Solve the equation $\left|\frac{x^2-8x+12}{x^2-10x+21}\right|=-\frac{x^2-8x+12}{x^2-10x+21}$

This equation has the form $\left|f\right(x\left)\right|=-f\left(x\right)$

when, $f\left(x\right)=\frac{x^2-8x+12}{x^2-10x+21}$

such an equation is equivalent to the collection of systems

$\left\{\begin{array}{l}f\left(x\right)=-f\left(x\right),\text{ if }f\left(x\right)\ge 0\\ f\left(x\right)=f\left(x\right),\text{ if }f\left(x\right)<0\end{array}\right.$

The first system is equivalent to f(x)=0 and the second system is equivalent to $f\left(x\right)<0$the combining both systems, we get

$$\begin{gathered} f\left(x\right) \le 0 \\ \therefore \frac{x^2-8x+12}{x^2-10x+21}\le 0 \\ \Rightarrow \frac{(x-2)(x-6)}{(x-3)(x-7)} \le 0 \end{gathered}$$

Hence, by Wavy curve method,

$x\in [2,3)\cup [6,7)$