Okay so I did a lot of research online and on stackexchange on solving absolute value inequalities.
I read in an answer here that on having different set of values in different cases, we take union when the inequality is greater than and take intersection when the inequality is less than. This logic solved almost all of my problems correctly however, I came across a question which seems to be an exception to this rule.
As we see, the denominator is necessarily always positive except for $x=3$, so it won't affect the inequality in general, on solving the two cases, one while opening the mod normally and the other while opening it up with a negative sign, I got two cases as follows.
$x\in (-2,-5)$ and $x\in (2,3) \cup (3,5)$
Now according to the rule the answer should be the intersection of the two cases since the inequality sign in the original problem is less than. However, the answer in my book arrives on taking the union.
Someone just please let me know what is going wrong and if this rules won't work everytime, is there any solid and purely logical rule that I can use blindfoldy for every problem without having to consider checking the values to figure out whether to take union or intersection? Please help me guys, I'm literally going mad over it.
At last, Thanks a lot for showing patience and interest in this querry, thanks for giving me your precious time.