<s>[INST] Solving $|a| < |b|$ I apologize if this question in general, but I've been having trouble finding solutions as Google discards absolute value signs and inequality symbols.
I am looking for a way to eliminate absolute value functions in $|a| < |b|$.
I can solve $|a| < b$ and $|a| > b$, but I am unsure what method / combination of methods to use to eliminate absolute value signs from both sides.
Thank you!
An example problem:
$$|x + 2| < |x - 4|$$
 [/INST] $|a| < |b| \iff -|b| < a < |b| \iff 
\begin{cases}
   -b < a < b & \text{If $b > 0$} \\
   \text{No solution.} &\text{If $b=0$} \\
   b < a < -b & \text{If $b < 0$}
\end{cases}$
E. G. $|x + 2| < |x - 4|$
\begin{align}
   x > 4 &\implies-x+4 < x+2 < x-4 \\
         &\implies -2x+4 < 2 < -4 \\
         &\implies \text{No solution.}
\end{align}
\begin{align}
   x > 4 &\implies -x+4 < x+2 < x-4 \\
         &\implies  4 < 2 < 2x-4 \\
         &\implies x < 1
\end{align}
\begin{align}
   x < 4 &\implies x-4 < x+2 < -x+4 \\
         &\implies  -4 < 2 < -2x+4 \\
         &\implies x < 1
\end{align}
Hence $x \in (-\infty, 1)$
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