Question

Solve the equation
$\left|\frac{x^2-8x+12}{x^2-10x+21}\right|=-\frac{x^2-8x+12}{x^2-10x+21}$, the ans is $x\in [2,a)\cup [b,7)$
Find b-a?

Answer 1

This equation has the form $\left|f\left(x\right)\right|=-f\left(x\right)$
where$f\left(x\right)=\frac{x^2-8x+12}{x^2-10x+21}$
Such an equation is equivalent to the collection of systems
$\{\begin{array}{c}\left|f\left(x\right)\right|=-f\left(x\right),iff\left(x\right)\ge 0\\ \left|f\left(x\right)\right|=f\left(x\right),iff\left(x\right)<0\end{array}$
The first system is equivalent to $f\left(x\right)=0$ and second system is equivalent to $f\left(x\right)<0$ then combining both systems we get
$f\left(x\right)\le 0$
$\therefore \frac{x^2-8x+12}{x^2-10x+21}\le 0\Rightarrow \frac{(x-2)(x-6)}{(x-3)(x-7)}\le 0$
From wavy curve method, we get
$x\in [2,3)\cup [6,7)$

$\therefore b-a=3$