1 ≤ |x - 2| and |x - 2| ≤ 3

When,

|x - 2| ≥ 1

Then,

x – 2 ≤ -1 and x -2 ≥ 1

Now when,

x – 2 ≤ - 1

Adding 2 to both the sides in above equation

x – 2 + 2 ≤ -1 + 2

x ≤ 1

Now when,

x – 2 ≥ 1

Adding 2 to both the sides in above equation

x – 2 + 2 ≥ 1 + 2

x ≥ 3

For |x – 2| ≥ 1: x ≤ 1 or x ≥ 3

When,

|x - 2| ≤ 3

Then,

x – 2 ≥ - 3 and x – 2 ≤ 3

Now when,

x – 2 ≥ -3

Adding 2 to both the sides in above equation

x – 2 + 2 ≥ -3 + 2

x ≥ -1

Now when,

x – 2 ≤ 3

Adding 2 to both the sides in above equation

x – 2 + 2 ≤ 3 + 2

x ≤ 5

For |x – 2| ≤ 3: x ≥ -1 or x ≤ 5

Combining the intervals:

x ≤ 1 or x ≥ 3 and x ≥ -1 or x ≤ 5

Merging the overlapping intervals:

-1 ≤ x ≤ 1 and 3 ≤ x ≤ 5

Therefore,

x ϵ [-1 ,1] Ս [3, 5]