the first card and drawing the second card, are independent, since what happens on the first draw does not affect the outcome of the second draw. However, if you draw a card from a deck and do NOT replace the card, then whatever happened on the first draw DOES affect the result of the second draw. These events are NOT independent.

Other famous examples are the tossing of a coin, which comes up equally heads or tails. If you toss a coin twice or three times or ten times, what happens on the subsequent tosses is INDEPENDENT of the outcomes on the first toss (that is ASSUMING it is a "fair" coin). Now, if the coin toss is performed 10 times, and it comes up heads all 10 times, what would you think about tossing the coin the 11th time? Would you expect it to come up tails because "tails is due"?? The answer is certainly NOT! Assuming that it is a fair coin, you still have a 50-50 chance of getting heads or tails. (If I saw a coin come up heads 10 times in a row, I would be suspicious that it is NOT a fair coin!!) This holds with tossing dice as well.

Now, the way you calculate probabilities of independent events, is you multiply the probability of the first event times the probability of the second event. If you tossed a coin three times, what is the probability of getting three heads? Since the coin tosses are independent, you multiply the probability of getting a head each time (which is 1/2), so this would be

The basic formula for probability of independent events A and B is


If the events are NOT independent, then we have to recalculate the probability of the second event given that the first event has occurred. Then, we multiply the (Probability of A) times the (Probability of B given that A has already occurred). This is called "conditional probability", which will in all "probability" be your next assignment!!

Hope that helps.

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To find the x intercept, ALWAYS let y = 0. In this case , if y = 0, then 3x= 3, so x = 1. That's the x-intercept, which is where the graph crosses the x axis.

To find the y intercept, ALWAYS let x = 0. In this case , if x = 0, then -y = 3, so y = -3. That's the y-intercept, which is where the graph crosses the y axis.

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WOW!! Somebody who can write an equation box!! Congratulations for that.

Start by factoring the numerator and denominator of the first fraction as the difference of two squares.

To lead into this, start with
, which is pretty obvious.
In the same way, the slightly more complicated difference of squares as
and
.

So, factor the first fraction, and it should look like this, giving you some factors that will divide out:

*

Now, both factors in the second fraction match up and divide out with one of the factors in the first fraction. Divide out the and the , which leaves:


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I would say that if you have a ratio of two numbers, it means to divide the two numbers!
2/3 : 5/8 really means 2/3 divided by 5/8.

This in turn means , so you would multiply 2 times 8 and divide by 3 times 5. Answer here is . Is that what you meant by cross multiply??

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I'm assuming that you would like to factor this?
15cd-16+12c-20d

Rearrange the terms and group the first two and the second two terms together:
15cd+ 12c - 20d - 16

From the first two terms take out a common factor of 3c, and from the second two terms take out a common factor of -4, and hope for the best--a common factor must result, or this doesn't work!!
15cd+ 12c - 20d - 16
3c(5d+ 4) -4(5d+ 4)

Glory Hallelujah!! We must be living right!! We DO have a common factor of (5d+4).

Take out the common factor of (5d+4):
(5d+4)(3c-4)

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If you want to solve for R, then you must eliminate or undo everything that has been done to R. In this problem, R is multiplied by P and T. In order to "undo" multiplication, you must DIVIDE both sides by P and T.



The P and T divide out, and you are left with


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Domain is always the set of x values, the first number in the ordered pair. So the answer is B, the set of all years that are given in the "function."

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This is a quadratic equation because of the term, so the first step must be to set it equal to zero. To avoid a troublesome negative coefficient of , I would move everything to the right side by adding to each side:



Hopefully this will factor:

However, it does NOT. So let's invoke a quadratic equation solver from algebra.com.

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=284 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 1.34659282473948, -1.06087853902519. Here's your graph:

I think the answer you should get from this is reduces to .

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This is a parabola that opens up, since the coefficient of is positive.

You can complete the square or use a formula to find the vertex and the line of symmetry.

Completing the square looks like this:


Take half of the x coefficient (which is -2) and square it (which is 4), and add and subtract 4 from the right side of the equation.





The line of symmetry will be the value of x that "zeros out the , which will be x=2. This will also be the x coordinate of the vertex. To find the y coordinate of the vertex, let x= 2, and you have y = -1.

Vertex = (2,-1)

The y intercept is where x = 0. If x = 0, then y =3.

The x intercept is where y = 0. If y = 0, then


So, the x intercepts will be at x =3 and x= 1.

Graph

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Wow!! That's a tough one to do by my usual trial and error method. So, let's try setting it equal to zero, and solve by the quadratic formula. The numbers will be HUGE so I'll try to use the Quadratic Equation solver:

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=144 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 0.866666666666667, 0.777777777777778. Here's your graph:

The solutions are x=0.866666 . . . and x= 0.7777777 . . .
With a calculator you can easily convert these repeating decimals to fractions, and you might be able to get the second one in your head. I get x=13/15, and the second one is x = 7/9.

Now what this means is
x=13/15
15x=13
(15x-13)=0

Also,
x=7/9
9x=7
(9x-7)=0

Therefore the factored form for the trinomial is as follows:



My other idea would be to graph this on a graphing calculator that can find the zeros (or roots) of a graph. Determine the decimal values of these roots and convert them to fractions, proceeding in the same way, but without solving by the quadratic formula.

What a GREAT problem! If you want to re-post this question, and see if someone else has any better ideas on it, I will refrain from answering it a second time in order to allow a second opinion. (And I will not have my feelings hurt when I see it posted again!!)

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The formula they gave you is g(x)= 2x+3, and it is read "g of x equals 2x+3". Then they give you g(9), which means that the number 9 represents the value of x. It means to substitute 9 in place of the x (or all the x's if there had been more x's!), and simplify. It looks like this:

g(x) = 2x+3
g(9) = 2(9) +3
g(9) = 18+3= 21

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Each face or surface of the prism will be 14 in. by 14 in. for an area of 196 square inches. There are a total of 6 faces (top, bottom, and four around the side), each of which is 196 square inches. Therefore the total surface area is 196 x 6 = 1176 square inches.

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By the first law of logarithms, the sum of two logs is the log of the product. See my Lesson Plans in algebra.com on logarithms if you need help with this.



Now, if the log of THIS = the log of THAT, then THIS = THAT, so
BUT you must remember the restriction that you can never have the log of a negative, so you are restricted to values of x > 6. Do you see why they (and we!) said that?

So this is a straight line starting at the point (6,0) with an open circle to indicate that this point is NOT included. From that point, draw a straight line with a slope of 1 (Rise=1, and Run = 1). I'll draw it on this grapher, but I don't know exactly how to limit the graph as I have described. I'll have to draw the whole line, and then I'm asking YOU to make an open circle at (6,0) and from this point go UP the graph. Everything below the x-axis should NOT be there.



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Graph , and see where the graph crosses the x-axis. These points will be the solutions of the equation.



As you can see, the graph crosses the x-axis at x= 2 and x=3. These are the solutions of the equation.

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solve x^2 - 7x +12 = 0 by factoring! This I can do!!
(x-4)(x-3) = 0
x = 4 or x= 3

I don't have a clue what you want to do in the rest of this problem. Maybe try again.

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x^2-16x+32=0

First add -32 to each side and leave a blank space after the x^2- 16x .


You must take half of the -16 (which is -4) and square it (which would be 16). Add +16 to each side of the equation on the blank spaces I provided you.


Now, notice that the left side is a perfect square trinomial:


Now you can take the square root of each side:


Remember that ??


Last add +4 to each side:


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Cube both sides to eliminate the cube root:


Take the ln of both sides to get into logarithmic form:



I never saw one like this before. Maybe this is what they had in mind?? What a day to forget to bring my calculator home!! Tsk!!!

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ln7 + 1/2 ln9






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First, raise both sides as a power of e, to undo the ln.

or

There are several ways to do the next step. I like to invert both sides of the equation. In other words, if two quantities are equal, then their reciprocals are also equal. Flip both sides:
or

Unfortunately, I left my calculator at the office today, so you'll have to do the calculation yourself.

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First, raise both sides as a power of e, to undo the ln.

or

There are several ways to do the next step. I like to invert both sides of the equation. In other words, if two quantities are equal, then their reciprocals are also equal. Flip both sides:
or

Unfortunately, I left my calculator at the office today, so you'll have to do the calculation yourself.

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I've just posted a brand new lesson plan explaining Metric System that you might be interested in seeing. It's in algebra.com under Geometry: Length and Distance. You need to know that Metric System is much easier to work with than the English system, and the vocabulary is much simpler as well. The lesson plan goes into this in great detail. For now, all you want to do is convert from one distance measure to another.

First, the basic unit of measure for distance is the meter. From here, everything is measured in terms of 10, 100, 1000, or 1/10, 1/100, 1/1000 th of a meter. Here are the other measures in order from largest to smallest. I call this the metric line:

____|_______|______|_______|_________|______|_______|______
----Kilo----Hecto ---- Deka ---- Basic Unit ----Deci ----Centi ----Milli
------ k ----- h -------- dk -------- m, l, g ------- d ------- c ------- m
-- 1000 --- 100 ------- 10 ----------- 1 -------- 1/10 ---- 1/100 ---- 1/1000

Now, to convert from basic unit of meters to centimeters, you must move to the left from the Basic Unit to the Centi which is 2 units to the left. Therefore, you must move the decimal 2 units to the left.

If you convert from say millimeters to centimeters, this is 1 unit to the right, so move the decimal 1 unit to the right.

To convert from centimeters to millimeters, this would be 1 unit to the left, so move the decimal 1 unit to the left.

To convert from kilometers to centimeters, this will be 5 units to the rigth, so move the decimal 5 units to the right.

To convert from cenimeters to kilometers, move 5 units to the right.

In summary all you have to do is figure out which direction you move, and how many places. That's it!!

There's the lesson plan. Here are your problems:

(1) 6.8cm= how many mm? Move 1 place to the right = 68 cm
(2) 10mm= how many km.? Move 6 units to the left =0.00001 km (four place holder zeros!)
(3) 0.14mm = how many m.? Move 3 places to the left = 0.00014 m
(4) 5dm = how many cm? Move 1 place to the right = 50 cm
(5) 0.89cm= how many km? Move 5 places to the left = 0.0000089 km.

Check my math--I make a LOT of mistakes, and someone may have to catch one for me here!!

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There are three laws of logarithms, and you probably need to practice them backwards and forwards. If you need to get some extra explanations and examples, see my lesson plans in algebra.com under Logarithms, especially the ones that say "In Living Color". Those might be helpful to you.

REMEMBER that a LOGARITHM is LIKE an EXPONENT!! The Laws of Logarithms are like the Laws of Exponents!

Now, Do you remember from the laws of exponents, when you multiply, you ADD EXPONENTS? Well, with logarithms, when you multiply, you add LOGARITHMS.

Likewise, Do you remember from the laws of exponents, when you divide, you SUBTRACT EXPONENTS? Well, with logarithms, when you divide, you subtract LOGARITHMS.

As I said earlier, you have to know these backwards and forwards. This means sometimes you have to write it backwards:


Which brings us to your problem:


Did I just answer your question, or was that a Lesson Plan???

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First step, let's undo the ln, by raising both sides to the power of e:



Next, undo the square root by squaring both sides:



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This equation factors as the difference of two squares:



The first factor factors again:


Now, set each factor equal to zero and solve:
or or
or or or
or or or


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Given

a. Any line parallel to this must have the SAME slope . For example, or

b. Any line perpendicular to this line must have a slope that is the NEGATIVE RECIPROCAL of this line. The negative reciprocal of is or . For example, or

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Easier formula to remember and use:
5P2 means a permutation of 5 things taken 2 at a time. Start with the first number (which in this case is 5), and count down for a total of 2 numbers:
5P2 = 5*4 = 20

No extra charge for a few more:

5P3 means to start with 5 and count down for a total of 3 numbers:
5P3 = 5*4*3 = 60

7P1= 7
7P2 = 7*6 = 42
10P2= 10*9=90
10P3 = 10*9*8 = 720
11P2= 11*10 = 110
6P4 = 6*5*4*3 = 360
etc.

Isn't that simple??

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An x intercept is always where y = 0. If y = 2 log(10x), then let y = 0, and solve for x.
2 log(10x) = 0
log(10x) = 0





A y intercept is always where x = 0. If y = 2 log(10x), then y = 2 log(0), which is undefined.

Conclusion: X-intercept (,0). There is NO y-intercept.

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This is a combination problem, since the order of selection does NOT matter. The number of ways of choosing 6 items from 12 is called a C(12, 6). There is a formula that a lot of people use, but I think it's easier to do it this way.

If it had been a permutation (in which the order DID matter), then it would be a P(12,6). With a P(12,6), start with 12*11*10*9*8*7 and multiply these 6 numbers together. If it had been P(12, 2), then the answer is 12*11. In other words the first number tells you where to start counting down, and the second number tells you how many numbers to multiply.

Now, the difference between a Combination and a Permutation is that with a Combination, you must multiply the same numbers as for a permutation, but divide by the factorial of the second number. Here are some examples of combinations:
C(5,2) =
C(6,2) =
C(12,2) =
C(5,3) =

Now here at last is YOUR question:
C(12,6)=

Got a calculator??? I got 924 for the answer. By the way, most of your calculators, even a $15 TI 30 will do this for you, so you don't have to understand this lesson!!

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The first problem is a F OI L problem. First write it like this:





The second question, any time you need the y intercept, just let x = 0 and solve for y.

2x - 3y = 6
2(0) - 3y = 6
-3y = 6

Divide both sides by -3:



Sometimes this is written in (x,y) form (0,-2). Remember in this form, it's always in alphabetical order, with the x first, and the y second.

Good luck on your test!! Then again, it's NOT luck--it's PREPARATION!!!

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