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The inequality given is:

\[
\frac{x + 5}{x + 3} \leq \frac{x + 1}{x - 1}
\]

### Step-by-Step Solution:

1. **Find the common denominator**:
   To solve the inequality, first cross-multiply to eliminate the denominators (but be cautious about the signs since cross-multiplication involves multiplication, which could flip the inequality sign depending on the values of \( x \)).

   \[
   (x + 5)(x - 1) \leq (x + 1)(x + 3)
   \]

2. **Expand both sides**:
   Expand both sides of the inequality:

   Left side:
   \[
   (x + 5)(x - 1) = x^2 - x + 5x - 5 = x^2 + 4x - 5
   \]

   Right side:
   \[
   (x + 1)(x + 3) = x^2 + 3x + x + 3 = x^2 + 4x + 3
   \]

3. **Simplify the inequality**:
   Subtract the terms on the right side from those on the left:

   \[
   x^2 + 4x - 5 - (x^2 + 4x + 3) \leq 0
   \]

   Simplify the expression:
   \[
   -8 \leq 0
   \]

   Since \(-8 \leq 0\) is always true, the inequality is **always true** for the intervals where the expressions are defined.

### Additional Considerations:
   However, we must also consider the intervals where the original inequality is defined. The denominators \( x + 3 \) and \( x - 1 \) cannot be zero:

   - \( x \neq -3 \)
   - \( x \neq 1 \)

### Final Solution:
Thus, the solution is **all real numbers except \( x = -3 \) and \( x = 1 \)**.

Would you like further details or have any questions? 

Here are some related questions you might find interesting:
1. How do you handle inequalities involving absolute values?
2. What happens if the inequality had strict inequality (i.e., \( < \) or \( > \)) instead of \( \leq \)?
3. Can you solve a similar inequality but involving a quadratic term in the numerator?
4. What are the conditions that could cause the inequality sign to flip when solving?
5. How would this inequality change if one of the terms in the numerators or denominators were squared?

**Tip:** Always consider the domain restrictions, such as where the denominator can be zero, when solving rational inequalities.

Learn how to solve the rational inequality (x + 5)/(x + 3) <= (x + 1)/(x - 1) with a detailed step-by-step solution. This problem involves rational expressions and domain restrictions. Suitable for advanced high school students.

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