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I have an inequality as follows

$$ y \log y - (1-y) \log(1-y) < k$$

where $k$ is a constant. I tried solving the equation using exponential on both sides, but it does not come to a proper form where I can get the solution. How can I proceed to solve this inequation?

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  • $\begingroup$ Such equations with the unknown outside and inside a transcendental function usually have no analytic solution. You need to resort to numerical methods, after solving the root separation issue. $\endgroup$ – Yves Daoust Mar 29 '16 at 9:34
  • $\begingroup$ I do not see an equation, but an inequality. $\endgroup$ – Julián Aguirre Mar 29 '16 at 9:48
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The given function is defined for $x\in[0,1]$ and takes values in $[-\ln(2),0]$. It reminds the shape of a parabola and is symmetric about $x=\dfrac12$.

Hence, for any $k\in(-\ln(2),0)$, you can start numerical resolution of the equation, for instance by regula falsi with $x_0=0,x_1=\dfrac12$.

After convergence to $\hat x$, the solution of the inequation is the open interval $(\hat x,1-\hat x)$.

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