Can we sketch the graph of $y = |x| + |x-1| + |x-4|$?

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If \(p\) is real the value of \(\big|p\big|\) is \(p\) if \(p \ge 0\) and is \(-p\) if \(p < 0\).

By considering the ranges \(x < 0\), \(0 \le x < 1\), \(1 \le x < 4\) and \(x \ge 4\), show that the graph of \[\begin{equation*} y = \big|x\big| + \big|x-1\big| + \big|x-4\big| \end{equation*}\]

consists of four line segments and give the equation of each segment. Sketch the graph.

Calculate the roots of the equations

1. \(\big|x\big| + \big|x-1\big| + \big|x-4\big| = 6 - \dfrac{1}{3}x\),
2. \(\big|x\big| + \big|x-1\big| + \big|x-4\big| = \dfrac{1}{x}\).