I'm trying to nail down the logic behind absolute value inequalities. Given an inequality like $$\Bigg\lvert\frac{2}{x+4}\Bigg\rvert>2,$$ the normal process is to use cases and the definition of absolute value to determine the solution set.
However, upon thinking about the process a little more I've confused myself. When faced with this problem the goal is to find the values of $x$ that make the relation true. But by using cases I am assuming that $\frac{2}{x+4}$ is nonnegative in one case and negative in another and then I proceed to find the solutions that satisfy each case. Is this logic circular since the sign of $\frac{2}{x+4}$ depends on $x$? It seems like I'm assuming something about $x$ before I find it. Can someone explain the logic here? Thanks for the help.
Best Answer
There are two ways for the given inequality to be true. Either $\left(\frac{2}{x+4}\geq 0\right) \land \left(\frac{2}{x+4}>2\right)$, or else $\left(\frac{2}{x+4} < 0\right) \land \left(-\frac{2}{x+4}>2\right)$.
That's a total of $4$ inequalities, joined together with "and"s and "or"s. The solution to each one is some subset of the number line, which can be boiled down to something about $x$. Because of the "and"s and "or"s involved, you want the union of two sets, one of which is the intersection of the solutions of the first two inequalities, and the other is the intersection of the solutions of the last two inequalities.
There's no circular reasoning, because we're not assuming anything about $x$ to get solutions. We're just writing down all of the conditions that have to be satisfied, and finding out where that happens.