Let A={x:x∈R,|x|<1};
B={x:x∈R,|x−1|≥1} and
A∪B=R−D, then the set D is
{x:1≤x<2}
Since A={x:x∈R,|x|<1}
⇒A={x:x∈R,−1<x<1}
and B={x:x∈R,|x−1|≥1}
⇒B={x:x∈R,x−1≤−1 or x−1≥1}
={x:x∈R,x≤0 or x≥2}
Now, A∪B={x:x<1 or x≥2}
It is given that A∪B=R−D,
using the above information, we can conclude that D={x:x∈R,1≤x<2}.