The equation || x-1|+a|=4 can have real solutions for x if 'a' belong…

We have, 

$|| x-1|+a|=4$

$⇒ |x-1|+a=±4$

$⇒ |x-1|=±4-a$

For real solutions, we must have

$±4-a>0$

$⇒ 4-a≥ 0$ and $-4-a≥0$

$⇒ a≤4$ and $a≤-4$

$⇒a≤-4⇒a∈(-∞, -4]$