Select The Correct Answer.Solve This Equation: X + 4 6 X = 1 X \frac{x+4}{6x} = \frac{1}{x} 6 X X + 4 ​ = X 1 ​ A. X = 0 , − 2 X = 0, -2 X = 0 , − 2 B. X = 2 X = 2 X = 2 C. X = 0 , 2 X = 0, 2 X = 0 , 2 D. X = − 2 X = -2 X = − 2

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Introduction

In this article, we will be solving the equation $\frac{x+4}{6x} = \frac{1}{x}$ and determining the correct answer from the given options. This equation involves fractions and variables, making it a challenging problem to solve. We will use algebraic techniques to simplify the equation and find the values of $x$ that satisfy the equation.

Step 1: Multiply Both Sides by $6x$

To eliminate the fractions in the equation, we can multiply both sides by $6x$. This will give us:

$$\frac{x+4}{6x} \cdot 6x = \frac{1}{x} \cdot 6x$$

Simplifying the left-hand side, we get:

$$x+4 = 6$$

Step 2: Subtract 4 from Both Sides

To isolate the variable $x$, we can subtract 4 from both sides of the equation:

$$x+4-4 = 6-4$$

This simplifies to:

$$x = 2$$

Step 3: Check for Extraneous Solutions

Before concluding that $x = 2$ is the only solution, we need to check if there are any extraneous solutions. An extraneous solution is a value of $x$ that makes the original equation undefined or false.

In this case, we need to check if $x = 2$ makes the original equation undefined. Substituting $x = 2$ into the original equation, we get:

$$\frac{2+4}{6(2)} = \frac{1}{2}$$

Simplifying the left-hand side, we get:

$$\frac{6}{12} = \frac{1}{2}$$

This is true, so $x = 2$ is not an extraneous solution.

Step 4: Check for Other Solutions

Now that we have found one solution, $x = 2$, we need to check if there are any other solutions. To do this, we can substitute $x = 2$ into the original equation and see if it is true.

Substituting $x = 2$ into the original equation, we get:

$$\frac{2+4}{6(2)} = \frac{1}{2}$$

Simplifying the left-hand side, we get:

$$\frac{6}{12} = \frac{1}{2}$$

This is true, so $x = 2$ is the only solution.

Conclusion

In conclusion, the correct answer is $\boxed{B. x = 2}$.

Final Answer

The final answer is $\boxed{2}$.

Discussion

This problem involves solving a rational equation, which can be challenging. However, by using algebraic techniques, such as multiplying both sides by the denominator and checking for extraneous solutions, we can find the correct solution.

Tips and Tricks

• When solving rational equations, it is often helpful to multiply both sides by the denominator to eliminate the fractions.
• Be careful when checking for extraneous solutions, as a value of $x$ that makes the original equation undefined or false is not a valid solution.
• When checking for other solutions, substitute the known solution into the original equation to see if it is true.

Related Problems

• Solving rational equations with multiple variables
• Finding extraneous solutions in rational equations
• Using algebraic techniques to solve rational equations

References

• [1] "Rational Equations" by Math Open Reference
• [2] "Solving Rational Equations" by Purplemath
• [3] "Rational Equations and Functions" by Khan Academy
  

Introduction

In our previous article, we solved the equation $\frac{x+4}{6x} = \frac{1}{x}$ and determined the correct answer. In this article, we will answer some frequently asked questions about solving rational equations.

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions, which are expressions that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are polynomials and $q$ is not equal to zero.

Q: How do I solve a rational equation?

A: To solve a rational equation, you can use the following steps:

1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
2. Simplify the resulting equation.
3. Solve for the variable.
4. Check for extraneous solutions.

Q: What is the least common multiple (LCM)?

A: The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. In the context of rational equations, the LCM is the smallest expression that is a multiple of each of the denominators.

Q: How do I find the LCM of two or more expressions?

A: To find the LCM of two or more expressions, you can use the following steps:

1. Factor each expression into its prime factors.
2. Identify the common factors and the unique factors.
3. Multiply the common factors together and multiply the unique factors together.
4. The result is the LCM.

Q: What is an extraneous solution?

A: An extraneous solution is a value of the variable that makes the original equation undefined or false. In other words, it is a solution that is not valid.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you can substitute the solution into the original equation and see if it is true. If the solution makes the original equation undefined or false, it is an extraneous solution.

Q: What are some common mistakes to avoid when solving rational equations?

A: Some common mistakes to avoid when solving rational equations include:

• Not multiplying both sides of the equation by the LCM.
• Not simplifying the resulting equation.
• Not checking for extraneous solutions.
• Not using the correct order of operations.

Q: Can you give an example of a rational equation?

A: Yes, here is an example of a rational equation:

$$\frac{x+2}{x-1} = \frac{3}{x+2}$$

Q: How do I solve this equation?

A: To solve this equation, you can use the following steps:

1. Multiply both sides of the equation by the LCM, which is $(x-1)(x+2)$.
2. Simplify the resulting equation.
3. Solve for the variable.
4. Check for extraneous solutions.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = 3$.

Conclusion

In conclusion, solving rational equations can be challenging, but by following the correct steps and avoiding common mistakes, you can find the correct solution. Remember to multiply both sides of the equation by the LCM, simplify the resulting equation, solve for the variable, and check for extraneous solutions.

Final Answer

The final answer is $\boxed{3}$.

Discussion

This article provides a comprehensive overview of solving rational equations, including frequently asked questions and examples. By following the steps outlined in this article, you can become proficient in solving rational equations and apply this skill to a wide range of mathematical problems.

Tips and Tricks

• When solving rational equations, make sure to multiply both sides of the equation by the LCM.
• Simplify the resulting equation to make it easier to solve.
• Check for extraneous solutions to ensure that the solution is valid.
• Use the correct order of operations to avoid mistakes.

Related Problems

• Solving rational equations with multiple variables
• Finding extraneous solutions in rational equations
• Using algebraic techniques to solve rational equations

References

• [1] "Rational Equations" by Math Open Reference
• [2] "Solving Rational Equations" by Purplemath
• [3] "Rational Equations and Functions" by Khan Academy