1. **State the problem:** We need to show that the curve given by the function $$y=\frac{2x+1}{3x+6}$$ is increasing for all $$x \neq -2$$. 2. **Find the derivative:** To determine where the function is increasing, we find its derivative $$y'$$ with respect to $$x$$. Using the quotient rule: $$y' = \frac{(2)(3x+6) - (2x+1)(3)}{(3x+6)^2}$$. 3. **Simplify the numerator:** $$ (2)(3x+6) - (2x+1)(3) = 6x + 12 - (6x + 3) = 6x + 12 - 6x - 3 = 9 $$. 4. **Write the derivative:** $$ y' = \frac{9}{(3x+6)^2} $$. 5. **Analyze the sign of the derivative:** The denominator $$ (3x+6)^2 $$ is always positive except at $$ x = -2 $$ where it is zero (and the function is undefined). Since the numerator is positive (9), the derivative $$ y' $$ is positive for all $$ x \neq -2 $$. 6. **Conclusion:** Because $$ y' > 0 $$ for all $$ x \neq -2 $$, the function $$ y=\frac{2x+1}{3x+6} $$ is increasing on its entire domain except at $$ x = -2 $$ where it is undefined.