solve the rational equation. If there is more than one answer, separate your answers with commas.  If there is no solution, enter "No Solution."

5 plus fraction numerator 9 over denominator x minus 2 end fraction equals 2 plus fraction numerator x plus 7 over denominator x minus 2 end fraction

To solve the rational equation:

\[
5 + \frac{9}{x - 2} = 2 + \frac{x + 7}{x - 2}
\]

we'll begin by eliminating the denominators. Since both fractions have the same denominator \(x - 2\), we can multiply each term by \(x - 2\) to clear the fractions.

### Step 1: Clear the Denominators
Multiply both sides by \(x - 2\):

\[
5(x - 2) + 9 = 2(x - 2) + (x + 7)
\]

### Step 2: Distribute and Simplify
Now, expand each term:

\[
5x - 10 + 9 = 2x - 4 + x + 7
\]

Combine like terms:

\[
5x - 1 = 3x + 3
\]

### Step 3: Solve for \(x\)
Isolate \(x\) by moving all terms involving \(x\) to one side and constants to the other:

\[
5x - 3x = 3 + 1
\]

\[
2x = 4
\]

\[
x = 2
\]

### Step 4: Check for Extraneous Solutions
Since the original equation has a denominator of \(x - 2\), we need to check if \(x = 2\) is a valid solution. Substituting \(x = 2\) into \(x - 2\) gives zero in the denominator, which makes the expression undefined. 

Thus, \(x = 2\) is not a valid solution.

### Conclusion
Since there are no other solutions, the answer is:

\[
\text{No Solution}
\]

Would you like a further breakdown of these steps?

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### Related Questions:
1. How do you solve rational equations with multiple denominators?
2. What are common methods for checking extraneous solutions in rational equations?
3. Why does setting the denominator to zero invalidate solutions in rational equations?
4. Can a rational equation have solutions if all potential answers are extraneous?
5. What are the differences between solving rational and polynomial equations?

#### Tip:
Always check for extraneous solutions by substituting answers back into the original equation, especially for rational equations, to ensure they don’t make any denominator zero.

solve the rational equation. If there is more than one answer, separate your answers with commas. If there is no solution, enter "No Solution." 5 plus fraction numerator 9 over denominator x minus 2 end fraction equals 2 plus fraction numerator x plus 7 over denominator x minus 2 end fraction

Explore a detailed step-by-step solution for the rational equation 5 + 9/(x - 2) = 2 + (x + 7)/(x - 2), covering key algebraic methods for clearing denominators and identifying extraneous solutions. Ideal for students in Grades 9-11.

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