# How to solve a quadratic inequality containing parameters?

I have the following inequality to solve for $R$ in terms of $\delta$ and $n$, with assumption that $n \geq 3$ and $\delta>0$:

$$-2 < 2 -\frac{16n\delta^{2}}{R^{2}\sin^{2}\frac{\pi}{n}} \left(nR^{2} - R\sin \frac{\pi}{n} - n\delta^{2}\right) < 2$$.

I try to use Reduce command to solve it. My Mathematica code is

Reduce[2 < 2 - 16*n*d^2*f[n, R]/(R^2*(Sin[Pi/n])^2) < 2, R]


which gives False. Trying to solve the two parts of the inequality separately also does not work, i.e. putting

Reduce[2 - 16*n*d^2*f[n, R]/(R^2*(Sin[Pi/n])^2) < 2, R]


gives the output that Mathematica is unable to solve "with the methods available to Reduce".

How could I modify my code to solve the above inequality?

• How did you define f[n, R]? Also, are you missing a minus sign in the first reduce expression (2 < ... should be -2 < ...)? Can you make any assumptions on the values of the parameters (e.g. real, positive, negative, integer)? – MarcoB Jun 22 '16 at 15:58
• Sorry, I defined $f(n,R) = \left(nR^{2} - R\sin \frac{\pi}{n} - n\delta^{2}\right)$ earlier in my code, so that is not an issue. The assumptions are stated above: $n \geq 3$, $\delta>0$. Also, $n$ is an integer (but that does not matter to me). @MarcoB – Alex Jun 23 '16 at 16:41