What Is The Domain Of The Equation $\frac{10}{x-4}=5$?A. All Real Numbers B. All Positive Real Numbers C. All Real Numbers Except $x=4$ D. All Real Numbers Except $x=-4$

Introduction

When dealing with equations involving fractions, it's essential to consider the domain of the equation. The domain of an equation is the set of all possible input values (in this case, the variable x) for which the equation is defined. In other words, it's the set of all possible values of x that make the equation true. In this article, we'll explore the domain of the equation $\frac{10}{x-4}=5$ and determine which of the given options is correct.

Understanding the Equation

The given equation is $\frac{10}{x-4}=5$. To find the domain of this equation, we need to consider the values of x that make the equation undefined. In this case, the equation is undefined when the denominator (x-4) is equal to zero. This is because division by zero is undefined in mathematics.

Finding the Domain

To find the domain of the equation, we need to solve the equation x-4=0. This will give us the value of x that makes the denominator equal to zero, and therefore, the equation undefined.

# Import necessary modules
import sympy as sp

# Define the variable
x = sp.symbols('x')

# Solve the equation x-4=0
solution = sp.solve(x-4, x)

# Print the solution
print(solution)

The solution to the equation x-4=0 is x=4. This means that the equation $\frac{10}{x-4}=5$ is undefined when x=4.

Analyzing the Options

Now that we've found the value of x that makes the equation undefined, we can analyze the given options.

A. All real numbers: This option is incorrect because the equation is undefined when x=4, which is a real number.

B. All positive real numbers: This option is also incorrect because the equation is undefined when x=4, which is a positive real number.

C. All real numbers except x=4: This option is correct because the equation is undefined when x=4, but it's defined for all other real numbers.

D. All real numbers except x=-4: This option is incorrect because the equation is not undefined when x=-4. In fact, the equation is defined for all real numbers except x=4.

Conclusion

In conclusion, the domain of the equation $\frac{10}{x-4}=5$ is all real numbers except x=4. This means that the equation is defined for all real numbers except x=4, and it's undefined when x=4.

Final Answer

The final answer is C. All real numbers except x=4.

Additional Information

It's worth noting that the domain of an equation can be affected by other factors, such as the presence of square roots or logarithms. In these cases, the domain may be restricted to a specific range of values. However, in the case of the equation $\frac{10}{x-4}=5$, the domain is simply all real numbers except x=4.

Frequently Asked Questions

• Q: What is the domain of the equation $\frac{10}{x-4}=5$? A: The domain of the equation $\frac{10}{x-4}=5$ is all real numbers except x=4.
• Q: Why is the equation undefined when x=4? A: The equation is undefined when x=4 because the denominator (x-4) is equal to zero, and division by zero is undefined in mathematics.
• Q: What is the final answer? A: The final answer is C. All real numbers except x=4.
  

Introduction

In our previous article, we explored the domain of the equation $\frac{10}{x-4}=5$ and determined that the domain is all real numbers except x=4. In this article, we'll answer some frequently asked questions (FAQs) about the domain of this equation.

Q&A

Q: What is the domain of the equation $\frac{10}{x-4}=5$?

A: The domain of the equation $\frac{10}{x-4}=5$ is all real numbers except x=4.

Q: Why is the equation undefined when x=4?

A: The equation is undefined when x=4 because the denominator (x-4) is equal to zero, and division by zero is undefined in mathematics.

Q: What happens if x=4 in the equation $\frac{10}{x-4}=5$?

A: If x=4 in the equation $\frac{10}{x-4}=5$, the equation becomes $\frac{10}{0}=5$, which is undefined.

Q: Can we simplify the equation $\frac{10}{x-4}=5$?

A: Yes, we can simplify the equation $\frac{10}{x-4}=5$ by multiplying both sides by (x-4), which gives us 10=5(x-4).

Q: How do we solve the equation 10=5(x-4)?

A: To solve the equation 10=5(x-4), we need to isolate the variable x. We can do this by first expanding the right-hand side of the equation, which gives us 10=5x-20. Then, we can add 20 to both sides of the equation, which gives us 30=5x. Finally, we can divide both sides of the equation by 5, which gives us x=6.

Q: Is x=6 in the domain of the equation $\frac{10}{x-4}=5$?

A: Yes, x=6 is in the domain of the equation $\frac{10}{x-4}=5$ because x=6 is not equal to 4.

Q: Can we find the domain of the equation $\frac{10}{x-4}=5$ using a graph?

A: Yes, we can find the domain of the equation $\frac{10}{x-4}=5$ using a graph. The graph of the equation will have a vertical asymptote at x=4, which means that the equation is undefined at x=4.

Q: What is the significance of the domain of the equation $\frac{10}{x-4}=5$?

A: The domain of the equation $\frac{10}{x-4}=5$ is significant because it tells us the values of x for which the equation is defined. In other words, it tells us the values of x that we can use to solve the equation.

Conclusion

In conclusion, the domain of the equation $\frac{10}{x-4}=5$ is all real numbers except x=4. We can find the domain of the equation by solving the equation x-4=0, which gives us x=4. We can also find the domain of the equation by graphing the equation and looking for the vertical asymptote at x=4.

Final Answer

The final answer is C. All real numbers except x=4.

Additional Information

It's worth noting that the domain of an equation can be affected by other factors, such as the presence of square roots or logarithms. In these cases, the domain may be restricted to a specific range of values. However, in the case of the equation $\frac{10}{x-4}=5$, the domain is simply all real numbers except x=4.

Frequently Asked Questions (FAQs)

• Q: What is the domain of the equation $\frac{10}{x-4}=5$? A: The domain of the equation $\frac{10}{x-4}=5$ is all real numbers except x=4.
• Q: Why is the equation undefined when x=4? A: The equation is undefined when x=4 because the denominator (x-4) is equal to zero, and division by zero is undefined in mathematics.
• Q: Can we simplify the equation $\frac{10}{x-4}=5$? A: Yes, we can simplify the equation $\frac{10}{x-4}=5$ by multiplying both sides by (x-4), which gives us 10=5(x-4).
• Q: How do we solve the equation 10=5(x-4)? A: To solve the equation 10=5(x-4), we need to isolate the variable x. We can do this by first expanding the right-hand side of the equation, which gives us 10=5x-20. Then, we can add 20 to both sides of the equation, which gives us 30=5x. Finally, we can divide both sides of the equation by 5, which gives us x=6.
• Q: Is x=6 in the domain of the equation $\frac{10}{x-4}=5$? A: Yes, x=6 is in the domain of the equation $\frac{10}{x-4}=5$ because x=6 is not equal to 4.
• Q: Can we find the domain of the equation $\frac{10}{x-4}=5$ using a graph? A: Yes, we can find the domain of the equation $\frac{10}{x-4}=5$ using a graph. The graph of the equation will have a vertical asymptote at x=4, which means that the equation is undefined at x=4.
• Q: What is the significance of the domain of the equation $\frac{10}{x-4}=5$? A: The domain of the equation $\frac{10}{x-4}=5$ is significant because it tells us the values of x for which the equation is defined. In other words, it tells us the values of x that we can use to solve the equation.