CPhill

1.  Here's how we can find the answer to the first one...if y varies directly with x...we have...

24  = k * 6        divide both sides by 6

k  = 4

So.....every  successive "y"  value  in the table should be  4 times of its associated  x value...is this true.....it looks like it....so....

k  = 4  and y  = 4x

2. For the second one...the answer is no

Note....if   x   = 3 , then y  = -13

And..if  x  = 6 , then y  = -20 

And if x  = 9, then y  = -27...... 

If this were a direct variation like the first problem, successive y values would be multiples of the first....but this isn't so   

[ Look at the first problem....x and y both change by multiples of  3   ]

3.  y = kx     .....so...... 

    2.52  = k  (8.4)    divide both sides by   8.4

    2.52 / 84  = k  =  3/10

So    when x  = 2.7....we have that

y  =  (3/10) 2.7

y  =  8.1 / 10    =   0 .81

Looks good......  

Yep....as NediaMaster says.....leave your problems up....others may want to see how to solve them, too.....!!!!!

875   - [  80 + 75 + 145 + 250 + 80 ]    = $245 is what he needs

We want to   find when  Costs =  Sales    where d  is the number of days it will take

32550  +  750d   =   1125d        subtract 750d from both sides

32550  =  375d                           divide both sides by  375

86.8  =  d   ≈    87 days

131.5 > −5.6𝑥       divide both sides by -5.6.....reverse the inequality sign since we're dividing by a negative

-131.5 / 5.6  <  x

a.   distributive property

b .  simplification of -36 + 12   [ not sure about the name of this property ] 

c.  addition property of equality

a.   distributive property

b.  addition property of inequality

c.  addition property of inequality

Solve for k.

𝑚 = −15𝑗 − 6𝑘 − 8      add 15j  and 8 to both sides

m + 15j + 8  = 6k      divide both sides by 6

[ m + 15j + 8 ] 

____________         =  k  

        6

Solve for t.

7 + 𝑧 = 6𝑡 − 8𝑛 − 15          add 8n    and 15 to both sides

[7 + z + 8n + 15  ]   =  6t        divide both sids by 6

[7 + z + 8n + 15  ]     =   t

______________

           6

−32 > −5 + 9𝑥        add 5 to both sides

-27  >  9x               divide both sides by 9

-3  >  x

𝑥 − 7.67 ≤ 9.4         add 7.67 to both sides

x ≤  17.07

[1/2] 𝑥 ≤ − 4/7        multiply both sides by 2

x ≤ - 8/7

Hey, TR.....good to see you......!!!!

Since  25 is a power of 5, the key here is to see if we can equate bases.....

25^x  = 1/5             note that 25  = 5²    and  1/5  =  5^-1  .....so we can write

(5²)^x   =  5^-1

5^2x  = 5 ^-1         since the bases are the same, we can equate exponents

2x  = -1              divide both sides by 2

x  =  -1/2

                                                   3​         11          18          33         65         123            216

Frst differences                               8           7           15          32          58             93

Second differences                                -1          8           17           26          35

Third differences                                         9             9            9            9

Since we have   a constant difference in the third differences, this series can be represented by a cubic polynomial =

an^2   +  bn^2  + cn  + d           where   "n"  is  is the nth value of the sequence

We have that

a      +      b       +      c      +     d     =    3      →  -a   - b   - c  - d   =  -3      (1)

8a    +    4b       +   2c       +     d     =   11      (2)

27a  +    9b       +   3c       +    d      =   18      (3)

64a  +    16b    +    4c       +    d      =   33      (4)

Add (1) and (2),   (1) and (3)   and  (1)  and (4)....and we have

7a    +  3b  +  c     =  8      (5)

26a  +  8b  +  2c   =  15    (6)

63a  + 15b  + 3c  =  30     (7)

Multiply   (5)  by -2 add it to (6)    and    (5)  by -3 and  add it to (7) 

And we have

12a  + 2b  =  - 1      (8)

42a  + 6b  =   6        (9)

Multiply  (8)  by  - 3 and add it to (9)

6a   =  9       →    a   = 9/6  = 3/2

And   to find b we have that       12 (3/2)  + 2b  = - 1   →   2b  = -19 →  b  = -19/2

And to find c we have  7(3/2) + 3(-19/2)  + c  = 8  →  -36/2  + c  = 8  → 

-18 + c  = 8       →  c  = 26

And to find d  we have that

(3/2)  + (-19/2)  +  26  + d   = 3

-16/2   +  26  + d   = 3

-8 +  26  +  d  = 3

18  + d  = 3

c  = -15

So....our generating  function is     (3/2)n^3  - (19/2)n^2  + 26n  -  15

Rosy  left for the summer to visit relatives in Ireland......whether, or not,  she made it back, I don't know.....

She was also in an area of Florida that was affected by Hurricane  Irma....don't know the story there, either.....