I'm a little confused as to what you're doing here. You say "assuming x = -2, you can solve this K(x) = -|x| as K(x) = -|2|". Why are you saying that K(x) = -|x|? If that were true, then according to your steps above, K(x) would have to be -2. But when finding x-intercepts, your goal is to find values of x when y is 0. You're graphing the function, so the output of the function is the y-coordinates. I sometimes find it helpful to begin the entire process with a step like this:
K(x) = y = -|x| + 2
Since K(x) = y, and we know y is 0 at the x-intercepts, then K(x) must also be 0 at those intercepts. And setting K(x) to 0 is what you did at the start of your work, which was correct. In fact, all of your work up to the last line where you say -2 = -|x| is fine. Now, if you graph the function, you can see that it crosses the x-axis twice. So you know you'll have two x-intercepts. Those x-values can be found by solving the equation:
-2 = -|x|
2 = |x| (Divide both sides by -1)
So, to rephrase what Stapel said in his previous post... if the absolute value of x is 2, then what are the possible values of x? Those are your x-intercepts.