Question

If |x – 1| + |x – 2| + |x – 3| ≥ 6 then ______.

पर्याय

• 0 ≤ x ≤ 4

• x ≤ –2 or x ≥ 4

• x ≤ 0 or x ≥ 4

• x ≤ –1 or x ≥ 3

Answer 1

If |x – 1| + |x – 2| + |x – 3| ≥ 6 then x ≤ 0 or x ≥ 4.

Explanation:

|x – 1| + |x – 2| + |x – 3| ≥ 6

∴ if x < 1, the inequation becomes

–(x – 1) – (x – 2) – (x – 3) ≥ 6

or –3x + 6 ≥ 6

∴ x ≤ 0

∴ x is such that x < 1 and x ≤ 0

⇒ x ≤ 0

If 1 ≤ x < 2, the inequation becomes

(x – 1) – (x – 2) – (x – 3) ≥ 6

or –x + 4 ≥ 6,

∴ x ≤ –2

No such values satisfy 1 ≤ x ≤ 2

If 2 ≤ x ≤ 3, the inequation becomes

(x – 1) + (x – 2) – (x – 3) ≥ 6

or x ≥ 6, no such values satisfy 2 ≤ x < 3

If x ≥ 3, the inequation becomes 

(x – 1) + (x – 2) + (x – 3) ≥ 6

or 3x – 6 ≥ 6 or 3 x ≥ 12

∴ x ≥ 4 which satisfy x ≥ 3

∴ The values of x satisfy x ≤ 0 or x ≥ 4

∴ Required solution is x ≤ 0 or x ≥ 4