Step 1:
Enumerate pairs of numbers whose product is 60
(1,60)
(2,30)
(3,20)
(4,15)
(5,12)
(6,10)
(10,6)
...
2. identify the pair whose components have a difference of 11.
(4,15)
3. attach a negative sign to the smaller component so that the sum is +11.
(-4,15)
Answer: the numbers are -4 and 15.