Solve the following system of equations in R.
$| x - 1 | + | x - 2 | + | x - 3 | \geq 6$

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Solution

We have,

$| x - 1 | + | x - 2 | + | x - 3 | - 6 \geq 0 . . . (i) C a s e I : | x - 1 | \geq 0 0 \geq 1 \Rightarrow x - 1 - (x - 2) - (x - 3) - 6 \geq 0 \Rightarrow - x + 4 - 6 \geq 0 \Rightarrow - x \geq 2 \Rightarrow x \leq - 2 \Rightarrow (- \infty, - 2] C a s e I I : | x - 2 | \geq 0 x \geq 2 \Rightarrow x - 1 + x - 2 - (x - 3) - 6 \geq 0 x - 6 \geq 0 \Rightarrow [6, \infty) . . . (i i i) C a s e I I I : | x - 3 | \geq 0 x \geq 3 \Rightarrow x - 1 + x - 2 + x - 3 - 6 \geq 0 \Rightarrow 3 x - 12 \geq 0 \Rightarrow 3 x \geq 12 \Rightarrow x \geq 4 \Rightarrow ∴ x ϵ [4, \infty)$

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