Consider the equation: $$y=\big||x|-1\big|$$ In what interval will $y$ lie, if more than two values of $x$ should satisfy the above equation?
My work:
I was able to solve this by plotting the graph of the given function. From the graph, we can identify that $y \in (0,1]$. However, I wish to avoid using graphical solutions such as this.
I’m unsure how to solve this algebraically. Any help is appreciated.
\begin{align} ||x|-1| = y &\to |x| - 1 = \pm y \\ &\to |x| = 1 \pm y \\ &\to x = \pm1 \pm y, \ \end{align} so we automatically have at least two solutions, and potentially as many as four. Now the first equation, $$|x| - 1 = \pm y,$$ has both signs achievable as distinct solutions iff $0 < y \leq 1$, since $|x| - 1 \geq -1$. So this condition, $y \in (0, 1]$, is also what is needed to guarantee three or more distinct solutions.