$(-7,1)$; see graph.
Work Step by Step
Step 1. From $3|2x+6|-9\lt 15$, we have $|2x+6|\lt 8$, which can be written as $-8\lt 2x+6 \lt 8$ Step 2. Solving the above inequality, we have $-14\lt 2x \lt 2$ and $-7\lt x \lt 1$, or in interval notation $(-7,1)$. Step 3. We can graph the above solution on a number line as shown.