1. **State the problem:** Determine whether the equation $x - 4 = -4 + x$ has one solution, no solutions, or infinitely many solutions. 2. **Simplify both sides:** The left side is $x - 4$ and the right side is $-4 + x$. 3. **Compare both sides:** Notice that $x - 4$ and $-4 + x$ are actually the same expression because addition is commutative. 4. **Rewrite the equation:** $x - 4 = x - 4$ 5. **Interpretation:** Since both sides are identical for all values of $x$, the equation holds true for every real number $x$. 6. **Conclusion:** The equation has infinitely many solutions. 7. **Support with two values:** For example, if $x=0$, then $0 - 4 = -4$ and $-4 + 0 = -4$, so both sides are equal. If $x=5$, then $5 - 4 = 1$ and $-4 + 5 = 1$, so both sides are equal. Therefore, the equation is true for all $x$. **Final answer:** The equation has infinitely many solutions.