PROBLEM Absolute Value Function and Coordinate Geometry

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I have tried solving it, but I'm not sure if I did it right. If someone can help me with the complete solution. I'd highly appreciate

The equation I found was $y = 2x + 14$. I'm not sure if that was right.

A straight line passing through the points $(-5, 4)$ and $(2, 18)$, intersects an absolute value function with a vertex at $(-6, 22)$ that passes through the point $(5, -11)$ at points $A$ and $B$. Find the distance between points $A$ and $B$.

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How to find the absolute value function? You are given the vertex $(-6,22)$ and another point $(5,-11)$

Since vertex is given, the function is of the form

$$y = k(|x+6|) + 22$$

Here, the $k$ will change the slope of the absolute

Now, we know that the point $(5,-11)$ lies on it

$$\implies -11 = 11k + 22$$

$$\implies k = -3$$

Therefore the absolute value function is given as

$$y = -3|x+6| + 22$$

Now can you proceed?