What is the sum of the roots of this equation: $\left\lvert \frac {2x-1}{6} \right\rvert + \left\lvert 2-4x \right\rvert = \frac {26}{3}$?

Question

What is the sum of the roots of this equation? $$\left\lvert \frac {2x-1}{6} \right\rvert + \left\lvert 2-4x \right\rvert = \frac {26}{3}.$$

I currently don't have any idea about this question.

Answer 1

Hint:

$$\iff26\cdot\dfrac63=|2x-1|+6|(-2)(2x-1)|=|2x-1|(1+6|-2|)=?$$ as $|a|=|-a|$ for any complex number $a$

Answer 2

Hint. The left-hand side is a function in $x$ which is symmetric with respect to $1/2$ (note that $|2-4x|=2|2x-1|$). Therefore the average of the roots is $1/2$.

Answer 3

it simplifies to $$\frac{1}{6}|2x-1|+2|2x-1|=\frac{26}{3}$$ and from here we get $$|2x-1|=4$$ Can you proceed?