Let $S$ be the set of all real roots of the equation, ${ 3 }^{ x }\left( { 3 }^{ x }-1 \right) +2=\left| { 3 }^{ x }-1 \right| +\left| { 3 }^{ x }-2 \right| $. Then $S$:

Substitute $y = 3^x$ where $y > 0$, which changes the equation to $y(y - 1) + 2 = |y - 1| + |y - 2|$. For the interval $1 < y < 2$, the equation simplifies to $y^2 - y + 2 = y - 1 + 2 - y$, leading to $y^2 - y + 1 = 0$, which has no real roots. For $y \ge 2$, the equation becomes $y^2 - y + 2 = y - 1 + y - 2$, resulting in $y^2 - 3y + 3 = 0$, which also has a negative discriminant. By testing $y = 1$, we find the equation holds true, giving $3^x = 1$ and the only solution $x = 0$. Therefore, the set $S$ is a singleton.

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