Question

The values of $k$ for which the equation $||x-2|-5|=k$ has $4$ distinct solutions is

• A. $k\in [0,5]$
• B. $k\in [0,5)$
• C. $k\in (0,5)$
• D. $k\in (1,5]$

Answer 1

The correct option is C $k\in (0,5)$

$|x-2|\to$

$|x-2|-5\to$

$||x-2|-5|\to$

Now for $4$ distinct roots, the graph of $y=k$ should intersect the above graph at $4$ distinct point
$\Rightarrow 0<k<5$
$\therefore k\in (0,5)$