Suppose one has an equation in one unknown that has three or more absolute value signs such as $$ |ax + b| + |cx + d| + |ex + f| = gx + h $$ Without invoking sophisticated techniques such as the CAD algorithm described in this question, is there an elementary approach other than case-by-case analysis that will yield a solution?
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The case-by-case method (with 4 cases, as I discussed in my answer in the question to which you linked) is the only approach to solving such an equation that I've ever seen within the context of pre-college mathematics.