1. **State the problem:** We are given the transformation of the absolute value function $f(x) = |x|$ to $g(x) = |x + 3| - 5$ and asked to solve the equation $|6x + 3| = 33$. We also need to determine the domain, range, and intervals of increase and decrease for the function $g(x)$. 2. **Transformation explanation:** The function $g(x) = |x + 3| - 5$ is a horizontal shift of $f(x) = |x|$ to the left by 3 units and a vertical shift down by 5 units. 3. **Solve the equation $|6x + 3| = 33$:** The absolute value equation $|A| = B$ implies $A = B$ or $A = -B$. So, $$6x + 3 = 33 \quad \text{or} \quad 6x + 3 = -33$$ 4. **Solve each linear equation:** For $6x + 3 = 33$: $$6x = 33 - 3$$ $$6x = 30$$ $$x = \frac{30}{6}$$ $$x = 5$$ For $6x + 3 = -33$: $$6x = -33 - 3$$ $$6x = -36$$ $$x = \frac{-36}{6}$$ $$x = -6$$ 5. **Domain and Range of $g(x)$:** - Domain of any absolute value function is all real numbers: $$(-\infty, \infty)$$ - Range is all $y$ values greater than or equal to the vertex's $y$-coordinate. Since vertex is at $(-3, -5)$ and the graph opens upwards, range is: $$[-5, \infty)$$ 6. **Intervals of increase and decrease:** - The graph decreases on the interval to the left of the vertex: $$(-\infty, -3)$$ - The graph increases on the interval to the right of the vertex: $$(-3, \infty)$$ **Final answers:** - Solutions to $|6x + 3| = 33$ are $x = 5, -6$ - Domain: $(-\infty, \infty)$ - Range: $[-5, \infty)$ - Decreasing on $(-\infty, -3)$ - Increasing on $(-3, \infty)$