Real Solutions for Equation |x-|x-|x-4||| = a

[anemone]

Find all real numbers $$a$$ such that the equation $$ |x-|x-|x-4||| =a
$$ has exactly three real solutions.

 

[zzephod]

Find all real numbers $$a$$ such that the equation $$ |x-|x-|x-4||| =a
$$ has exactly three real solutions.

Spoiler

The curve

$$y= |x-|x-|x-4||| $$

is piece-wise linear with knots at $$x=4/3,\ 2,\ 4$$ and so considering the shape of the curve $$a=y(2)$$ is the value sought.

.

 

[MarkFL]

Spoiler

The curve

$$y= |x-|x-|x-4||| $$

is piece-wise linear with knots at $$x=4/3,\ 2,\ 4$$ and so considering the shape of the curve $$a=y(2)$$ is the value sought.

.

The OP is having trouble logging on to MHB, and has asked me to congratulate you on her behalf on finding the correct answer, and says the method you have used is nearly identical to the method she used in finding the solution.