When solving equations with absolute values, such as $\lvert x \rvert = 1$, you break into two cases, $x=1$ and $-x=1$, isolate the variable, and take the union of the two resulting solution sets. The final solution is $x=1$ or $x=-1.$ It makes sense that the solution is a union, because absolute value is a two-to-one function onto positive numbers.
If you were instead going to solve an inequality of the form $\lvert x\rvert \neq 1,$ then it would make sense by De Morgan's laws that the answer should instead be an intersection, since we are negating the disjunction of the two cases. $x\neq 1$ and $x\neq -1.$
But how do we decide for $<,>,\leq,\geq$ inequalities? Why is the solution to $\lvert x\rvert <1$ the intersection of the two cases, (namely, $x<1$ and $x>-1$), while the solution of $\lvert x\rvert>1$ is the union of the cases?
Best Answer
When you consider $x<0$ or $x\ge0$, they are really both "or"; it's just that one simplifies to "and".
Specifically, $|x|<1⟺x<0$ and $−x<1$ (i.e., $x>−1$) or $x≥0$ and $x<1$
$\iff-1<x<0$ or $0\le x<1$$\iff -1<x$ and $x<1$,
whereas $|x|>1⟺x<0$ and $−x>1$ (i.e., $x<−1$) or $x≥0$ and $x>1$
$\iff x\lt-1$ or $x>1.$