Start by factoring $2$ from inside the absolute value:

Using the property $|ab|=|a||b|$, this becomes

So the whole expression is

That matches $2|x-3|+4$.

Points of intersection occur where the two expressions for $y$ are equal:

Subtract $1$ from both sides:

An absolute value equation of the form $|\text{expression}|=6$ has two solutions, because the expression can be $6$ or $-6$:

So $x=10$ or $x=-2$. That means the horizontal line $y=7$ intersects the graph of $y=|x-4|+1$ at two points. The correct answer is $2$.

The equation $|d-k|=m$ has solutions that are the same distance from $k$. Since $d=3$ and $d=11$ both satisfy the equation, $k$ must be the midpoint of $3$ and $11$:

Then the distance from $k$ to either solution is $m$:

So,

An absolute value equation of the form $|x-4|=c$ has exactly two distinct real solutions when $c>0$. Here, $c=k-1$, so the condition is $k-1>0$. Solving gives $k>1$. If $k=1$, then $|x-4|=0$, which has only one solution, $x=4$. If $k<1$, then $k-1<0$, and an absolute value cannot equal a negative number, so there are no real solutions. Therefore, the statement that must be true is $k>1$.

Distance on a number line is represented with absolute value. The drone's distance from the base station at $0$ is $|x|$, and its distance from the weather sensor at $18$ is $|x-18|$. The rule says the distance from the base station is exactly $6$ kilometers less than twice the distance from the weather sensor. Twice the sensor distance is $2|x-18|$, and $6$ less than that is $2|x-18|-6$. So the correct modeling equation is