help with absolute value question $||x-3|-2|\leq 1$

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could anyone help in validating this absolute value question that my son is working on? $|(|x-3|-2)|\leq 1$

$-1\leq|x-3|-2\leq 1$

$1\leq|x-3|\leq 3$

$1\leq x-3\leq 3$

$4\leq x\leq 6$

or

$-3\leq x-3\leq -1$

$0\leq x\leq 2$

thus

$x\in [4,6]\cup [0,2]$

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It looks okay to me. ${}{}{}{}{}{}$