I am just wondering, is there a general method to solve equalities and inequalities involving absolute valued functions?
Case i) $|x\pm k|= |x \pm l| + c$
Case ii) $|x \pm k| \ge |x \pm l| +c$
I would appreciate some detailed solution to solving the general cases I am confused with above.
For either case we can assume that $k, l \in \mathbb{R} $ so the problem becomes: $$|x+k| = |x+l| + c$$ There are 2 possible cases for each of the absolute values, giving 4 possible solutions for $x$: $$x+k = x+l + c \Rightarrow x \in \mathbb{R}, k = l+c$$ $$x+k = -(x+l) + c \Rightarrow x = \frac{c-k-l}{2} $$ $$-(x+k) = x+l + c \Rightarrow x = \frac{-c-k-l}{2} $$ $$-(x+k) = -(x+l) + c \Rightarrow x \in \mathbb{R},k = l-c $$