This one is an interesting question.
Lets solve first the equation by isolation of the unknown x:
6x + 6 +2 = (3k / 5x )+3 +3
6x = (3k/5x) + 6 -6-2
6x = (3k/5x) -2
6x = (3k / 5x) - (10x / 5x)
6x = (3k-10x)/5x
6x*5x = 3k - 10x
30x² = 3k - 10x
30x² + 10x = 3k
30x² + 10x - 3k = 0
This is really interesting because this looks exactly like a polynomial function. So we know that there is no solution for a polynomial once the discriminant delta is strictly less than 0.
But delta = b² - 4ac where b = 10, a = 30 and c = -3k
So we need to solve delta < 0
100 - 4*30* (-3k) < 0
100 - 120*(-3)*k < 0
100 - (-360)k < 0
100 + 360k < 0
360k < -100
k < -100/360= -10/36
So there is no solution for all k strictly less than -10/36
Good luck
