CAT Exam Model Paper 3 with solutions for free online practice

(b) We have the inequality $\left|\mathit{x}-1\right|+|\mathit{x}-2|+|\mathit{x}-3|\ge 6$
Here, four cases arise,
Case I When x < 1, then
− ( x − 1) − ( x − 2) − ( x − 3) ≥ 6
⇒ − 3x + 6 ≥ 6
⇒ − 3x ≥ 0
⇒ x ≤ 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