Let's substitute -3 for \( x \) in the equation \( x = x + 1 \):
\[ -3 = -3 + 1 \]
Now simplify the right side:
\[ -3 = -2 \]
This statement is false.
The equation \( x = x + 1 \) does not hold true for any value of \( x \), including -3. Therefore, the equation has no solutions.