Basic Mathematics — JEE Main · 2022 · 25 Jun 2022 · Shift 2 Question with Solution

Let

A = {x \in R : ∣ x + 1∣ < 2}

and

B = {x \in R : ∣ x - 1∣ \geq 2}

. Then which one of the following statements is NOT true ?

Answer: (B)

Solution

First, solve the inequalities to find sets

A

and

B

.

For

A

∣ x + 1∣ < 2 \Rightarrow - 2 < x + 1 < 2 \Rightarrow - 3 < x < 1

, so

A = (- 3, 1)

.

For

B

∣ x - 1∣ \geq 2 \Rightarrow x - 1 \leq - 2

x - 1 \geq 2 \Rightarrow x \leq - 1

x \geq 3

, so

B = (- \infty, - 1] \cup [3, \infty)

.

Now check each option:

(A) $A - B = (- 3, 1) ∖ [(- \infty, - 1] \cup [3, \infty)] = (- 1, 1)$ ✓
(B) $B - A = [(- \infty, - 1] \cup [3, \infty)] ∖ (- 3, 1) = (- \infty, - 3] \cup [3, \infty) = R - (- 3, 3)$ , NOT $R - (- 3, 1)$ ✗
(C) $A \cap B = (- 3, 1) \cap [(- \infty, - 1] \cup [3, \infty)] = (- 3, - 1]$ ✓
(D) $A \cup B = (- 3, 1) \cup [(- \infty, - 1] \cup [3, \infty)] = R - [1, 3)$ ✓

Option (B) is NOT true.

JEE Main · 2022 · 25 Jun 2022 · Shift 2 · Single correct option