https://file.aiquickdraw.com/mathbot/file/17254578699923uiieyhc.jpg

The equation provided in the image is:

\[
\frac{2}{x + 3} + \frac{5}{x^2 - 9} = \frac{1}{x - 3}
\]

Let's solve this step by step.

### Step 1: Factorize the denominator
Notice that the second term \(\frac{5}{x^2 - 9}\) can be factorized, as \(x^2 - 9\) is a difference of squares:

\[
x^2 - 9 = (x - 3)(x + 3)
\]

So the equation becomes:

\[
\frac{2}{x + 3} + \frac{5}{(x - 3)(x + 3)} = \frac{1}{x - 3}
\]

### Step 2: Find a common denominator
The common denominator for the left-hand side is \((x + 3)(x - 3)\). So, we rewrite the equation:

\[
\frac{2(x - 3)}{(x + 3)(x - 3)} + \frac{5}{(x + 3)(x - 3)} = \frac{1}{x - 3}
\]

This simplifies to:

\[
\frac{2(x - 3) + 5}{(x + 3)(x - 3)} = \frac{1}{x - 3}
\]

### Step 3: Simplify the numerator
Now, simplify the numerator of the left-hand side:

\[
2(x - 3) + 5 = 2x - 6 + 5 = 2x - 1
\]

So the equation becomes:

\[
\frac{2x - 1}{(x + 3)(x - 3)} = \frac{1}{x - 3}
\]

### Step 4: Cross-multiply and solve for \(x\)
Cross-multiply to get rid of the denominators:

\[
2x - 1 = 1 \cdot (x + 3)
\]

Now, distribute and solve for \(x\):

\[
2x - 1 = x + 3
\]

Subtract \(x\) from both sides:

\[
x - 1 = 3
\]

Add 1 to both sides:

\[
x = 4
\]

### Step 5: Verify the solution
Let's verify \(x = 4\) by substituting it back into the original equation to ensure both sides are equal.

Substituting \(x = 4\):

\[
\frac{2}{4 + 3} + \frac{5}{4^2 - 9} = \frac{1}{4 - 3}
\]

This simplifies to:

\[
\frac{2}{7} + \frac{5}{16 - 9} = 1
\]

Further simplification:

\[
\frac{2}{7} + \frac{5}{7} = 1
\]

\[
\frac{7}{7} = 1
\]

So, the solution is correct.

### Final Answer:
The solution to the equation is \( \boxed{x = 4} \).

Would you like more details on any of the steps or have any questions?

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Here are five related questions:

1. How can you determine the common denominator for a given rational equation?
2. What are the steps involved in solving a rational equation with multiple fractions?
3. How does the factorization of quadratic expressions help in solving rational equations?
4. Can you explain the difference between cross-multiplication and finding a common denominator?
5. What should be checked after solving a rational equation to ensure the solution is valid?

**Tip:** Always remember to check for extraneous solutions when dealing with rational equations, as the solution might make a denominator zero, which is undefined.

Solve the equation: 2/(x + 3) + 5/(x^2 - 9) = 1/(x - 3)

This guide provides a step-by-step solution to the rational equation 2/(x + 3) + 5/(x^2 - 9) = 1/(x - 3), covering key algebraic concepts such as factorization of quadratic expressions and solving rational equations. Ideal for students in Grades 10-12.

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