Hi I really have no idea how to deal with this - it should reduce to some kind of quadratic. Any help would be much appreciated.
(2+3x)/(4-x) < 2x
I wrote out a detailed explanation to a similar problem here: Trying to go back to school to learn calculus (x-2) / (x+4) <7
, so
, so
, so
, so
, so
, so
.
Many ways to proceed now, by the easiest is using the Intermediate Value Theorem. Basically, on intervals where it's continuous (so doesn't include 4), and not 0 (so doesn't include 2 or 1/2), it must remain entirely positive or entirely negative throughout the entire interval. Test a point in there, and you'll know the inequality for the entire interval.
Intervals are: .
At . No good. (Were checking .)
At . So interval is part of the solution.
At . No good. (Were checking .)
At . So interval is part of the solution.
Since this was a strick inequality, we don't have to check the endpoints.
Thus the solution is .