Number of solutions for a given logical equation I came across the following question while studying logic and cannot find a solution for it anywhere. I am studying by myself and think I just don't know exactly the right terms to search for it online (I'm not sure it is called a logical equation so excuse the title of this question in case it isn't):

Given the proposition $P$, it's logical value is defined as $[P] = 0$, in case $P$ is false, and $[P] = 1$, in case $P$ is true.
Consider the following open sentences defined in the set of integers:
$ P_i(x): x \le 5$
$ P_{ii}(x): x \ge 3$
$ P_{iii}(x): $ x is odd 
$ P_{iv}(x): x \ge 6$
How many solutions does the following equation have?
$ x = [P_i(x)] + 2 \cdot[P_{ii}(x)]+3\cdot[P_{iii}(x)]+4\cdot[P_{iv}(x)]$

I've made this jsfiddle and from there, I can count the number of solutions through a loop. In this case I've looped from 0 to 1000 and it yields 2 solutions. Though I can clearly reason it wouldn't be possible for a very large number to work here, since these are all sums of multiplications of 0s or 1s, I am having a hard time articulating exactly why. How would you go about finding the largest number possible, in this very specific case? So you wouldn't have to loop through values of X too far off from it?

In addition to @Rushabh Mehta's answer:
note that $[P_i]$ can be seen as a "classic" (aka Pre-Calculus) function. For example,  $f_1=[P_1]$ is simply the function $$f_1(x)=[P_1(x)]=\left\{ \begin{array}{rl} 1, & x \leqslant 5 \\ 0, &x>5 \end{array} \right.$$
Therefore, you can study the given equation $x=f_1(x)+2f_2(x)+3f_3(x)+4f_4(x)$ using all techniques you already knows for this kind of problems: considering cases (as in Rushabh Mehta's answer), drawing the graph carefully,...
