https://file.aiquickdraw.com/mathbot/file/17267630930162ouz1z67.jpg

The inequality shown in the image is:

\[
\frac{x + 12}{x + 2} > 2
\]

### Steps to Solve the Inequality:

1. **Rewrite the inequality:**
   Subtract 2 from both sides to get everything on one side:
   
   \[
   \frac{x + 12}{x + 2} - 2 > 0
   \]

2. **Simplify the expression:**
   Combine the terms on the left-hand side by finding a common denominator:
   
   \[
   \frac{x + 12}{x + 2} - 2 = \frac{x + 12 - 2(x + 2)}{x + 2}
   \]
   Simplify the numerator:
   
   \[
   \frac{x + 12 - 2x - 4}{x + 2} = \frac{-x + 8}{x + 2}
   \]
   
   So the inequality becomes:
   
   \[
   \frac{-x + 8}{x + 2} > 0
   \]

3. **Sign Analysis:**
   Now, we need to perform sign analysis for the expression \(\frac{-x + 8}{x + 2}\).

   The critical points occur when either the numerator or the denominator equals zero:
   - Numerator: \(-x + 8 = 0 \Rightarrow x = 8\)
   - Denominator: \(x + 2 = 0 \Rightarrow x = -2\)

### Sign Analysis Table:
We analyze the sign of \(\frac{-x + 8}{x + 2}\) over the intervals determined by the critical points \(x = -2\) and \(x = 8\).

| Interval      | Test Point | Numerator \((-x + 8)\) | Denominator \((x + 2)\) | Expression \(\frac{-x + 8}{x + 2}\) | Sign of Expression |
|---------------|------------|------------------------|------------------------|-------------------------------------|--------------------|
| \(x < -2\)    | \(x = -3\) | \(11\)                 | \(-1\)                 | \(\frac{11}{-1} = -11\)             | Negative           |
| \(-2 < x < 8\)| \(x = 0\)  | \(8\)                  | \(2\)                  | \(\frac{8}{2} = 4\)                 | Positive           |
| \(x > 8\)     | \(x = 9\)  | \(-1\)                 | \(11\)                 | \(\frac{-1}{11} = -\frac{1}{11}\)   | Negative           |

### Conclusion:
The expression \(\frac{-x + 8}{x + 2} > 0\) is positive in the interval:

\[
(-2, 8)
\]

Thus, the solution to the inequality is:

\[
x \in (-2, 8)
\]

Note that \(x = -2\) is excluded because it makes the denominator zero.

Would you like further details or have any questions?

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### Related Questions:
1. How do you solve inequalities involving rational expressions in general?
2. What happens if we multiply both sides of an inequality by a negative number?
3. How can you graph the solution set of this inequality on a number line?
4. What are critical points, and why are they important in sign analysis?
5. Can this method be applied to other types of inequalities, such as polynomial inequalities?

### Tip:
Always remember to check critical points when solving rational inequalities to avoid dividing by zero!

Solve the rational inequality \frac{x + 12}{x + 2} > 2 using sign analysis. Follow step-by-step instructions, including critical points, test intervals, and sign analysis table. Suitable for students in Grades 9-11.

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