Analyzing the Domain of 2x+2/9x+6: Understanding the Function's Limits

To solve this problem, we will determine the domain of the rational function by following these steps:

• Step 1: Identify the denominator of the function, which is $9x + 6$.
• Step 2: Set the denominator equal to zero to find values of $x$ that need to be excluded from the domain: $9x + 6 = 0$.
• Step 3: Solve the equation $9x + 6 = 0$ for $x$.
• Step 4: To solve, subtract 6 from both sides to get $9x = -6$.
• Step 5: Divide each side by 9 to solve for $x$, resulting in $x = -\frac{2}{3}$.
• Step 6: The domain of the function excludes the value $x = -\frac{2}{3}$ since it makes the denominator zero.

Thus, the domain of the given function is all real numbers except $x = -\frac{2}{3}$, expressed as $x \ne -\frac{2}{3}$.

Therefore, the correct choice for the domain is: $x\ne-\frac{2}{3}$.