(b) We have the inequality |x−1|+|x−2|+|x−3|≥6 Here, four cases arise, Case I When x < 1, then − ( x − 1) − ( x − 2) − ( x − 3) ≥ 6 ⇒ − 3x + 6 ≥ 6 ⇒ − 3x ≥ 0 ⇒ x ≤ 0 Common condition to x < 1 and x ≤ 0 is x ≤ 0 ... (i) Case II When 1≤ x < 2, then ( x −1) − ( x − 2) − ( x − 3) ≥ 6 x − 1 − x + 2 − x + 3 ≥ 6 ⇒ − x + 4 > 6 ⇒ − x ≥ 2 ⇒ x ≤ − 2 There is no common condition between 1≤ x < 2 and x ≤ − 2 Case III When 2 ≤ x < 3, then ⇒ ( x − 1) + ( x − 2) − ( x − 3) ≥ 6 ⇒ x − 1 + x − 2 − x + 3 ≥ 6 ⇒ x ≥ 6 There is not common condition between 2 ≤ x < 3 and x ≥ 6 Case IV When x ≥ 3, then ( x − 1) + ( x − 2) + ( x − 3) ≥ 6 ⇒ 3x − 6 ≥ 6 ⇒ 3x ≥ 12 ⇒ x ≥ 4 ... (ii) Common condition for x ≥ 3 and x ≥ 4 is From Eqs(i) and (ii), x ≤ 0, x ≥ 4