Please go through the following link: Why is equating one of the bracks to zero in this equation correct?
Now, the expression given there is $(x+1)(x+3)$, I understand now why we take either of these two be zero and then find out the value. However, suppose I have the inequality: $(x + 2)(x - 3)\lt0$. In thise case, I'm expected to find out two values by evaluation $(x + 2)=0$ and $(x - 3)=0$ and then divide the real number line in 4 intervals and figure out for which intervals the inequality holds by figuring out whether the expression on the LHS is less than $0$ or greater than $0$.
However, what I don't get is, why are we allowed to use the evaluations $(x + 2)=0$ and $(x - 3)=0$? I understand that in the first case (in case of the equation), either of those values can be zero. But in this case, neither of values need to be zero in order to satisfy the inequality. Then how is equating to zero and finding out the value(s) of $x$ a valid operation?