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

k \in [0, 5]

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k \in [0, 5)

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k \in (0, 5)

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k \in (1, 5]

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Solution

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
$\Rightarrow 0 < k < 5$
$∴ k \in (0, 5)$

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