CAT Exam Model Paper 3 with solutions for free online practice

(b) We have the inequality $\left| x - 1\right| + \left| x - 2\right| + \left| x - 3\right| \ge 6$ Here, four cases arise, Case I When x < 1, then− ( x − 1) − ( x − 2) − ( x − 3) ≥ 6⇒ − 3x + 6 ≥ 6⇒ − 3x ≥ 0⇒ x ≤ 0Common condition to x < 1 and x ≤ 0 isx ≤ 0 ... (i)Case II When 1≤ x < 2, then( x −1) − ( x − 2) − ( x − 3) ≥ 6x − 1 − x + 2 − x + 3 ≥ 6⇒ − x + 4 > 6 ⇒ − x ≥ 2⇒ x ≤ − 2There is no common condition between 1≤ x < 2 and x ≤ − 2Case III When 2 ≤ x < 3, then⇒ ( x − 1) + ( x − 2) − ( x − 3) ≥ 6⇒ x − 1 + x − 2 − x + 3 ≥ 6⇒ x ≥ 6There is not common condition between2 ≤ x < 3 and x ≥ 6Case 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 isFrom Eqs(i) and (ii), x ≤ 0, x ≥ 4