The minimum value of the expression $y = | x + 3 | + | x + 1 | + | x - 5 |$ is

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Solution

$y = | x + 3 | + | x + 1 | + | x - 5 |$
Here the turning points are $- 3, - 1, 5$

$| x + 3 | + | x + 1 | + | x - 5 | = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - (x + 3) - (x + 1) - (x - 5), & x < - 3 (x + 3) - (x + 1) - (x - 5), & - 3 \leq x < - 1 (x + 3) + (x + 1) - (x - 5), & - 1 \leq x < 5 (x + 3) + (x + 1) + (x - 5), & x \geq 5 \end{matrix}$

$\Rightarrow | x + 3 | + | x + 1 | + | x - 5 | = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} 1 - 3 x, & x < - 3 7 - x, & - 3 \leq x < - 1 9 + x, & - 1 \leq x < 5 3 x - 1, & x \geq 5 \end{matrix}$

Hence, the minimum value of the given expression is $8$ .

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