How can I expand a inequality with Abs

Question

I want

Abs[x] + Abs[y] <= 1

to be convert to

x + y <= 1 && x + y >= -1 && x - y <= 1 && x - y >= -1  

How could I get it with Mathematica?

Answer 1

expandAbs[eq_] := And @@ Flatten@Module[{case},
eq //. 
 x_?((case = Cases[#, Abs[_], Infinity, 1]) =!= {} &) :> (case = 
    First[case]; {x /. case -> First[case], 
    x /. case -> -First[case]})]
expandAbs[Abs[x] + Abs[y] <= 1]

Answer 2

In case you want to visualize the answer:

RegionPlot[Evaluate@Reduce[Abs[x] + Abs[y] <= 1, {x, y}, Reals], {x,-2,2}, {y,-2,2}]