1. The problem is to verify the correctness of the solution intervals on the number line with critical points at $-3$, $-\frac{3}{2}$, $3$, and $5$. 2. The solution intervals given are: $$x < -3 \quad \text{or} \quad -\frac{3}{2} < x < 3 \quad \text{or} \quad x > 5$$ 3. To check this, we consider the sign chart around the critical points $-3$, $-\frac{3}{2}$, $3$, and $5$. 4. The intervals are: - $(-\infty, -3)$ positive - $(-3, -\frac{3}{2})$ negative - $(-\frac{3}{2}, 3)$ positive - $(3, 5)$ negative - $(5, \infty)$ positive 5. This matches the given solution intervals where the function or expression is positive. 6. Therefore, the result is correct. Final answer: $$x < -3 \quad \text{or} \quad -\frac{3}{2} < x < 3 \quad \text{or} \quad x > 5$$