The set of real values of x satisfying |x-1| ≤ 3 and |x-1| ≥ 1 is

Question

The set of real values of x satisfying |x-1| ≤ 3 and |x-1| ≥ 1 is (A) [2,4] (B) (-∞, 2] ∪[4,+∞) (C) [-2,0] ∪[2,4] (D) None of these

Tardigrade Admin · Accepted Answer

| x -1| ≤ 3 ⇒ -3 ≤ x -1 ≤ 3 ⇒ -2 ≤ x ≤ 4 and | x -1| ≥ 1 ⇒ x -1 ≤-1 or x -1 ≥ 1 ⇒ x ≤ 0 or x ≥ 2 Taking the common values of x, we get x ∈[-2,0] ∪[2,4]

Q. The set of real values of $x$ satisfying $∣ x - 1∣ \leq 3$ and $∣ x - 1∣ \geq 1$ is

$[2, 4]$

$(- \infty, 2] \cup [4, + \infty)$

$[- 2, 0] \cup [2, 4]$

None of these

$∣ x - 1∣ \leq 3$
$\Rightarrow - 3 \leq x - 1 \leq 3$
$\Rightarrow - 2 \leq x \leq 4$
and $∣ x - 1∣ \geq 1$
$\Rightarrow x - 1 \leq - 1$ or $x - 1 \geq 1$
$\Rightarrow x \leq 0$ or $x \geq 2$
Taking the common values of $x$ , we get
$x \in [- 2, 0] \cup [2, 4]$

Q. The set of real values of x satisfying ∣x−1∣≤3 and ∣x−1∣≥1 is

[2,4]

(−∞,2]∪[4,+∞)

[−2,0]∪[2,4]