5. Given the function $f(x)=\left|x^{2}+3 x\right|, x \in R$. If the equation $f(x)-a|x-1|=0$ has exactly 4 distinct real roots, then the range of the real number $a$ is $\qquad$

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5. $(0,1) \cup(9,+\infty)$

Analysis: Clearly, $x=1$ is not a root of the equation, so we transform the equation $f(x)-a|x-1|=0$ into $a=\left|\frac{x^{2}+3 x}{x-1}\right|$. Since $a=\left|\frac{x^{2}+3 x}{x-1}\right|=\left|x-1+\frac{4}{x-1}+5\right|$, from the graph, we can see that to have 4 distinct roots, then $a \in(0,1) \cup(9,+\infty)$.