How to tell Mathematica that a variable represents a real number, is greater than zero, or similar conditions

assumptions.nb

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Notebook[{

Cell[CellGroupData[{
Cell["Assumptions", "Title"],

Cell[TextData[{
  "How can I tell ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " that a variable represents a real number, is greater than zero, or \
simmilar things?"
}], "Text"],

Cell[TextData[{
  "Certain commands such as ",
  StyleBox["Integrate",
    FontSlant->"Italic"],
  ", ",
  StyleBox["Simplify",
    FontSlant->"Italic"],
  ", etc. use the environment variable ",
  StyleBox["$Assumptions",
    FontSlant->"Italic"],
  " to get information on the variables used in the expression. By default \
is"
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Cell[TextData[{
  "For single commands one can use ",
  StyleBox["Assuming",
    FontSlant->"Italic"],
  ", this modifies ",
  StyleBox["$Assumptions",
    FontSlant->"Italic"],
  " temporarily, i.e. only for this single expression.\nThe option ",
  StyleBox["Assumptions",
    FontSlant->"Italic"],
  " of ",
  StyleBox["Integrate",
    FontSlant->"Italic"],
  " is similar, it points to ",
  Cell[BoxData[
      FormBox[
        StyleBox["$Assumptions",
          FontSlant->"Italic"], TraditionalForm]]],
  " by default but can be changed to any set of assumptions.\nTo modify the \
assumptions not just temporarily for one command, you can write directly to \
the environment variable.\n\nExample:"
}], "Text"],

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Cell[BoxData[
    \(Sign[x]\)], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
    \($Assumptions\  = \((x\  > \ 0)\)\)], "Input"],

Cell[BoxData[
    \(x > 0\)], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
    \($Assumptions\  = \ $Assumptions\  && \ \((x\  \[Element] \ 
            Reals)\)\)], "Input"],

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