How to find the number of solutions of $6|\cos x|=x$? Now, I think the only way to solve this problem for a high school student was graphically.
However using pen and paper to draw the graph, it was virtually impossible to justify or refute the existence of the "Fourth" solution.
Using desmos, I realised that we must must work on the the existence of the Fourth solution analytically because $ y=x$ is really close to $6|\cos x|$ at the potential Fourth solution.
You can see for yourself.

So, this being a high school problem, is there any way to predict whether $y=x$ will intersect $6|\cos x|$ or not?
With enough zoom we can see that $y=x$ does not intersect $y=6|\cos x|$. However, how could have I predicted this with a pen and paper?
I am acquainted with calculus but had no clue how to go about it.
Thanks for your time!

I know this not necessarily relevant, but some high school students take calc. Keeping this in mind what I would suggest to a student is that this is the same as solving $6|\cos(x)| - x =0$ which itself is equal to two problems for each positive and negative side of the absolute values. 
For example the positive $6\cos (x) - x = 0$ then there is a Taylor expansion of the cosine and take it to the forth power for example and solve that polynomial for its roots. I think does not solve the problem with pen and paper but it is an approach that also is valid. (Also you do not have to really solve the polynomial you. just have to say something about it's roots)
