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.