Finding solutions using graph

Question

Find the number of solutions of the equation $$|\ln|x|\;|=\sin(\pi x)$$

I know how to draw roughly the graph of $|\ln|x|\;|$. However, at $x=1$, both the graphs will pass through $(1,0)$. How do I know whether they will intersect before that point? After $x=1$, graph of $\sin(\pi x)$ will go in the fourth quadrant, while that of $|\ln|x|\;|$ will continue towards infinity. So, I don't think they will intersect later, though I am not sure. How to be sure in such cases where the solutions are to be found using graphs?

Answer 1

Hint: note that $$|\ln|x|\;|\ge 1 \iff x\le -e \lor x\ge e$$ and since $\sin(\pi x)$ is bounded between $-1$ and $1$, all the possible solutions are to be found in $[-e,+e]$. Moreover, from a sketch of the graph we actually see that all the $6$ intersections of the functions involved ($|\ln|x|\;|$ and $\sin(\pi x)$) are in $[-e,+e]$.