After multiplying your equation by q times (r+1) we obtain:

It's equivalent to:

Because q has no divisors there are only following possibilities:

(r+1 can't equal 1 or 2 because r>1, p-q can't equal q so I've rejected 3 possibilities)

i)If difference between two prime numbers is 1 one of them have to be 2 (it's only even prime).

ii)If 3 numbers q, q+2 and 2q-1 are prime the reminder from devison q by 3 cannot be 1 (in this case q+2 would be divisible by 3) or 2 (2q-1 would be divisible by 3). So q is divisible by 3 and q=3 (if q+2 =3 or 2q-1=3 one of numbers p, r is not prime).

iii)