Interpretation of a fundamental inequality

Question

How do you interpret $0 < \vert x-a\vert < d$ with algebra? (I understand what this means geometrically but am struggling to understand this through algebra.)

Answer 1

The absolute value is defined as $$\vert x\vert= \begin{cases} x, \quad\text{if } x\geq0\\ -x, \quad\text{if } x<0 \end{cases} .$$ If you apply this definition to your inequality, you obtain two equations: $$ 0<x-a<d\quad\text{ for }x-a\geq0,\\0<-(x-a)<d\quad\text{ for }x-a<0 $$ Solving this leads to $$a<x<d+a\;\vee\;a-d<x<a.$$ Note that the conditions $x-a\geq0$ respectively $x-a<0$ are implicitly contained in the final solution.