# Combinations of variables that satisfy my inequality

This question is a continuation to one answered by kglr here.

In the following situation, I find that given my assumptions, it is indeed possible for $g$ to be negative.

g = f - (-1 + f) P + (-2 f + 2 f P) w
Assuming[0 <= w <= 1 && 0 <= P <= 1 && 0 <= f <= 1 , FullSimplify@Reduce[g < 0]]
P + f (-1 + P) (-1 + 2 w) < 0


Given this finding, can Mathematica output combinations of regions of P, f, and w, that give the specified result $P + f (-1 + P) (-1 + 2 w) < 0$ ?

• Thanks for accepting my answer, but I think you were too hasty doing that. While accepting is one of the things to do after your question is answered, we recommend that users should test answers before voting and wait 24 hours before accepting the best one. That allows people in all timezones to answer your question and an opportunity for other users to point alternatives, caveats or limitations of the available answers. – rhermans Aug 10 '18 at 13:29
• @rhermans Thank you. I will keep that in mind. – user120911 Aug 10 '18 at 13:32

## 2 Answers

FullSimplify @ Reduce[{P + f (-1 + P) (-1 + 2 w) < 0,
0 <= w <= 1 && 0 <= P <= 1 && 0 <= f <= 1}, { P, f, w}]

0 <= P < 1/2 &&
-(P/(-1 + P)) < f <= 1 &&
(f + P - f P)/(2 f - 2 f P) < w <= 1

region = ImplicitRegion[
P + f (-1 + P) (-1 + 2 w) < 0
, {P, f, w}
]

RegionPlot3D[
P + f (-1 + P) (-1 + 2 w) < 0
, {P, -3, 3}
, {f, -3, 3}
, {w, -3, 3}
]