#### Answer

all real numbers

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inqeuality, $
|x+2|\gt x
,$ use the definition of a greater than (greater than or equal to) absolute value inequality and solve each resulting inequality. Finally, graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
x+2\gt x
\\\\\text{OR}\\\\
x+2\gt -x
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
x+2\gt x
\\\\
x-x\gt -2
\\\\
0\gt -2
\text{ (TRUE)}
\\\\\text{OR}\\\\
x+2\gt -x
\\\\
x+x\gt -2
\\\\
2x\gt -2
\\\\
x\gt -\dfrac{2}{2}
\\\\
x\gt -1
.\end{array}
Hence, the solution is the set of all real numbers.