Question:

Step 1: Recognize the algebraic relationship between the expressions. We observe that the left side can be decomposed as: $$x^2 + 6x + 6 = (x^2 + 4x + 9) + (2x - 3)$$ This allows us to rewrite the original equation in the form $|A + B| = |A| + |B|$, where: - $A = x^2 + 4x + 9$ - $B = 2x - 3$ Step 2: Determine the sign of $A = x^2 + 4x + 9$. We complete the square for $A$: $$A = x^2 + 4x + 9 = (x + 2)^2 + 5$$ Since $(x + 2)^2 \geq 0$ for all real $x$, we have: $$A = (x + 2)^2 + 5 \geq 5 > 0$$ Therefore, $A$ is always positive for all real values of $x$. Step 3: Apply the condition for $|A + B| = |A| + |B|$. The equation $|A + B| = |A| + |B|$ holds if and only if $A$ and $B$ have the same sign (or one of them is zero). This is equivalent to the condition: $$AB \geq 0$$ Step 4: Solve for the required condition on $x$. Since we established that $A > 0$ for all $x$, the condition $AB \geq 0$ becomes: $$A \cdot B \geq 0 \implies B \geq 0$$ Therefore: $$2x - 3 \geq 0$$ $$x \geq \frac{3}{2}$$ Step 5: State the final answer. The solution set is $\left[\dfrac{3}{2}, \infty\right)$. The answer is **Option 1**.