> or= to 0

The critical numbers are numbers that cause 
the expression to be 0 or undefined.  x=-1 
causes the numerator to be 0, so -1 is one 
critical number  x=4 causes the denominator 
to become 0, and the expression to be 
undefined. So 4 is the other critical number. 
Place the critical numbers on a number line:

---------o--------------o--------------
        -1              4

That divides the real number into
three regions and two critical values.

(-oo,-1), {-1}, (-1,4), {1}, (1,oo}

choose a convenient point in (-oo,-1)
Substitute in -2

 > or = to 0

 > or= to 0

 > or= to 0

That is true so we shade (-oo,-1)

<========○--------------○-----------
        -1              4

Next we test the critical value -1

  (x+1)/(x-4) > or= to 0
(-1+1)/(-1-4) > or= to 0
            0 > or= to 0

So we darken the circle at -1,
and so the solution set contains 
(-oo,-1]

<========●--------------o--------------
        -1              4

choose a convenient point in (-1,4)
Substitute in 0

 > or = to 0

 > or= to 0

 > or= to 0

That is false so we do not shade 
(-1,4)

Next we test the critical value 4

  (x+1)/(x-4) > or= to 0
(4+1)/(4-4) > or= to 0
        5/0 > or= to 0

That is undefined so we no not darken
the circle at 4.

<========●--------------o--------------
        -1              4
choose a convenient point in (4,oo)
Substitute in 5

 > or = to 0

 > or= to 0

 > or= to 0

That is true so we shade (4,oo)

<========●--------------o=============>
        -1              4

So the solution is

(-oo,-1] U (4,oo)

You do the other one.

Edwin