The number of integers which do not satisfy the given equation $| x - 1 | + | x - 2 | + | x - 3 | \geq 6$ is

3

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4

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2

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0

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Solution

The correct option is A $3$
$⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} - x + 1 - x + 2 - x + 3, & x \leq 1 x - 1 - x + 2 - x + 3, & 1 < x \leq 2 x - 1 + x - 2 - x + 3, & 2 < x \leq 3 x - 1 + x - 2 + x - 3, & x > 3 \end{matrix}$
$⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} - 3 x + 6, & x \leq 1 - x + 4, & 1 < x \leq 2 x, & 2 < x \leq 3 3 x - 6, & x > 3 \end{matrix}$

So for inequality $- 3 x + 6 \geq 6 \Rightarrow x \leq 0 3 x - 6 \geq 6 \Rightarrow x \geq 4$

so, $x = 1, 2, 3$ do not satisfy the inequality.
$∴ n = 3$

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