Choosing a threshold based on the likelihood ratio test

Question

I am stuck with a task of ratio test. Please help me with some advice. We are studying the Linear Discriminant Analysis. After projecting all the points on to "best" line the entire 2D dataset becomes two univariate normal distributions: $$ N(w^T\mu_1,w^TC_1w), N(w^T\mu_2,w^TC_2w) $$ I am asked to find an optimal threshold value (aka intersection of two univariate normal curves) by using the likelihood ratio test. I understand that all the terms of both normal distributions are just scalars after projection. The likelihood ratio test assumes that: $$ \frac{N(w^T\mu_1,w^TC_1w)}{N(w^T\mu_2,w^TC_2w)} > 1 $$ I suppose that if these "dot(w,mu)" etc are just scalars, so I can rewrite that as: $$ \frac{N(\mu_1,\sigma_1)}{N(\mu_2,\sigma_2)} > 1 $$ and expand as the regular Gaussian. My last step was: $$ ln(\frac{\sigma_2}{\sigma_1})-\frac{1}{2}(\sum_0^{N-1}\frac{(x-\mu_1)^2}{\sigma_1^2}-\sum_0^{N-1}\frac{(x-\mu_2)}{\sigma_2^2}) > ln(\gamma) $$ and it reduces to: $$ \sum_i^{N-1}\sigma_2^2(x_i-\mu_1)^2-\sigma_1^2(x_i-\mu_2)^2 > ln(\gamma)-\frac{2\sigma_1^2\sigma_2^2}{ln(\frac{\sigma_2}{\sigma_1})} $$ Please explain me how it works. Is this right direction? How to proceed with my results?