I am working with hydrology in GIS systems, and so i am a little bit rusty in mathematics (so sorry if this question is simple). Trying to solve the following inequality:
$$A_1B_1 + (1-A_1)C \ge k[A_1B_1 + (1-A_1)A_2B_2 + (1-A_1)(1-A_2)C],$$
where $B_1$, $B_2$ and $C$ are positive numbers, and A1,A2 are between 0 and 1.
I want to calculate the value of $B_2$ in relation to $A_1$, $C$, $A_2$ and $C$. I have solved the inequality, without the $k$ factor, but when trying to solve it with the k factor, I seem to be a little bit stuck. Any ideas - help? Thank you!
If $A_2=0$ or $A_1=1$, then $B_2$ vanish from the inequality.
We assume that $A_2 > 0$ and $A_1 < 1$ onwards. Also, assuming $k>0$,
$$\frac{A_1B_1 + (1-A_1)C}k \ge A_1B_1 + (1-A_1)A_2B_2 + (1-A_1)(1-A_2)C$$
$$\frac{A_1B_1 + (1-A_1)C}k - A_1B_1 - (1-A_1)(1-A_2)C \ge(1-A_1)A_2B_2 $$
and we have
$$B_2 \le \frac{\frac{A_1B_1 + (1-A_1)C}k - A_1B_1 - (1-A_1)(1-A_2)C }{(1-A_1)A_2} $$
Remark: when $k=1$, we can remove $A_1B_1$ but hence $B_1$ doesn't appear in your easier formulation. This is not the case when $k \ne 1$.