Example 4 Solve the inequality $(x-a)(x-2a)<0$ for $x$, where the constant $a \in \mathbf{R}$.

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Solution (1) When $a > 2a$, i.e., $a < 0$, the solution set of the original inequality is $(a, 2a)$.
Key Points for Solving The key to this problem is to discuss the size of the two roots $a$ and $2a$ of the equation $(x-a)(x-2a)=0$.