Unusual Partition

$|3x-|1-2x||=2$ has two solutions, viz.

1. $3x-|1-2x|=2$ $\Rightarrow 3x-2=|1-2x|...(A)$

2. $3x-|1-2x|=-2$ $\Rightarrow 3x+2=|1-2x|...(B)$ .

Further more, each equation - $(A)$ & $(B)$ will provide us with 2 solutions each for $x$ . Let's name them $x_1$ , $x_2$ , $x_3$ & $x_4$ .

(A) $\Rightarrow x_1=\frac{3}{5}, x_2=1$

(B) $\Rightarrow x_3=\frac{-1}{5}, x_4=-3$

$x_1$ & $x_4$ are rejected as they do not satisfy the main equation : $|3x-|1-2x||=2$

Therefore, the values of $x$ satisfying the given equation are $x_2$ & $x_3$ , which are $1, \frac{-1}{5}$ respectively.

$\Rightarrow x_2+x_3=\frac{4}{5}=\frac{m}{n}$

Therefore, $m+n =4+5=\boxed{9}$ .

Let $f(x) = |3x-|1-2x||$ .

Case 1: For $x < \dfrac 12$

$f(x) = |3x-1+2x| = |5x-1| \implies \begin{cases}\small \text{For } x < \dfrac 15 & \implies f(x) = 1-5x = 2 & \implies x = - \dfrac 15 < \dfrac 15 & \small \color{#3D99F6} \text{Accepted} \\ \small \text{For } \dfrac 15 \le x < \dfrac 12 & \implies f(x) = 5x - 1 & \implies 0 \le f(x) < \dfrac 32 < 2 & \small \color{#D61F06} \text{No solution} \end{cases}$

Case 2: For $x \ge \dfrac 12$

$f(x) = |3x-2x+1| = |x+1| = x+1 = 2 \implies x = 1 \ge \dfrac 12 \quad \small \color{#3D99F6} \text{Accepted}$

Therefore, the sum of solution is $1-\dfrac 15 = \frac 45$ $\implies m+n = 4+5 = \boxed 9$ .

I will here make 2 cases

i) $x\geq\dfrac{1}{2}$ .

ii) $x<\dfrac{1}{2}$ .

Solving case 1 we get $|3x+1-2x|=2$ $x+1=2,x+1=-2$ $x=1,x=-3$ Here we will reject $x=-3$ as a solution since it is not in the chosen domain and keep $x=1$ .

Now solving case 2 we get $|3x-1+2x|=2$ $5x-1=2,5x-1=-2$ $x=\dfrac{3}{5} , x={-1}{5}$ Here we will reject $x=\dfrac{3}{5}$ and keep $x=\dfrac{-1}{5}$ .

So we get the sum of $\boxed{1-\dfrac{1}{5}} = \dfrac{4}{5}$ .

Assume x has a positive value, the equation would then solve for us to obtain x=1.

Assume x has a negative value, the equation would then solve for us to obtain x= -0.2.

Then, add both possible values together and we would obtain 4/5.

Hence, the answer is 4+5=9.