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Isosceles triangle    soo    7x-8 = 3x-2    solve for x     then distance  between  =   7x-8    +   3x-2

Could be   ADxx  AxDx AxxD  xADx  xAxD  xxAD    six ways for A before D

    the other two slots have two ways to arrange

6 x 2 = 12 ways

2 should be the maximum. And that would occur when n =1 and n=2. 1!+1=2 and 2!+2 =4 and the GCD(2, 4)=2. Any two consecutive terms after 2 would have a GCD of 1. So: 3!+3 =9. 4!+4=28 and GCD(9, 28)=1...and so on.

1  / (1/6 + 1/2 + 1/3) = days for all three = .....day(s)

Draw a Venn diagram

  Here is my quick sketch  Venn diagram..... add up all of the numbers  then subtract from 680 to find runners who participated in none...

6x + 16         +     42       =   11x+3       solve for x       then   TGH =  6x+ 16

Give students names or initials:
A=Alice, B=Bob, C=Cathy, D=David
Suppose Alice and David do not want to stand next to each other, then you have the following arrangements: 4! / 2! = 12. A,D cannot stand next to each other
{A, B, C, D} | {A, B, D, C} | {A, C, B, D} | {A, C, D, B}  {B, A, C, D} | {B, D, C, A} | {C, A, B, D} | {C, D, B, A}  {D, B, A, C} | {D, B, C, A} | {D, C, A, B} | {D, C, B, A} 

First, we find the slope of the line. This is y_1-y_2/x_1-x_2, which is 3-7/0-2=-4/-2=2. Now, we need the y-intercept. This is fairly easy, as we have a graph. We see (0, 3) is the y-intercept. Therefore, we have the equation y=2x+3. Here is the graph:

https://www.desmos.com/calculator/eenucyoqak

:)

Oh~that makes a lot more sense. Thanks!

$x^2 + y^2 = 2x - 6y + 6\\ (x^2 -2x)+( y^2 -6y)= 6\\ (x^2 -2x+1)+( y^2 -6y+9)= 6+1+9\\ (x-1)^2+(y-3)^2=16$

This is a circle with a radius of 3

you can find the area from there.

25 men  x  20 days = 500 man-days to complete the job

   at the end of 10 days  they have completed 250 man days   leaving  250 yet to be completed

       in 10 more days   they complete  20 x10 = 200 man-days     leaving  15 men to complete 50 man -days of work

              3 1/3 more days

10 days + 10 days + 3 1/3 days = ~ 24 days

The purple line is the asymptote.

I treated it as a graphing exercise.

Your answer was very great too! Although I am a bit confused on what the purple line means...

Hi  ilorty,

For this specific question your way is MUCH better than mine.

I have only been introduced to your method very recently, perhaps by you, and it will take me a while to integrate it into my own memory and use.

Thanks very much for reminding me   

There are 3 ways to stand so they do not have to stand next to each other

1001

1010

0101             the other two slots have 2 ways to be arranged     3 x 2 = 6 ways 

Guest answer is correct..... I neglected the fact that you could switch the two students around for a total of SIX ways they would not have to stand next to each other.....then the other two slots have two ways     soooo     6 x 2 = 12  ways

Thanx, Guest !

ok thank you!!!! I also got the answer eventually this way:

-x + 2y + 1 > 0 is the answer. :) 

Let's find the equation of the dashed line first. From the graph, we can see that it has a slope of 1/2 and a y-intercept of -1/2 so using slope-intercept form, the equation of the line is y = 1/2x - 1/2. Since the line is dashed and not solid, we know that we will use either < or > instead of ≤ or ≥, and because we can see that the shaded region is above the dashed line, we know that the linear equation is y > 1/2x - 1/2. However, we want the answer to be in the form ax + by + c > 0 where a, b, and c are integers so therefore:

y > 1/2x - 1/2

= 2y > x - 1

= -x + 2y + 1 > 0

SQUARE room of 225 ft²   means each side is 15 ft      15 ft will require THIRTY  1/2 foot tiles

We need to find the distance from A to C   that is   2 /(2+1) = 2/3 of the way to C

x:   from  3   to -5   is   -8          2/3 * (-8)   added to 3 = ........

y :  from  -6  to 11  is +17         2/3 * (17)   added to -6 = .........

Huh?  Are you SERIOUS? 

You have   7 to start with        Bitfu has 5 

x = yours    y = b's

x+1 = 2(y-1)       and

x-1 = y+1               these are given in the description

    re-arrange the second equation  x = y+2    sub this into the first equation

       y+2   +1     =  2y-2

       y + 3          = 2y-2

           y = 5                                  then   x = y+2    = 7 

First one        the graph is misleading because the bottom of the graph does not start at 0.....this makes the differences look much more striking

second one      greatest increase was may to june ....see how much it goes UP

                         lowest median sales price WAS below 215 000  in Sept and Oct

                         C  the median prices in Jan and June WERE the same @ 240 000

Great!

I did it differently, but Melody's answer is still very good. (Or maybe they are the same, and I just did not see...)

The first thing to do, as Melody did, is to set f(x) to y. This gives us:

y=(x^2 +14x+9)/(x^2+2x+3).

Now, we can multiply both sides by x^2+2x+3, and get:

y(x^2+2x+3)=x^2 +14x+9.

Now, we must solve for the discriminant:

x^2 + 14x + 9=yx^2+2yx+3y

x^2 + 14x + 9-(yx^2+2yx+3y)=0

(1-y)x^2 + (14-2y)x + 9-3y=0      <---- regrouping to get the coefficients

(14-y)^2-4(1-y)(9-3y)=196-56y+4y-(36-48y+12y^2)=-8y^2-8y+160      <--------discriminant formula

Now that we got our discriminant, we can say:

-8y^2-8y+160>=0   <------- or else it would be imaginary, which is no good

y^2+y-20 <= 0

(y+5)(y-4) <= 0

We see that for any number between -5 and 4, it works, so our range is [-5, 4]

:)