The maximum value of $y = | x - 4 | - | x - 7 |$ is

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Solution

$y = | x - 4 | - | x - 7 |$
Here the turning points are $4, 7$
$y = ⎧ ⎨ ⎩ \begin{matrix} - (x - 4) + (x - 7) & x < 4 (x - 4) + (x - 7) & 4 \leq x < 7 (x - 4) - (x - 7) & x \geq 7 \end{matrix} \Rightarrow y = ⎧ ⎨ ⎩ \begin{matrix} - 3 & x < 4 2 x - 11 & 4 \leq x < 7 3 & x \geq 7 \end{matrix}$

Hence, the maximum value is $3$ .

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