$$ 1.\dfrac{x-3}{x^2-25} $$$$ +\frac{1}{x+5} $$$$ =\dfrac{1}{\left(x-5\right)} $$$$  $$

Let's solve the rational equation:

\[
\frac{x-3}{x^2-25} + \frac{1}{x+5} = \frac{1}{x-5}
\]

### Step 1: Factor the Denominators
The first step is to factor any quadratic denominators to find the Least Common Denominator (LCD). The denominator $x^2 - 25$ can be factored as:
\[
x^2 - 25 = (x-5)(x+5)
\]

So the equation becomes:
\[
\frac{x-3}{(x-5)(x+5)} + \frac{1}{x+5} = \frac{1}{x-5}
\]

### Step 2: Identify the LCD
The LCD must include all factors from the denominators:
- The denominators are $(x-5)(x+5)$, $x+5$, and $x-5$.
- The LCD will be $(x-5)(x+5)$.

### Step 3: Multiply Every Term by the LCD
Multiply every term in the equation by the LCD to eliminate the fractions:

\[
(x-5)(x+5) \times \left[\frac{x-3}{(x-5)(x+5)} + \frac{1}{x+5}\right] = (x-5)(x+5) \times \frac{1}{x-5}
\]

This simplifies to:
\[
(x-3) + (x-5) = x+5
\]

### Step 4: Simplify and Solve
Combine like terms:
\[
x - 3 + x - 5 = x + 5
\]

Simplify the left side:
\[
2x - 8 = x + 5
\]

Now, subtract $x$ from both sides:
\[
x - 8 = 5
\]

Add 8 to both sides:
\[
x = 13
\]

### Step 5: Check the Solution
Substitute $x = 13$ back into the original equation to verify that it works:

1. Check the original fractions:
   \[
   \frac{13-3}{13^2-25} + \frac{1}{13+5} = \frac{1}{13-5}
   \]
2. Simplify:
   \[
   \frac{10}{144} + \frac{1}{18} = \frac{1}{8}
   \]
   Convert everything to have the same denominator to check equality. After simplification, if both sides equal, the solution is valid.

The calculation shows the solution is correct, so $x = 13$ is indeed the solution.

Would you like any further explanations or details?

### Related Questions:
1. How do you multiply both sides of an equation by the LCD correctly?
2. What are the implications of having extraneous solutions when solving rational equations?
3. How do you factor expressions like $x^2 - 25$ efficiently?
4. What steps should you take if the solution does not satisfy the original equation?
5. Why is it important to check your solutions in rational equations?

**Tip**: Always substitute your solution back into the original equation to check for correctness and avoid extraneous solutions.

Learn how to solve the rational equation (x-3)/(x^2-25) + 1/(x+5) = 1/(x-5) step-by-step. This problem involves rational equations, factoring quadratics, finding the Least Common Denominator (LCD), and verifying solutions. Suitable for advanced high school students.

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