Which Of The Following Represents The Solution Of $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$?A. $x=\frac{1}{5}$B. $x=\frac{5}{9}$C. All Real NumbersD. No Solution

$$

Introduction

In this article, we will be solving a mathematical equation involving fractions. The equation is $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$. Our goal is to find the value of $x$ that satisfies this equation. We will use algebraic techniques to solve for $x$ and determine which of the given options represents the solution.

Step 1: Multiply both sides of the equation by the least common multiple (LCM) of the denominators

To eliminate the fractions, we need to multiply both sides of the equation by the LCM of the denominators. The LCM of $2$, $2x$, and $5x$ is $10x$. Multiplying both sides of the equation by $10x$, we get:

$$10x \cdot \frac{3}{2} = 10x \cdot \left(\frac{3}{2x} - \frac{6}{5x}\right)$$

Step 2: Simplify the equation

Simplifying the equation, we get:

$$15x = 15 - 12x$$

Step 3: Add $12x$ to both sides of the equation

Adding $12x$ to both sides of the equation, we get:

$$15x + 12x = 15$$

Step 4: Combine like terms

Combining like terms, we get:

$$27x = 15$$

Step 5: Divide both sides of the equation by 27

Dividing both sides of the equation by 27, we get:

$$x = \frac{15}{27}$$

Step 6: Simplify the fraction

Simplifying the fraction, we get:

$$x = \frac{5}{9}$$

Conclusion

The solution to the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$ is $x = \frac{5}{9}$. This means that option B is the correct answer.

Discussion

In this article, we used algebraic techniques to solve a mathematical equation involving fractions. We multiplied both sides of the equation by the LCM of the denominators, simplified the equation, added $12x$ to both sides of the equation, combined like terms, divided both sides of the equation by 27, and simplified the fraction to find the value of $x$. The solution to the equation is $x = \frac{5}{9}$, which means that option B is the correct answer.

Final Answer

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

Additional Information

In this article, we solved a mathematical equation involving fractions. We used algebraic techniques to find the value of $x$ that satisfies the equation. The solution to the equation is $x = \frac{5}{9}$, which means that option B is the correct answer. This article demonstrates the importance of using algebraic techniques to solve mathematical equations involving fractions.

Related Topics

• Solving linear equations
• Simplifying fractions
• Multiplying and dividing fractions
• Adding and subtracting fractions

References

• [1] "Algebra" by Michael Artin
• [2] "Elementary and Intermediate Algebra" by Marvin L. Bittinger
• [3] "College Algebra" by James Stewart

Keywords

• Algebra
• Fractions
• Linear equations
• Simplifying fractions
• Multiplying and dividing fractions
• Adding and subtracting fractions

Categories

• Mathematics
• Algebra
• Fractions
• Linear equations

Tags

• Algebra
• Fractions
• Linear equations
• Simplifying fractions
• Multiplying and dividing fractions
• Adding and subtracting fractions
  

$$

Introduction

In our previous article, we solved the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$ and found that the solution is $x = \frac{5}{9}$. In this article, we will answer some frequently asked questions about solving this equation.

Q: What is the least common multiple (LCM) of the denominators in the equation?

A: The LCM of the denominators $2$, $2x$, and $5x$ is $10x$.

Q: Why do we need to multiply both sides of the equation by the LCM of the denominators?

A: We need to multiply both sides of the equation by the LCM of the denominators to eliminate the fractions. This makes it easier to solve for $x$.

Q: How do we simplify the equation after multiplying both sides by the LCM of the denominators?

A: After multiplying both sides by the LCM of the denominators, we simplify the equation by combining like terms. In this case, we get $15x = 15 - 12x$.

Q: How do we solve for $x$ in the equation $15x = 15 - 12x$?

A: To solve for $x$, we add $12x$ to both sides of the equation, which gives us $27x = 15$. Then, we divide both sides of the equation by 27, which gives us $x = \frac{15}{27}$.

Q: Can we simplify the fraction $\frac{15}{27}$?

A: Yes, we can simplify the fraction $\frac{15}{27}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us $x = \frac{5}{9}$.

Q: What is the solution to the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$?

A: The solution to the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$ is $x = \frac{5}{9}$.

Q: Why is it important to use algebraic techniques to solve mathematical equations involving fractions?

A: It is important to use algebraic techniques to solve mathematical equations involving fractions because it allows us to eliminate the fractions and solve for the variable. This makes it easier to find the solution to the equation.

Q: What are some related topics to solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$?

A: Some related topics to solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$ include solving linear equations, simplifying fractions, multiplying and dividing fractions, and adding and subtracting fractions.

Q: What are some references for learning more about solving mathematical equations involving fractions?

A: Some references for learning more about solving mathematical equations involving fractions include "Algebra" by Michael Artin, "Elementary and Intermediate Algebra" by Marvin L. Bittinger, and "College Algebra" by James Stewart.

Q: What are some keywords related to solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$?

A: Some keywords related to solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$ include algebra, fractions, linear equations, simplifying fractions, multiplying and dividing fractions, and adding and subtracting fractions.

Q: What are some categories related to solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$?

A: Some categories related to solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$ include mathematics, algebra, fractions, and linear equations.

Q: What are some tags related to solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$?

A: Some tags related to solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$ include algebra, fractions, linear equations, simplifying fractions, multiplying and dividing fractions, and adding and subtracting fractions.

Conclusion

In this article, we answered some frequently asked questions about solving the equation $\frac{3}{2}=\frac{3}{2x}-\frac{6}{5x}$. We covered topics such as the least common multiple of the denominators, simplifying the equation, solving for $x$, and related topics. We also provided some references and keywords related to solving mathematical equations involving fractions.