Question

If x satisfies |x − 1|+ |x − 2| + |x − 3| ≥ 6, then

• A. 0 ≤ x ≤ 4
• B. x ≤−2 or x ≥ 4
• C. x ≤ 0 or x ≥ 4
• D. None of these

Answer 1

Answer: (C)

x ≤ 0 or x ≥ 4

Answer 2

|x − 1| + |x −2| + |x − 3| ≥ 6

Consider f(x) = |x − 1| + |x + 2| + | x − 3|

$f(x) = \left\{ \begin{matrix} 6 - 3x, & x < 1 \\ \begin{matrix} 4 - x, \\ x, \\ 3x - 6, \end{matrix} & \begin{matrix} 1 \leq x < 2 \\ 2 \leq x < 3 \\ x \geq 3 \end{matrix} \end{matrix} \right.\$

Graph of f(x) shows f(x) ≥ 6 for x ≥ 0 or x ≥ 4