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.

*

*If $ -\infty <x≤2$, then you have

$$k=2(2-x)-3(3-x)=x-5 \Longrightarrow - \infty <k=x-5≤-3$$

*

*If $2<x≤3$, then you have

$$k=2(x-2)-3(3-x)=5x-13 \Longrightarrow -3<k=5x-13≤2$$

*

*If $3<x<\infty$, then you have

$$k=2(x-2)-3(x-3)=-x+5 \Longrightarrow 2>k=-x+5> - \infty \Longrightarrow -\infty <k<2$$
Finally, you get

$$k \in (- \infty, -3] ∪ (-3, 2] ∪ (-\infty, 2)= (-\infty,2].$$

