https://file.aiquickdraw.com/mathbot/file/1730062560663ldmwqcb9.jpg

Let's solve **Question 4** from the image.

The equation given is:

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

### Step 1: Combine the Terms on the Left Side
Since both terms on the left have a common denominator \( x - 2 \), we can combine them:

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

Simplifying the numerator:

\[
\frac{14}{x - 2} = \frac{x + 7}{x - 2}
\]

### Step 2: Cross Multiply
To remove the denominator, we can cross-multiply:

\[
14 = x + 7
\]

### Step 3: Solve for \( x \)
Subtract 7 from both sides:

\[
x = 7
\]

### Step 4: Check for Restrictions
Since the original equation has a denominator of \( x - 2 \), \( x \) cannot equal 2, as this would make the denominator zero. However, \( x = 7 \) is valid.

### Solution
The solution to the equation is:

\[
x = 7
\]

Would you like details on this solution or have any questions?

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Here are some related questions you might find helpful:

1. What happens if we have different denominators in a rational equation?
2. How would we handle an equation where the numerator is a polynomial rather than a constant?
3. What does it mean if, after cross-multiplying, the equation has no solutions?
4. Can we solve a rational equation if the variable appears in multiple denominators with different terms?
5. How do restrictions in rational equations affect the solution?

**Tip:** Always check for values that make any denominator zero in rational equations, as these values cannot be solutions.

Solve the rational equation. If there is more than one answer, separate your answers with commas. If there is no solution, enter "No Solution." \( \frac{5}{x - 2} + \frac{9}{x - 2} = \frac{x + 7}{x - 2} \)

Explore how to solve the rational equation \( \frac{5}{x - 2} + \frac{9}{x - 2} = \frac{x + 7}{x - 2} \) using cross-multiplication. This guide covers key algebraic techniques to solve rational equations, suitable for students in Grades 8-10.

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