Solving Inequalities - a curious observation This question is about an interesting observation that comes up whenever one solves an inequality with a variable in the denominator.
eg.
1/x < 1
Why is it that, in the solution to these inequalities, x can be a values that are 1) on the outsides of the two critical values or 2) in between the two critical values?
I have never seen inequalities where x can take values in between both values AND on the outside of one critical value.
Another query: what is the formal, purely algebraic definition of a critical value (if there is one)?
 A: I'm a little unclear about your question, as it mentions two critical values (I'm assuming you mean restrictions or non-permissible values) but the inequality you included only has one. However, here are two examples that might address what you're asking:
$$\frac{1}{(x+1)(x-1)^2} > 0 \quad \text{and} \quad \frac{1}{(x+1)^2(x-1)^2} > 0$$
Both the above inequalities have the restrictions $x \neq \pm 1\,$.
The solution for the first is $\,-1 < x < 1\,$ or $\,x>1\,$, i.e. in between the two non-permissible values and to the right of the larger one.
The solution to the second is $\,x<-1\,$ or $\,-1 < x < 1\,$ or $\,x>1\,$, i.e. in between the two non-permissible values and outside both.
You'll no doubt notice that the difference between the two examples is the power of the $\,(x+1)\,$ factor in the denominator. In general, for  inequalities involving polynomial or rational functions, 'the solutions "change" around zeroes that come from factors with odd exponents (or zeroes with odd 'multiplicity', if you're familiar with that term), whereas they don't "change" around zeroes that come from factors with even exponents.
