Solve for x, the inequalities in |x – 1| ≤ 5, |x| ≥ 2

|x – 1|≤ 5

There are two cases,

Case 1:

x – 1 ≤ 5

⇒ x ≤ 6 [adding 1 both side]

Case 2:

⇒ -(x – 1) ≤ 5

⇒ -x + 1 ≤ 5

⇒ -x ≤ 4 [Subtracting 1 both side]

⇒ x ≥ -4

From both cases, we have

⇒ -4 ≤ x ≤ 6 [1]

Also,

|x| ≥ 2

⇒ x ≥ 2 and

⇒ -x ≥ 2

⇒ x ≤ -2

⇒ x ∈ (∞, -2] ∪ [2, ∞) [2]

Combining [1] and [2] we get

x ∈ [-4, -2] ∪ [2, 6]