1. **State the problem:** Solve the equation $ (3x+7)! = 1 $ for $x$. 2. **Recall the factorial definition:** The factorial of a non-negative integer $n$, denoted $n!$, is the product of all positive integers from 1 to $n$. By definition, $0! = 1$ and $1! = 1$. 3. **Analyze the equation:** Since factorial is defined for non-negative integers, $3x+7$ must be a non-negative integer. 4. **Set factorial equal to 1:** The factorial equals 1 only when the argument is 0 or 1, i.e., $$3x + 7 = 0 \quad \text{or} \quad 3x + 7 = 1$$ 5. **Solve each case:** - For $3x + 7 = 0$: $$3x = -7$$ $$x = -\frac{7}{3}$$ - For $3x + 7 = 1$: $$3x = 1 - 7 = -6$$ $$x = -2$$ 6. **Check domain:** Factorial is defined only for non-negative integers. Since $3x+7$ must be an integer $\geq 0$, check if these values satisfy this: - For $x = -\frac{7}{3}$, $3x+7 = 0$ which is valid. - For $x = -2$, $3(-2)+7 = 1$ which is valid. 7. **Final answer:** $$x = -\frac{7}{3} \quad \text{or} \quad x = -2$$