Solve the following rational equation.

StartFraction 3 Over x EndFraction plus StartFraction 2 Over x minus 4 EndFraction3x+2x−4equals=StartFraction x minus 2 Over x minus 4 EndFractionx−2x−4

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Part 1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

The solution set is 

StartSet nothing EndSetenter your response here.

​(Simplify your answer. Use a comma to separate answers as​ needed.)

B.

The solution is the empty set.

To solve the rational equation:

\[
\frac{3}{x} + \frac{2}{x - 4} = \frac{x - 2}{x - 4}
\]

### Step 1: Identify the restrictions
Before solving, identify the restrictions by finding the values of \(x\) that would make the denominators zero.

- For \(x\), the denominator in \(\frac{3}{x}\), we must have \(x \neq 0\).
- For \(x - 4\), the denominator in \(\frac{2}{x - 4}\) and \(\frac{x - 2}{x - 4}\), we must have \(x \neq 4\).

So, \(x \neq 0\) and \(x \neq 4\) are the restrictions.

### Step 2: Multiply through by the common denominator
The common denominator between the terms is \(x(x - 4)\). Multiply both sides of the equation by this to eliminate the denominators:

\[
x(x - 4) \left( \frac{3}{x} + \frac{2}{x - 4} \right) = x(x - 4) \left( \frac{x - 2}{x - 4} \right)
\]

This simplifies as follows:

\[
(x - 4) \cdot 3 + x \cdot 2 = x(x - 2)
\]

\[
3x - 12 + 2x = x^2 - 2x
\]

### Step 3: Simplify the equation
Now, combine like terms:

\[
5x - 12 = x^2 - 2x
\]

Move all terms to one side of the equation:

\[
0 = x^2 - 2x - 5x + 12
\]

\[
x^2 - 7x + 12 = 0
\]

### Step 4: Solve the quadratic equation
Now, solve the quadratic equation \(x^2 - 7x + 12 = 0\). We can factor this quadratic:

\[
(x - 3)(x - 4) = 0
\]

Thus, the solutions are:

\[
x = 3 \quad \text{or} \quad x = 4
\]

### Step 5: Check for extraneous solutions
Recall that \(x \neq 4\) is a restriction. Therefore, \(x = 4\) is an extraneous solution and must be discarded.

The only valid solution is \(x = 3\).

### Final Answer:
The solution set is:

\[
\{ 3 \}
\]

### Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How do you factor quadratic equations with different methods?
2. What happens when rational equations have no solutions?
3. How do restrictions affect the solution of rational equations?
4. Can you explain how to clear denominators in rational equations?
5. How do you handle extraneous solutions in general?

**Tip:** Always check for extraneous solutions by considering the restrictions from the original denominators!

Solve the following rational equation.

StartFraction 3 Over x EndFraction plus StartFraction 2 Over x minus 4 EndFraction3x+2x−4equals=StartFraction x minus 2 Over x minus 4 EndFractionx−2x−4

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.
The solution set is StartSet nothing EndSet.
(Simplify your answer. Use a comma to separate answers as needed.)

B.
The solution is the empty set.

Learn how to solve a rational equation with quadratic terms by eliminating the denominators and factoring the quadratic expression x^2 - 7x + 12. This step-by-step solution covers key algebraic concepts, including restrictions and extraneous solutions, suitable for Grades 9-12.

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