10. In the Cartesian coordinate system, the graph of the function $y=\frac{x+1}{|x|+1}$ has three different points on the line $l$, and the sum of the x-coordinates of these three points is 0. Find the range of the slope of $l$.

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10. In the Cartesian coordinate system, the graph of the function $y=\frac{x+1}{|x|+1}$ has three different points on the line $l$, and the sum of the x-coordinates of these three points is 0. Find the range of the slope of $l$.
Let the line $l: y=kx+b$, combining $\left\{\begin{array}{l}y=kx+b, \\ y=\frac{x+1}{-x+1}\end{array} \Rightarrow kx^{2}+(b-k+1)x-b+1=0\right.$ $\Rightarrow x_{1}+x_{2}=-\frac{b-k+1}{k}$, let $kx+b=1 \Rightarrow x_{3}=\frac{1-b}{k}$,
thus $x_{1}+x_{2}+x_{3}=\frac{k-2b}{k}=0 \Rightarrow b=\frac{1}{2}k$.
Substituting into the equation, it can be simplified to: $kx^{2}-\left(\frac{1}{2}k-1\right)x-\frac{1}{2}k+1=0$, the equation has two negative roots,
so $\left\{\begin{array}{l}\frac{\frac{1}{2}k-1}{k}<0\end{array} \quad \Rightarrow 0<k<\frac{2}{9}\right.$.