First slide

Introduction to sets

Question

$If x satisfies | x - 1 | + | x - 2 | + | x - 3 | \geq 6, then$

Moderate

A

$0 \leq x \leq 4$
B

$x \leq - 2 or x \geq 4$
C

$x \leq 0 or x \geq 4$
D

None of these

Solution

$\begin{array}{l} Let | x - 1 | + | x - 2 | + | x - 3 | < 6 \\ \Rightarrow | (x - 1) + (x - 2) + (x - 3) | < | x - 1 | + | x - 2 | + | x - 3 | < 6 \\ \Rightarrow | 3 x - 6 | < 6 \end{array}$
$\begin{array}{ll} \Rightarrow | x - 2 | < 2 \\ \Rightarrow - 2 < x - 2 < 2 \\ \Rightarrow 0 < x < 4 \end{array}$
$Hence, for | x - 1 | + | x - 2 | + | x - 3 | \geq 6, x \leq 0 or x \geq 4$

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