TITLE: What is the best way to solve an equation involving multiple absolute values?
QUESTION [20 upvotes]: An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases}
ax-b & \text{ if } x\ge \frac{b}{a} \\ 
b-ax & \text{ if } x< \frac{b}{a} 
\end{cases}$, so an equation with $n$ separate absolute value expressions can be split up into $2^n$ cases, but is there a better way?
For example, with $|2x-5|+|x-1|+|4x+3|=13$, is there a better way to handle all the possible combinations of $x\ge\frac{5}{2}$ versus $x<\frac{5}{2}$, $x\ge 1$ versus $x< 1$, and $x\ge-\frac{3}{4}$ versus $x<-\frac{3}{4}$?

REPLY [9 votes]: This is merely a very special case of the powerful CAD (cylindrical algebraic decomposition) algorithm for quantifier elimination in real-closed fields, e.g. see Jirstrand's paper [1] for a nice introduction.
[1] M. Jirstrand. Cylindrical algebraic decomposition - an introduction. 1995
Technical report S-58183, Automatic Control group, Department of Electrical Engineering
Linkoping University, Linkoping, Sweden.
Freely available here or here.