Question

Let $A=\left[ x:x\in R,|x|<1 \right];B=\left[ x:x\in R,|x-1|\ge 1 \right]$ and $A\cup B=R-D$ then the set D is
A. $\left[ x:1 < x\le 2 \right]$
B. $\left[ x:1\le x < 2 \right]$
C. $\left[ x:1\le x\le 2 \right]$
D. None of these

Answer 1

We will first find the domain of ‘x’ In set ‘A’ and Set ‘B’ which are the elements of sets, from where we will find out the $A\cup B$ , which will help us to find the set D by comparing the result we got from $A\cup B$ and $R-D$ .

Complete step by step answer:
Moving ahead with the question in step wise manner,
We need to find out the value of set D, so let us first find out the set A and Set B elements, so first in set A we are given with the info $A=\left[ x:x\in R,|x|< 1 \right]$ to get the domain of ‘x’ we need to simplify $|x|< 1$ by using the mode property. So we have $|x|< 1$ by simplifying it using the mode inequality we will get;
$-1 < x < 1$ .
Similarly simplifying the set B we have $B=\left[ x:x\in R,|x-1|\ge 1 \right]$ , so to find the elements of set B, we need to simplify $|x-1|\ge 1$ , which on simplifying using property of mode we will get;
$\begin{aligned} & |x-1|\ge 1 \\\ & x\ge 2\cup x\le 0 \\\ \end{aligned}$
So we got set A and set B which are equal to $-1 < x < 1$ and $x\ge 2\cup x\le 0$ .
So now we need to find out the $A\cup B$ . So let us draw it on the number line and find out the union of both sets, so let us plot the number line and mark the points $-1,0,1$ and 2 as shown in figure.

As we can see, if we take the union of two sets A and B then all numbers get covered on the number line except the numbers from 1 to 2, including 1 and excluding 2. We can also say that $A\cup B=R-[1,2)$ so by comparing it with $R-D$ we can say that set D is $[1,2)$ which can also be written as $1\le x < 2$ .
So set D is $1\le x < 2$ .

So, the correct answer is “Option B”.

Note: As in the set B the number 2 is included, so in the set $A\cup B$ the number will be included, same is with 1; as in set A number 1 is not included in set so it will be not present in the number line, hence we can say that in the number line $A\cup B$ all number will be there except the number 2 which is included and the number 1 will be not be counted in $A\cup B$ because it is not in set A.