On the interval $-2\le x\le 4$, the quantity $x-4\le 0$, so $|x-4|=4-x$. Also, $x+2\ge 0$, so $|x+2|=x+2$. Therefore,

so $k=6$.

Now rewrite the target expression on the same interval. Since $2x-8=2(x-4)$ and $x-4\le 0$,

Also, $|x+2|=x+2$. So

Because $k=6$, this becomes

So the equivalent expression is $k+4-x$.

The expression $|x-a|+|x-b|$ represents the sum of the distances from $x$ to $a$ and $b$ on a number line. A key fact is that for any $x$ between $a$ and $b$, this sum is constant and equals $|a-b|$. Since the equation is true for every $x$ in the entire interval $[2,8]$, every value of $x$ from $2$ to $8$ must lie between $a$ and $b$. Therefore, the whole interval $[2,8]$ is contained between $a$ and $b$, and the constant value of the sum on that interval must be $|a-b|$. Because the given sum equals $10$ for all those $x$, it follows that

This must be true.

Distance from the submersible at position $x$ to the sensor at $-5$ is $|x-(-5)|=|x+5|$. Since the submersible is $12$ meters from that sensor, one condition is $|x+5|=12$. The second sensor is at $9$, so its distance from the submersible is $|x-9|$. The problem says this distance is $4$ meters greater than the distance to the first sensor, so $|x-9|=|x+5|+4$, which is equivalent to $|x+5|+4=|x-9|$. Therefore the system that models the situation is $|x+5|=12$ and $|x+5|+4=|x-9|$.

Distance on a number line is represented by absolute value. Since the campsite is at mile 12 and the hiker is 5 miles away, the distance between the hiker's position $x$ and 12 is 5. That gives the equation $|x-12|=5$. This equation has two solutions, $x=17$ and $x=7$, which makes sense because the hiker could be 5 miles to the right or 5 miles to the left of the campsite.

The equation $|r-6|=3$ means the distance between $r$ and $6$ is $3$. So $r-6=3$ or $r-6=-3$. Solving gives $r=9$ or $r=3$. The sum of all possible values is $9+3=12$, so the correct answer is $12$.