Indefinite integral

To solve this problem and find the domain for the expression $2x + \frac{6}{x}$, we apply the following steps:

• Step 1: Identify when the fraction $\frac{6}{x}$ is undefined. This occurs when the denominator $x$ equals zero.
• Step 2: To find the restriction, set the denominator equal to zero: $x = 0$.
• Step 3: Solve for $x$ to find the values excluded from the domain. Here, $x \neq 0$.

Since $\frac{6}{x}$ is undefined for $x = 0$, the value $x = 0$ must be excluded from the domain.
Hence, the domain of the equation is all real numbers except zero.

Therefore, the solution to the problem, indicating the domain of the expression, 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.

Since the unknown variable is in the denominator, we should remember that the denominator cannot be equal to 0.

In other words, $x\ne0$

The domain of the function is all those values that, when substituted into the function, will make the function legal and defined.

The domain in this case will be all real numbers that are not equal to 0.

To determine the domain of the function $\frac{2x+20}{\sqrt{2x-10}}$, we must ensure that the expression under the square root is non-negative, because the square root of a negative number is not defined in the real numbers.

We start by analyzing the denominator, specifically the square root, $\sqrt{2x-10}$. For the square root to be valid (for real numbers), we require:

• $2x-10 \geq 0$

Now, solve the inequality $2x - 10 \geq 0$:

• Add 10 to both sides: $2x \geq 10$
• Divide both sides by 2: $x \geq 5$

However, since the expression $2x-10$ also prohibits zero in the denominator (as the square root in the denominator cannot be zero), we strictly have:

• $x > 5$

Thus, the domain of the function is all $x$ such that $x > 5$.

Therefore, the domain of the function $\frac{2x+20}{\sqrt{2x-10}}$ is $x > 5$.

To find the domain of the expression $\frac{5x+8}{2x-6} = 30$, we need to identify values of $x$ that make the denominator of the fraction zero.

Step 1: Identify the denominator of the fraction, which is $2x - 6$.

Step 2: Set the denominator equal to zero to find the values to exclude:

• Solve the equation $2x - 6 = 0$.
• Add 6 to both sides: $2x = 6$.
• Divide both sides by 2: $x = 3$.

Therefore, $x = 3$ is the value that makes the denominator zero, so it must be excluded from the domain.

Given the choices, the correct answer is $x \neq 3$.

Therefore, the domain of the expression is all real numbers except $x = 3$.

This implies that the correct choice is:

$x \neq 3$