Question

Solve for x, the inequality given below.

|x − 1| ≤ 5, |x| ≥ 2

Answer 1

x – 1| ≤ 5

There are two cases,

x – 1 ≤ 5

Adding 1 to L.H.S. and R.H.S

⇒ x ≤ 6

⇒ –(x – 1) ≤ 5

⇒ –x + 1 ≤ 5

Subtracting 1 from L.H.S. and R.H.S.,

⇒ –x ≤ 4

⇒ x ≥ –4

From cases 1 and 2, we have

⇒ –4 ≤ x ≤ 6 ......[i]

Also,

|x| ≥ 2

⇒ x ≥ 2 and

⇒ – x ≥ 2

⇒ x ≤ –2

⇒ x ∈ (`oo`, –2] ∪ [2, `oo`)  ......[ii]

Combining equation [i] and [ii], we get

x ∈ [–4, –2] ∪ [2, 6]