Indefinite integral

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: $x\ne-\frac{2}{3}$.

To determine the domain of the function $\frac{5-x}{2-x}$, we need to identify and exclude any values of $x$ that make the function undefined. This occurs when the denominator equals zero.

• Step 1: Set the denominator equal to zero:
  $2-x = 0$
• Step 2: Solve for $x$:
  Adding $x$ to both sides gives $2 = x$. Hence, $x = 2$.

This means that the function is undefined when $x = 2$. Therefore, the domain of the function consists of all real numbers except $x = 2$.

Thus, the domain is: $x \ne 2$.

The correct answer choice is:

Yes, $x\ne2$

To determine the domain of the function $\frac{24}{21x-7}$, we need to ensure that the denominator is not equal to zero.

Step 1: Set the denominator equal to zero and solve for $x$:

• $21x - 7 = 0$

• $21x = 7$

• $x = \frac{7}{21}$

• $x = \frac{1}{3}$

The function is undefined when $x = \frac{1}{3}$ because it would cause division by zero.

Step 2: The domain of the function is all real numbers except $x = \frac{1}{3}$.

Therefore, the domain of the function is all $x$ such that $x \neq \frac{1}{3}$.

Thus, the correct answer is $\boxed{ x\ne\frac{1}{3}}$.

To find the domain of the given function $22\left(\frac{2}{x} - 1\right) = 30$, follow these steps:

• Identify critical terms: The term $\frac{2}{x}$ is undefined when $x = 0$ because division by zero is undefined.
• We need to exclude $x = 0$ from the domain to ensure the function remains defined.
• The correct domain for the equation is all real numbers except $x = 0$.

Thus, the domain of the equation is $x \neq 0$.

Therefore, the solution to the problem is $x \neq 0$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the fraction's denominator.

• Step 2: Determine where this denominator equals zero.

• Step 3: Exclude this value from the domain.

Now, let's work through each step:

Step 1: The given equation is $2x - 3 = \frac{4}{x}$. Notice that the fraction $\frac{4}{x}$ has a denominator of $x$.

Step 2: Set the denominator equal to zero to determine where it is undefined.

$\begin{aligned} x &= 0 \end{aligned}$

Step 3: Since the expression is undefined at $x = 0$, we must exclude this value from the domain.

Therefore, the domain of the expression is all real numbers except 0, formally stated as $x \neq 0$.

The correct solution to the problem is: x ≠ 0.