10. The solution to the equation ||$x-3|+3 x|=1$ is
10. -2 or -1.

Square both sides of the equation, we get
$$
(x-3)^{2}+6 x|x-3|+9 x^{2}=1 \text {. }
$$

When $x \geqslant 3$, we have
$$
x^{2}-6 x+9+6 x^{2}-18 x+9 x^{2}=1 \text {. }
$$

Simplifying, we get $16 x^{2}-24 x+8=0$.
Solving, we get $x=\frac{1}{2}$ or $x=1$ (which does not satisfy $x \geqslant 3$, so we discard it).
When $x<3$, we have
$$
x^{2}-6 x+9-6 x^{2}+18 x+9 x^{2}=1 \text {. }
$$

Simplifying, we get $4 x^{2}+12 x+8=0$.
Solving, we get $x=-2$ or $x=-1$.
Since both are less than 3, the solutions to the equation are -2 or -1.