How Many Extraneous Solutions Does The Equation Below Have?$\frac{2m}{2m+3} - \frac{2m}{2m-3} = 1$A. 0 B. 1 C. 2 D. 3

Introduction

In this article, we will delve into the world of algebra and solve a seemingly complex equation. The equation in question is $\frac{2m}{2m+3} - \frac{2m}{2m-3} = 1$. Our goal is to determine the number of extraneous solutions that this equation has. To do this, we will first solve the equation and then analyze the solutions to determine which ones are extraneous.

Step 1: Simplify the Equation

The first step in solving this equation is to simplify it by finding a common denominator. The common denominator of the two fractions is $(2m+3)(2m-3)$. We can rewrite the equation as follows:

$$\frac{2m(2m-3) - 2m(2m+3)}{(2m+3)(2m-3)} = 1$$

Step 2: Expand and Simplify

Next, we can expand and simplify the numerator of the fraction:

$$\frac{2m(2m-3) - 2m(2m+3)}{(2m+3)(2m-3)} = \frac{4m^2 - 6m - 4m^2 - 6m}{(2m+3)(2m-3)}$$

Step 3: Combine Like Terms

We can now combine like terms in the numerator:

$$\frac{4m^2 - 6m - 4m^2 - 6m}{(2m+3)(2m-3)} = \frac{-12m}{(2m+3)(2m-3)}$$

Step 4: Factor the Denominator

The denominator of the fraction can be factored as follows:

$$(2m+3)(2m-3) = 4m^2 - 9$$

Step 5: Rewrite the Equation

We can now rewrite the equation as follows:

$$\frac{-12m}{4m^2 - 9} = 1$$

Step 6: Solve for m

To solve for $m$, we can multiply both sides of the equation by the denominator:

$$-12m = 4m^2 - 9$$

Step 7: Expand and Simplify

Next, we can expand and simplify the equation:

$$-12m = 4m^2 - 9$$

$$0 = 4m^2 + 12m - 9$$

Step 8: Factor the Quadratic

The quadratic equation can be factored as follows:

$$0 = (4m - 9)(m + 1)$$

Step 9: Solve for m

We can now solve for $m$ by setting each factor equal to zero:

$$4m - 9 = 0 \Rightarrow m = \frac{9}{4}$$

$$m + 1 = 0 \Rightarrow m = -1$$

Step 10: Check for Extraneous Solutions

To determine which solutions are extraneous, we need to check if they satisfy the original equation. We can plug each solution back into the original equation to see if it is true.

For $m = \frac{9}{4}$:

$$\frac{2m}{2m+3} - \frac{2m}{2m-3} = \frac{2(\frac{9}{4})}{2(\frac{9}{4})+3} - \frac{2(\frac{9}{4})}{2(\frac{9}{4})-3} = \frac{9}{9+3} - \frac{9}{9-3} = \frac{9}{12} - \frac{9}{6} = \frac{3}{4} - \frac{3}{2} = -\frac{3}{4}$$

Since this is not equal to 1, $m = \frac{9}{4}$ is an extraneous solution.

For $m = -1$:

$$\frac{2m}{2m+3} - \frac{2m}{2m-3} = \frac{2(-1)}{2(-1)+3} - \frac{2(-1)}{2(-1)-3} = \frac{-2}{-2+3} - \frac{-2}{-2-3} = \frac{-2}{1} - \frac{-2}{-5} = -2 - \frac{2}{5} = -\frac{12}{5}$$

Since this is not equal to 1, $m = -1$ is also an extraneous solution.

Conclusion

In conclusion, the equation $\frac{2m}{2m+3} - \frac{2m}{2m-3} = 1$ has two extraneous solutions: $m = \frac{9}{4}$ and $m = -1$. Therefore, the correct answer is C. 2.

Final Answer

Q: What is the first step in solving the equation $\frac{2m}{2m+3} - \frac{2m}{2m-3} = 1$?

A: The first step in solving the equation is to simplify it by finding a common denominator. The common denominator of the two fractions is $(2m+3)(2m-3)$.

Q: How do you simplify the numerator of the fraction?

A: To simplify the numerator, we can expand and combine like terms. The numerator can be rewritten as $4m^2 - 6m - 4m^2 - 6m$, which simplifies to $-12m$.

Q: What is the next step in solving the equation?

A: The next step is to factor the denominator of the fraction. The denominator can be factored as $4m^2 - 9$.

Q: How do you rewrite the equation after factoring the denominator?

A: After factoring the denominator, we can rewrite the equation as $\frac{-12m}{4m^2 - 9} = 1$.

Q: What is the next step in solving the equation?

A: The next step is to solve for $m$ by multiplying both sides of the equation by the denominator.

Q: How do you solve for $m$?

A: To solve for $m$, we can expand and simplify the equation. The equation can be rewritten as $0 = 4m^2 + 12m - 9$. We can then factor the quadratic equation as $0 = (4m - 9)(m + 1)$.

Q: How do you find the solutions to the equation?

A: We can find the solutions to the equation by setting each factor equal to zero. The solutions are $m = \frac{9}{4}$ and $m = -1$.

Q: How do you check for extraneous solutions?

A: To check for extraneous solutions, we need to plug each solution back into the original equation to see if it is true.

Q: What is the final answer to the equation?

A: The final answer to the equation is C. 2, which means that the equation has two extraneous solutions: $m = \frac{9}{4}$ and $m = -1$.

Q: What is the importance of checking for extraneous solutions?

A: Checking for extraneous solutions is important because it ensures that the solutions we find are valid and not just a result of the algebraic manipulations.

Q: Can you provide an example of how to check for extraneous solutions?

A: Yes, let's take the solution $m = \frac{9}{4}$ as an example. We can plug this value back into the original equation to see if it is true:

$$\frac{2m}{2m+3} - \frac{2m}{2m-3} = \frac{2(\frac{9}{4})}{2(\frac{9}{4})+3} - \frac{2(\frac{9}{4})}{2(\frac{9}{4})-3} = \frac{9}{9+3} - \frac{9}{9-3} = \frac{9}{12} - \frac{9}{6} = \frac{3}{4} - \frac{3}{2} = -\frac{3}{4}$$

Since this is not equal to 1, $m = \frac{9}{4}$ is an extraneous solution.

Q: Can you provide another example of how to check for extraneous solutions?

A: Yes, let's take the solution $m = -1$ as an example. We can plug this value back into the original equation to see if it is true:

$$\frac{2m}{2m+3} - \frac{2m}{2m-3} = \frac{2(-1)}{2(-1)+3} - \frac{2(-1)}{2(-1)-3} = \frac{-2}{-2+3} - \frac{-2}{-2-3} = \frac{-2}{1} - \frac{-2}{-5} = -2 - \frac{2}{5} = -\frac{12}{5}$$

Since this is not equal to 1, $m = -1$ is also an extraneous solution.

Conclusion

In conclusion, checking for extraneous solutions is an important step in solving equations. By plugging the solutions back into the original equation, we can determine which solutions are valid and which ones are extraneous. In this case, the equation $\frac{2m}{2m+3} - \frac{2m}{2m-3} = 1$ has two extraneous solutions: $m = \frac{9}{4}$ and $m = -1$.