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I have an equation as follows: $$[\gamma(\beta+y)+1]\exp(-(\beta+y)\gamma)>\{1-[\gamma(\beta+y)+1]\exp(-(\beta+y)\gamma)\}\alpha$$ where all parameters are positive.$$$$How can I a closed form for this equation( so as y> or < parameters of equation). I want to extract $y$ from inequality. Thank you in advance!

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How can you have a closed form for this equation? :) –  B. S. Nov 13 '12 at 13:38
    
I need to have domain of y to solve my main problem! –  qyapo Nov 13 '12 at 13:46
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I do not see how to obtain a closed-form solution, but your inequality is clearly related to the Lambert W function. Let $w=-\gamma(\beta+y)-1$. Then \begin{align} [\gamma(\beta+y)+1]e^{-(\beta+y)\gamma} &>\{1-[\gamma(\beta+y)+1]e^{-(\beta+y)\gamma}\}\alpha,\\ -we^{1+w} &> \alpha[1+we^{1+w}],\\ -(\alpha+1)we^{1+w} &> \alpha,\\ we^w &< \frac{-\alpha/e}{\alpha+1}. \end{align} So, using notations from the Wikipedia page, we must have $W_{-1}\left(\frac{-\alpha/e}{\alpha+1}\right) < w < W_0\left(\frac{-\alpha/e}{\alpha+1}\right)$ (where $W_{-1}$ and $W_0$ are respectively the lower and upper branches of the double-valued Lambert $W$ function). To evaluate $W_{-1}\left(\frac{-\alpha/e}{\alpha+1}\right)$ and $W_0\left(\frac{-\alpha/e}{\alpha+1}\right)$ numerically, see the "Numerical Evaluation" and "Software" sections of the wiki page.

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