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I thought I got it, but he has a lot more evidence...

Solve for x:
4 + 2/x = 5 + 5/(3 x)

5 + 5/(3 x) = 5 + 5/(3 x):
4 + 2/x = 5 + 5/(3 x)

Bring 4 + 2/x together using the common denominator x. Bring 5 + 5/(3 x) together using the common denominator 3 x:
(2 (2 x + 1))/x = (5 (3 x + 1))/(3 x)

Cross multiply:
6 x (2 x + 1) = 5 x (3 x + 1)

Expand out terms of the left hand side:
12 x^2 + 6 x = 5 x (3 x + 1)

Expand out terms of the right hand side:
12 x^2 + 6 x = 15 x^2 + 5 x

Subtract 15 x^2 + 5 x from both sides:
x - 3 x^2 = 0

Factor x and constant terms from the left hand side:
-(x (3 x - 1)) = 0

Multiply both sides by -1:
x (3 x - 1) = 0

Split into two equations:
x = 0 or 3 x - 1 = 0

Add 1 to both sides:
x = 0 or 3 x = 1

Divide both sides by 3:
x = 0 or x = 1/3

4 + 2/x ⇒ 2/0 + 4 = ∞^~
5 + 5/(3 x) ⇒ 5/(3 0) + 5 = ∞^~:
So this solution is incorrect

4 + 2/x ⇒ 4 + 2/(1/3) = 10
5 + 5/(3 x) ⇒ 5 + 5/(3/3) = 10:
So this solution is correct

The solution is: x = 1/3

Have you ever heard of google?

a)   Was there ever a time when the first function  =  the second function?

      Was there ever a time when         16x/x^2+2   =   12x/x^2+1       ?

16x / x² + 2   =   12x / x² + 1       I am guessing that this is supposed to be...

16x / (x² + 2)   =   12x / (x² + 1)        Cross-multiply.

16x(x² + 1)  =  12x(x² + 2)                 Distribute.

16x³ + 16x   =   12x³ + 24x               Subtract  12x³  from both sides.

4x³  + 16x  =  24x                             Subtract  24x  from both sides.

4x³ - 8x  =  0               Factor  4x  out of both terms.

4x(x² - 2)  =  0             Set each factor equal to zero and solve for  x .

4x  =  0          or          x² - 2  =  0

x  =  0            or          x²  =  2

                                   x  =  ± √2

Yes, they are the same when  x = 0 ,  x = √2 ,  and  x = -√2 .

b)   For what values of  x  is  16x/x^2+2  greater than  12x/x^2+1    ?

16x / (x² + 2)   >   12x / (x² + 1)

Here is the answer, but all of the work may take a little to type in.

Thank you so much! :D

The circled points all have a  y , or  f(x) , value of  -1 . There are  4  of them.

The point  ( -2, f(-2) ) lies on the leftmost line segment.

The leftmost line segment contains the points  (-4, 4)  and  (-1, -1) .

So the slope of this line segment  =  [ -1 - 4 ] / [ -1 - -4 ]   =   -5 / 3  .

We know that the slope between  (-2, f(-2) )  and  (-1, -1)  will also be  -5/3 .

Using the slope  -5/3  and the points  (-1, -1)   and   (-2, f(-2) )

-5/3  =  [ f(-2) - -1 ] / [-2 - -1]

-5/3  =  [ f(-2) + 1 ] / [ -1 ]         Multiply both sides by  -1 .

5/3  =  f(-2) + 1                     Subtract  1  from both sides.

5/3 - 1  =  f(-2)

2/3  =  f(-2)

Thank you geno3141.

no it does not tell us the candy bars per week

The "mod" value is the remainder after division.

To find N in the equation:  111,119 mod N = 954

     first, subtract 954 from 111,119     --->     111,119 - 954  =  110,165

     now, find the factors of 110,165     --->     110,165  =  5 x 11 x 2003

     we need a factor larger than 954     --->     N = 2003

To find M in the equation:  111,119 mod M = 82

     first, subtract 82 from 111,119     --->     111,119 - 82  =  111,037

     now, find the factors of 111,037      --->     111,037  =  37 x 3001

     we need a factor larger than 82     --->     M = 3001

Thanks!

1 - kx  =  -3y

If this line passes through the point  (4, -3) , then we know that  x = 4  and  y = -3  are solutions to the equation. We know that this is true...

1 - k(4)  =  -3(-3)     Now solve this for  k .

1 - 4k  =  9

-4k  =  8

k  =  -2

You can also do it a bit simpler this way:

1/j+1/10=1/6
1/j=1/6 - 1/10
1/j=[10 - 6] / 60
1/j = 4/60
1/j =1/15
J = 15 hours that will take Jim to mow the lawn alone.

The function  f(x)  =  3x + b  can be written as  y  =  3x + b.

To find the inverse of this function:

1)  interchange the x and y terms:  x  =  3y + b

2)  solve for y:                            x - b  =  3y     --->     3y  =  x - b     --->     y  =  (x - b) / 3

To find where they intersect, set the two functions equal to each other:  3x + b  =  (x - b) / 3

and, since they intersect at the point (-3, a), replace x by -3:               3(-3) + b  =  (-3 - b) / 3

                                                                                                                  -9 + b  =  (-3 - b) / 3

                                                                                                              -27 + 3b  =  -3 - b

                                                                                                                       4b  =  24

                                                                                                                         b  =  6

So, the original function is  y  =  3x + 6  while the inverse is  y  =  (x - 6) / 3

Replace the x in either function with -3 and you get that y = -3, so  a = -3.

There is the point  (1, 7)  , so  f(1)  =  7

There is the point  (7, 9)  , so  f(7)  =  9

f( f(1) )  =  f( 7 )  =  9          Continuing this...

f( f(2) )  =  f( 4 )  =  8

f( f(3) )  =  f( 1 )  =  7

f( f(4) )  =  f( 8 )  =  6

f( f(5) )  =  f( 5 )  =  5

f( f(6) )  =  f( 2 )  =  4

f( f(7) )  =  f( 9 )  =  3

f( f(8) )  =  f( 6 )  =  2

f( f(9) )  =  f( 3 )  =  1

9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1   =   45

no sorry thats all the info I got :I

1/J + 1/10 =1/6

Solve for J:
1/10 + 1/J = 1/6

Bring 1/10 + 1/J together using the common denominator 10 J:
(J + 10)/(10 J) = 1/6

Cross multiply:
6 (J + 10) = 10 J

Expand out terms of the left-hand side:
6 J + 60 = 10 J

Subtract 10 J + 60 from both sides:
-4 J = -60

Divide both sides by -4:
J = 15 - hours that will take Jim to mow the lawn alone.

I'm sorry, but I don't understand this problem. If there was some more information, like what the function for f(x) was, I could solve it. ;-;

I believe I have gotten the answer to this riddle! Using the slope formula, one can find the value for a, if a solution exists.

$m=\frac{y_2-y_1}{x_2-x_1}$

Just plug in the appropriate values for the x- and y-corrdinates and solve.

hmmmm thats tricky...the equation to make those dots is needed to find the solution you ask for

is there any extra details you can give?

Thats what we have to figure out

i know i saw the solution