To solve the inequality 

\[
\frac{x + 1}{x + 3} \leq 2,
\]

we need to determine the values of \(x\) for which the inequality holds true.

### Step-by-step Solution

1. **Rewrite the inequality**:

   \[
   \frac{x + 1}{x + 3} \leq 2.
   \]

2. **Subtract 2 from both sides** to bring the inequality to a single fraction:

   \[
   \frac{x + 1}{x + 3} - 2 \leq 0.
   \]

3. **Combine into a single fraction**:

   To combine, we need a common denominator. The common denominator is \(x + 3\). Rewriting the expression:

   \[
   \frac{x + 1 - 2(x + 3)}{x + 3} \leq 0.
   \]

4. **Simplify the numerator**:

   Simplify inside the fraction:

   \[
   \frac{x + 1 - 2x - 6}{x + 3} \leq 0.
   \]

   This simplifies to:

   \[
   \frac{-x - 5}{x + 3} \leq 0.
   \]

5. **Find the critical points**:

   The critical points occur when the numerator or the denominator is zero:

   - **Numerator zero**: \(-x - 5 = 0 \implies x = -5\).
   - **Denominator zero**: \(x + 3 = 0 \implies x = -3\).

   These points, \(x = -5\) and \(x = -3\), divide the number line into intervals.

6. **Test the intervals** to determine where the inequality holds:

   The intervals to check are: \((- \infty, -5)\), \((-5, -3)\), and \((-3, \infty)\).

   - For \(x < -5\): Choose \(x = -6\):
     \[
     \frac{-(-6) - 5}{-6 + 3} = \frac{6 - 5}{-3} = \frac{1}{-3} = -\frac{1}{3} < 0.
     \]
     (Satisfies the inequality)

   - For \(-5 < x < -3\): Choose \(x = -4\):
     \[
     \frac{-(-4) - 5}{-4 + 3} = \frac{4 - 5}{-1} = \frac{-1}{-1} = 1 > 0.
     \]
     (Does not satisfy the inequality)

   - For \(x > -3\): Choose \(x = 0\):
     \[
     \frac{-0 - 5}{0 + 3} = \frac{-5}{3} = -\frac{5}{3} < 0.
     \]
     (Satisfies the inequality)

7. **Include critical points in the solution**:

   - \(x = -5\) makes the numerator zero: \( \frac{-(-5) - 5}{-5 + 3} = 0\) (satisfies the inequality).

   - \(x = -3\) makes the denominator zero: The fraction is undefined, so we **exclude** \(x = -3\).

8. **Combine the intervals**:

   The solution set for the inequality is:

   \[
   x \in (-\infty, -5] \cup (-3, \infty).
   \]

### Final Answer

\[
x \in (-\infty, -5] \cup (-3, \infty).
\]

Would you like more details on any part of the solution? 

Here are 5 related questions you might find interesting:
1. How do you solve a quadratic inequality?
2. What are the steps to solve rational inequalities involving quadratic expressions?
3. How does multiplying or dividing by a negative number affect inequalities?
4. What are the common mistakes to avoid when solving rational inequalities?
5. How can you graphically represent the solution of a rational inequality?

**Tip:** When solving inequalities, always consider the domain restrictions, especially where the expressions are undefined.

Learn how to solve the rational inequality (x + 1)/(x + 3) ≤ 2 with detailed step-by-step instructions. This problem involves inequalities and rational expressions. Suitable for students in Grades 10-12.

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