To solve the inequality \( \frac{x + 1}{x + 3} \leq 2 \), let's follow these steps:

### Step 1: Rewrite the Inequality
We start by moving all terms to one side of the inequality:

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

### Step 2: Combine the Fractions
Next, express the inequality as a single fraction:

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

Simplify the numerator:

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

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

### Step 3: Find the Critical Points
The critical points occur where the numerator and the denominator are equal to zero:

1. \( -x - 5 = 0 \) ⟹ \( x = -5 \)
2. \( x + 3 = 0 \) ⟹ \( x = -3 \)

So, the critical points are \( x = -5 \) and \( x = -3 \).

### Step 4: Test the Intervals
We need to test the sign of the fraction in the intervals determined by the critical points:

1. **Interval \( (-\infty, -5) \)**: 
   - Choose \( x = -6 \): \( \frac{-(-6) - 5}{-6 + 3} = \frac{6 - 5}{-3} = \frac{1}{-3} \), which is negative.
   
2. **Interval \( (-5, -3) \)**: 
   - Choose \( x = -4 \): \( \frac{-(-4) - 5}{-4 + 3} = \frac{4 - 5}{-1} = \frac{-1}{-1} = 1 \), which is positive.
   
3. **Interval \( (-3, \infty) \)**: 
   - Choose \( x = 0 \): \( \frac{-(0) - 5}{0 + 3} = \frac{-5}{3} \), which is negative.

### Step 5: Determine the Solution Set
The inequality \( \frac{-x - 5}{x + 3} \leq 0 \) is satisfied when the fraction is non-positive (negative or zero). This occurs in the intervals where the fraction is negative or where it is equal to zero:

- The fraction is negative on \( (-\infty, -5) \cup (-3, \infty) \).
- The fraction is zero at \( x = -5 \).

However, \( x = -3 \) makes the denominator zero, so \( x = -3 \) is not in the domain.

### Final Answer:
The solution set is:

\[
(-\infty, -5] \cup (-3, \infty)
\]

Would you like more details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How do you solve inequalities involving absolute values?
2. What are the steps to solve quadratic inequalities?
3. Can you explain how to solve compound inequalities?
4. How do you determine the domain of a rational function?
5. What is the significance of critical points in solving inequalities?

**Tip:** Always check the critical points and intervals carefully when solving rational inequalities, as they determine where the expression changes sign.

Learn how to solve the rational inequality (x + 1)/(x + 3) ≤ 2 step-by-step. This problem involves inequalities and rational functions, including critical points and testing intervals. Suitable for Grades 11-12 mathematics.

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