This question took me a long time but I have figured it out. I looked at my graph and imagined what is the line with the least gradient that gives me A as a solution, I realised that it would be the line 4x+y=36 because when the line is moved across the page it gives me A and D as solutions as minimums and maximums. I then looked at the maximum gradient for a line and found 2x+y=36 worked.
You can then recognise that these equations if divided across are the same as P=x+ky, so I ended up with x + y/4 = 9 and x + y/2 = 18. I have found my k values of 1/4 and 1/2 but I also had to recognise that the lines I found had two minimum points so the inequality does not include these values, hence 1/4 < k < 1/2.
This method is purely graphical and I would recommend instead comparing the points A and B when they are subbed into P. You then say that the value of P must be less for A than B so you solve the inequality A < B. Then do the same with the points C and D with C < D.