Question

Find the value of 'x' for which it satisfies the inequality |x - 2| + |x - 3| + |x - 4| $\ge$ 9.

• A. $x\le -2orx\ge 3$
• B. $-2\le x<3$
• C. $0\le x\le 6$
• D. $x\le 0orx\ge 6$

Answer 1

The correct option is D $x\le 0orx\ge 6$

Best way to solve this problem is by checking for different values of x.
For x = 0, |x - 2| + |x - 3| + |x - 4| = 9 =9 is true.
And for x = 7, |x - 2| + |x - 3| + |x - 4| = 12 is greater than 9.
Hence, option (a), (b), (c) can be eliminated.
$\therefore$ Option (d) is correct answer.

Alternatively:
Case I : If x $\ge$ 4
x - 2 + x - 3 + x - 4 $\ge$ 9
or, 3x $\ge$ 18 or, x $\ge$ 6 . . . (1)
Case II : 3 $\le$ x < 4
x - 2 + x - 3 - x + 4 $\ge$ 9
x $\ge$ 10; not possible.
Case III: If 2 $\le$ x < 3
x - 2 - x + 3 - x + 4 $\ge$ 9
-x $\ge$ 4
x $\le$ -4, not possible.
Case IV: If x < 2
- x - 2 - x + 3 - x + 4 $\ge$ 9
- 3x $\ge$ 0
$\Rightarrow$ x $\le$ 0 . . . (4)
So, from equation (1), (2) (3) and (4), x $\le$ or x $\ge$ 6