Let g(x)=x^3-6x^2+9x-2. The number of distinct real solutions of the equation |g(x)|=1 is

The equation splits into $g(x)=1$ and $g(x)=-1$. Since $g'(x)=3(x-1)(x-3)$, $g$ has a local maximum $g(1)=2$ and a local minimum $g(3)=-2$, and it rises from $-\infty$ to $+\infty$. Any horizontal level strictly between $-2$ and $2$ therefore meets the graph in exactly three points. Both levels $1$ and $-1$ lie in this open interval, so each equation has three distinct real roots. The two equations cannot share a root because $g$ cannot equal $1$ and $-1$ at once. Hence the total number of distinct solutions is $3+3=6$. Numerically the roots are $0.468,\,1.653,\,3.879$ for $g=1$ and $0.121,\,2.347,\,3.532$ for $g=-1$, all distinct, which confirms the count.