CPhill

y >  -2

x + y ≤ 4

Note  that  (0, -4)  = (x,y)   can't be correct  because    - 4  >  -2   is false

And (-2, -3)  can't be correct for the same  reason   because  -3  > -2  is false

Note that (1,5) can't be correct because 1 + 5 ≤ 4   is false

So  (1,3)  is the correct choice  because

3 > -2    is true     ....and.....

3 + 1 ≤  4   is also true

OK....  since M is the mid-point of  QR then MR  =  17

This triangle is isosceles so  if  we let QS  be the altitude....then this altitude will bisect PR..

So....SR  = 16

And  the  cosine of angle QRS  = SR/ QR  = 16/34  =  8/17

Using the Law of  Cosines

PM^2  = MR^2  + PR^2  - 2(MR)(PR)cos(QRS)

PM^2  = 17^2 + 32^2  - 2(17)(32) (8 /17)

PM^2  = 1313  - 16*32

PM^2 = 1313 - 512

PM  = sqrt (1313 - 512)

PM =sqrt (801)  = sqrt ( 9 * 89)  =  3  sqrt (89) ≈ 28.3  units

Here's a pic :

Similarly...for the second one...let's pick  ( 4 , 0)  = (x , y)

Look  at the last answer..... plug in  4 for x   and 0  for y  and we have

0 > -2(4) - 2

0 > -8 - 2

0 >  -10     which is true

Also

0  ≤ -(1/2)(4) + 6

0 ≤ -2 + 6

0 ≤ 4     which is also true

Lastly

0 ≤ 4     which is also true

Since  all  three inequalities are true, this must be the correct answer

Check for yourself that at least one equation in each of  the other three answers gives an incorrect inequality

For example in the first two answers

y ≤ -2x + 6     will not be true for  (x, y)  =  (4,0)

And in the third answer

y ≥ x  will not be true  for  (x, y)  = ( 4,0)

Hope that helps.....!!!!

First one...here's one way to do this....pick a point in the shaded area...I'll choose (1,2)  = (x , y)

Note that this satisfies   y  ≥   1      and    y  - x  >  0

So...the second answer is correct for the first one

(a - b)^3 + (b - c)^3  + ( c -a)^3  =

[( a - b ) + (b - c)] [ ( a - b)^2 - ( a - b)(b - c) + ( b - c)^2 ] + (c -a)^3 =

( a - c) [ a^2 - 2ab + b^2  - [ ab - b^2 - ac + bc] + b^2 - 2bc + c^2 ] + (c -a)^3 =

(a - c)  [ a^2 - 2ab + b^2  - ab + b^2 + ac- bc  + b^2 - 2bc + c^2 ]  + (c -a)^3  =

(a - c) [ a^2 - 3ab + 3b^2 + ac - 3bc  + c^2 ]  + (c - a)^3 =

(a - c) [ a^2 - 3ab + 3b^2 + ac - 3bc  + c^2 ] + ( c - a)^3]=

(a - c) [ a^2 - 3ab + 3b^2 + ac - 3bc  + c^2 ] + [ (c - a) (c - a)^2 ]  =

- (c - a) [ a^2 - 3ab + 3b^2 + ac - 3bc + c^2 ] [ (c - a)( c^2 - 2ac + a^2 ]  =

(c - a) [ -a^2 + 3ab - 3b^2 - ac + 3bc - c^2 + c^2 - 2ac + a^2 ] =

(c -a) [ 3ab  - 3b^2 - 3ac + 3bc] =

(c - a) [ 3ab - 3ac - 3b^2 + 3bc ] =

(c - a) [ 3a ( b - c) - 3b ( b - c) ]  =

(c - a) [3(a - b)) ( b - c)]  = 

3(a - b)(b - c) (c - a)

OK, Lightning....glad I could help   !!!

A polynomial is an expression in one  (ore more) variables

The  exponent on each variable must be a whole number  [ 0, 1, 2, 3....]

The polynomial must have a finite number of terms

Examples

6x^2  +  2x  +  8

(4/5)x^3y^5 + 2y^2  + 3x  -  9/13

The  red entries  are coefficients  [ real numbers associated with each variable term]

The blue  entries  are constants

b )     rx + s

        _____   =  f(x)

        2x + t

If the horizontal asymptote  =  -4, then   r / 2  = -4 ⇒  r  = -8

If the vertical asymptote = 3, then  2(3) + t  = 0  ⇒  6 + t  = 0  ⇒  t   = -6

If  (1,0)  is an x intercept, then  r(1) + s  = 0 ⇒  -8(1) + s  = 0 ⇒ - 8 + s  = 0 ⇒

s  = 8

Here's a graph :  https://www.desmos.com/calculator/cown9bo626

A function f has a horizontal asymptote of y = -4, a vertical asymptote of x = 3, and an x-intercept at (1,0).   Find an expression for f(x)....

a)   ax + b

     ______  =  f(x)       

       x + c

If the vertical asymptote is 3, then , in the denominator,  3 + c  = 0  and  c   =  -3

If the hrizontla asymptote is -4, then  ax / x  = -4  ⇒  a   = -4

And if the x intercept = ( 1,0), this implies that   a(1) + b  = 0  ⇒  -4(1) + b  = 0  ⇒

-4 + b  = 0  ⇒   b  = 4

Here is a graph : https://www.desmos.com/calculator/ohx3qy9kvt

7.   

 y  = √[2x - 3]     square both sides

y^2  = 2x  - 3

y^2 + 3   = 2x     

[ y^2 + 3 ] / 2  =x

[x^2  + 3] / 2    = y

8. 

y = √[4x] - 7

y + 7  =√[4x]   square both sides

(y + 7)^2  = 4x

( y + 7)^2  / 4   = x

(x + 7)^2 / 4   = y