Problem 515

Solve by discussing the relationship between the intersection points of images and the solutions of equations. Let $y=f_{1}(x)=\left|x^{2}-1\right|$ and $y=f_{2}(x)=x+k$. Sketch the graphs of these two functions, and note that $y=x+k$ is a set of lines parallel to $y=x$. Thus, we can obtain a partition of the set of real numbers:
$$\begin{array}{ll} (-\infty,-1), \\ \{-1\}, & (-1, \end{array}$$
1), $\{1\},\left(1, \frac{5}{4}\right),\left\{\frac{5}{4}\right\},\left(\frac{5}{4}, \infty\right)$.

The number of solutions to the original problem in each part is 0, 1,
$$2,3,4,3,2$$