Plotting your curve is difficult, since it is a function of 3 dimensions, you would need to do a ContourPlot3D. But plotting the region in question is not too difficult. The straightforward way to do this is with RegionPlot3D

RegionPlot3D[
 ImplicitRegion[
  1 <= z <= 2 && z <= y <= 4 z && z/y <= x <= 2 z/y, {x, y, z}], 
 PlotPoints -> 100]

But the trouble is that it looks awful, so you can use a function created by Simon Woods, detailed here,

contourRegionPlot3D[region_, {x_, x0_, x1_}, {y_, y0_, y1_}, {z_, z0_, z1_}, 
  opts : OptionsPattern[]] := Module[{reg, preds},
  reg = LogicalExpand[region && x0 <= x <= x1 && y0 <= y <= y1 && z0 <= z <= z1];
  preds = Union@Cases[reg, _Greater | _GreaterEqual | _Less | _LessEqual, -1];
  Show @ Table[ContourPlot3D[
     Evaluate[Equal @@ p], {x, x0, x1}, {y, y0, y1}, {z, z0, z1}, 
     RegionFunction -> Function @@ {{x, y, z}, Refine[reg, p] && Refine[! reg, ! p]},
     opts], {p, preds}]]
contourRegionPlot3D[
 1 <= z <= 2 && z <= y <= 4 z && z/y <= x <= 2 z/y, {z, 1, 2.5}, {y, 
  0, 7}, {x, 0, 2.5}, AxesLabel -> {"x", "y", "z"}]

Apparently ImplicitRegion is new to version 10 - in your original code you had ImplicitRegion, you should notice that it is in blue, meaning it isn't defined.

This will also work,

RegionPlot3D[
 1 <= z <= 2 && z <= y <= 4 z && z/y <= x <= 2 z/y, {z, 1, 2.5}, {y, 
  0, 7}, {x, 0, 2.5}, PlotPoints -> 100]