<s>[INST] What is the domain of k? $2|x-2|-3|x-3|=k$ What is the domain of k? 
$2|x-2|-3|x-3|=k$
Using the definition of absolute function, I obtain: 
$|x-2|= \begin{cases}
        x-2, & x\ge2 \\
        2-x, & x<2\\
        \end{cases}\\$
$|x-3|= \begin{cases}
        x-3, & x\ge3 \\
        3-x, & x<3 \\
        \end{cases} \\$
For $x<2$
$x-5=k$
For $2\le x<3$
$5x-13=k$
For $x \ge 3$
$-x+5=k$
But I need to find the domain of k.
 [/INST] From the $3$ rules that you worked out, the graph of $2|x-2|-3|x-3|$ increases on $(-\infty, 3]$ and then decreases from $[3, \infty)$.
Hence the possible values of $k$ are $(-\infty, f(3)) = (-\infty, 2]$.
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