Question

3/(1-∆)>4+2∆/1-∆

3/(1-∆)>(4-4∆ +2∆)/1-∆. how do you get the -4∆

Answer 1

Beats me. You've changed 4+2∆ to 4-2∆

How about some parentheses on the right side to make it clear just what you intend? If you meant
3/(1-∆)>(4+2∆)/(1-∆)
then if ∆≠1 just multiply by (1-∆) and you have
3 > 4+2∆
or
-3 < 4+2∆
depending on whether ∆-1 is positive or negative
so -1/2 < x < 1