Answer: \(x\lt 2\) or \(x\gt 6\).

The key point is to split the absolute-value expression into the correct cases before solving.

The graph of \(y=|x-5|\) is a V-shape with vertex \((5,0)\) and \(y\)-intercept \((0,5)\).

The graph of \(y=6-|2x-7|\) is an inverted V-shape. Its vertex occurs when \(2x-7=0\), so

\(x=\frac72.\)

The vertex is therefore

\(\left(\frac72,6\right).\)

When \(x=0\),

\(y=6-|-7|=-1,\)

so the \(y\)-intercept is \((0,-1)\).

The two graphs intersect at \(x=2\) and \(x=6\).

The inequality

\(|x-5|\gt 6-|2x-7|\)

means that the graph of \(y=|x-5|\) is above the graph of \(y=6-|2x-7|\).

This happens outside the two intersection points, so

\(x\lt 2\quad\text{or}\quad x\gt 6.\)