The number of real solutions of the equation $|x^2-3x+2|=x-1$ is:

Solution

1. Factor the equation as $|(x-1)(x-2)|=x-1$.
2. If $x-1=0$, then $x=1$ is a solution.
3. If $x-1>0$, we divide by $(x-1)$ to get $|x-2|=1$, which implies $x-2=1$ or $x-2=-1$, yielding $x=3$ or $x=1$.
4. If $x-1<0$, the equation becomes $-(x-1)(x-2)=x-1$, which simplifies to $-(x-2)=1$, resulting in $x=1$, which contradicts the condition $x<1$.
5. The distinct real solutions are $$\begin{array}{rl}x& =1,x\\ & =2,\end{array}$$ and $x=3$.
6. Hence the answer is (C).