CPhill

These are the correct constraints :

x ≥ 4y

x >  0

y > 0

7.50x  +  4.75y ≤  125

So we have that

[Bx  - 11 ] / [ x^2 -7x + 10]   =  A / [ x -2 ] + 3 / [x - 5]

Using partial fractions we can simplify this as

Bx  - 11   =  A[x - 5] + 3 [ x - 2]       simplify more

Bx  - 11  =  Ax - 5A  + 3x  - 6

Bx  - 11  = [ A + 3 ] x + [ -5A - 6 ]     equate coefficients

B  =  A +  3      

-11  = -5A - 6   ⇒  -5  =  -5A  ⇒    A  = 1

So....B  = 1 + 3   =  4

So....A + B  =   1  +  4  = 5

Thank you, hectictar......nice cake and poem....!!!!

And thanks to my good Aussie friend,  Melody......without her....I wouldn't be here

And thanks to all those - too numerous to mention - who contribute to this forum....!!!!

We can solve this with a proportion

(1/2) / (2/3)  =  x  / 1     where x is the gallons  to fill the whole bucket

(1/2)(3/2)  =  x

(3/4)  =  x

𝑓(𝑥) = −2(0.8)^x+1

-2(0.8)^{0+ 1} =  -1.6       not true

-2(0.8)^{-1+ 1}  =  -2 (0.8)⁰  =  -2(1)  =  -2   true

-2(0.8)=^{-1.6+ 1}  =  -2(0.8)^-.6  = -2.286  not true

-2(0.8)^{2.8 + 1}  =  -2 (0.8)^3.8  = -0.856  not true

So......only b is correct

For the second one, we have

11.5x  -  2.3y ≤  20

11.5 (-10)  - 2.3(10) ≤ 20    ⇒ -138  ≤ 20   which is true

11.5 (10)  - 2.3(-10) ≤ 20    ⇒ 138  ≤ 20   which is false

11.5 (0)  - 2.3(5) ≤ 20    ⇒ -11.5  ≤ 20   which is true

11.5 (1)  - 2.3(0) ≤ 20    ⇒ 11.5  ≤ 20   which is true

So....a, c and d are correct

OK.......here's the first one

f(x)  =  -5x  -  8

a. (2,-18)

b. (10, -42)

c. (12,-68)

d. (4,28)

Putting 2 into the funnction we get   -5(2) - 8  =  -18

Putting 10 into the function we have  -5(10) - 8  =  -58

Putting 12 into the function we have  -5(12) - 8  = -68

Putting 4 into the function we have  -5(4) - 8  = -28

So......a and c are correct

f(5)  =  2(2)^5-4   =  2(2)¹  =  2*2   =  4

The cycle appears to be

Multiply by 3

Add 1

Divide by 2

So.......the series is

3       9     10       5     15     16      8     24      25        12.5

   *3     +1      /2     *3      +1     /2     *3      +1       /2 

For the first one:

 𝑓(𝑥) = −5𝑥 − 8

a. (2,-18)

c. (12,-68)

For the second one :

11.5𝑥 − 2.3𝑦 ≤ 20.8 

a. (-10,10)

c. (0,5)

d. (1,0)

For the third one :

𝑓(𝑥) = −2(0.8)^x+1  

b. (-1,-2)

The explicit rule is  for the nth term  is  :     a_n  =  6*(2)^n-1