How to remove the sign before the element

Question

I want to get rid of the sign in front of the element, but the symbol variables are involved, so I can't use the Abs function to get rid of the sign.

I want this list {-1, -2, +3, 4, -x, +x, -Fx[3], +Fx[3]} to be processed to {1, 2, 3, 4, x, x, Fx[3], Fx[3]}.

In addition, What should I do when the list contains a ± sign? For example, when the list is {-1, -2, +3, 4, -x, +x, -Fx[3], +Fx[3], -Fx[-3], ±Fy[-2] ,±Fy[±x]}.

Answer 1

I am sure there are many ways to do this. One possible way could be

ClearAll[x,Fx];
expr = {-1, -2, +3, 4, -x, +x, -Fx[3], +Fx[3]};
Times[Internal`SyntacticNegativeQ /@ expr /. {True -> (-1), False -> 1}, expr]

gives

 {1, 2, 3, 4, x, x, Fx[3], Fx[3]}

Answer 2

What about

expr   /. y_ /; y < 0 -> -y 
(*{1, 2, 3, 4, x, x, Fx[3], Fx[3]}*)

Answer 3

Not as terse as Ulrich's replacement, but perhaps more robust:

list = {-1, -2, +3, 4, -x, +x, -Fx[3], +Fx[3], -Fx[-1], +Fx[-1]};

Replace[a : -_ | _?Negative :> -a] /@ list

{1, 2, 3, 4, x, x, Fx[3], Fx[3], Fx[-1], Fx[-1]}

Updated for your latest example:

list2 = {-1, -2, +3, 4, -x, +x, -Fx[3], +Fx[3], -Fx[-1], +Fx[-1], ±Fy[-2], ±Fy[±x]};

Replace[{a : -_ | _?Negative :> -a, ±a_ :> a}] /@ list2

{1, 2, 3, 4, x, x, Fx[3], Fx[3], Fx[-1], Fx[-1], Fy[-2], Fy[±x]}

Answer 4

ClearAll[removeSigns]

removeSigns = # Sign[#] /. Sign -> (1 &) &;

removeSigns @ {-1, -2, +3, 4, -x, +x, -Fx[3], +Fx[3]}

{1, 2, 3, 4, x, x, Fx[3], Fx[3]}

Also

ClearAll[removeSigns2, removeSigns3, removeSigns4, removeSigns5]

removeSigns2 = Abs[#] /. Abs -> (# &) &

removeSigns3 = ReplaceAll[Abs -> Identity] @* Abs

removeSigns4 = Map[Replace[Abs[x_] :> x]]@*Abs (*thanks: Mr.Wizard *)

removeSigns5 = Function[{x}, If[Internal`SyntacticNegativeQ[x], -x, x], Listable];

Answer 5

Sqrt[lst lst]//PowerExpand

{1, 2, 3, 4, x, x, Fx[3], Fx[3]}

(Originally posted as a comment)