Question

3/(1-∆)>4+2∆/1-∆

Answer 1

Assuming the usual carelessness with parentheses, I have
3/(1-∆)>4+2∆/(1-∆)
1/(1-∆) > 4
if ∆ < 1, then (1-∆) > 0, and
1/4 > 1-∆
∆ > 3/4
Since ∆ < 1, 3/4 < ∆ < 1

If ∆ > 1 then (1-∆) < 0, and
1/4 < 1-∆
∆ < 3/4
But ∆ > 1, so no solution here

Answer 2

my bad. I missed that ∆ in the numerator.
But I'm sure you can follow my logic, so make the necessary adjustment and redo the problem.