Inequality plot like Wolfram Alpha

I'd like to reproduce in Mathematica the output of this Wolfram Alpha query. Of course I can just use WolframAlpha but I'd like to manipulate the graphics (add another function, add legends, etc.) so I'm looking for a 'native' function.

I know about RegionPlot but I don't understand how to use it for this kind of inequality (if I just put it as first argument, it outputs a rectangular plot, which makes sense). Moreover I'd like to keep the graphs of the functions in the plot.

• Use the Filling option of Plot. – Szabolcs Oct 28 '19 at 9:16

Plot[{Sin[x], 1/x}, {x, 0, 4},