The line intersects the x axis at x = -7.5
The base length of the triangle will be k - (-7.5) = k + 7.5
The height of the triangle will be (2/3)k + 5
So...we are trying to solve this
(1/2) base * height = 20
(1/2) ( k + 7.5) [ (2/3)k + 5] = 20
(k + 7.5) (2/3k + 5) = 40
(2/3)k^2 + 5k + 5k + 37.5 = 40
(2/3)k^2 + 10k - 2.5 = 0 multiply through by 3
2k^2 + 30k - 7.5 = 0 multiply through by 10
20k^2 + 300k - 75 = 0
4k^2 - 60k - 15 = 0
The soutions to this are k ≈ -15.246 and k ≈ .245
So....the integer values for k that make this triangle have an area < 20 are the integers -15 to 0 inclusive
And the sum of these = - ( 15) (16) / 2 = -120