[Coda]

You found the by-inspection solution! Good job! Yes, you are correct: It's 45 degrees because a 1.0 coefficient of friction means that the (magnitude of the) force of friction is equal to the (magnitude of the) normal force. And the only angle where the horizontal component and the vertical component of gravity are equal is 45 degrees.

Nothing wrong with F_x' being an arrow. Arrows mean vectors, and F_x' is certainly a vector. (I mean, yes, it's just a mathematical construct for the purposes of analysis and not a representation of any individual force, but that doesn't mean it's not a vector.)

Is M used for the coefficient of friction in your textbook or something? I had to figure it out by context because I'm used to seeing mu being written as "u" if you can't actually write "μ".

Your math looks pretty good there. Try solving the problem with a coefficient of 0.5 and see if your strategy is sound. (Preliminary thing to think about: Does a lower coefficient mean it starts sliding on a higher angle, or a lower one?)