https://file.aiquickdraw.com/mathbot/file/17243678658245ei47nlb.jpg

The image is a bit small, but I can still assist with the problem. Based on what I can see, the problem involves a mathematical inequality. 

The problem appears to ask to solve the inequality:

\[
\frac{x - 5}{x + 2} \leq 1
\]

### Step-by-Step Solution:

1. **Isolate the variable expression:**

   First, subtract 1 from both sides of the inequality:

   \[
   \frac{x - 5}{x + 2} - 1 \leq 0
   \]

2. **Combine the fractions:**

   Combine the fractions by getting a common denominator:

   \[
   \frac{x - 5 - (x + 2)}{x + 2} \leq 0
   \]

   Simplify the numerator:

   \[
   \frac{x - 5 - x - 2}{x + 2} \leq 0
   \]

   \[
   \frac{-7}{x + 2} \leq 0
   \]

3. **Solve the inequality:**

   The inequality \(\frac{-7}{x + 2} \leq 0\) only holds when the denominator is positive since \(-7\) is a constant negative value.

   Thus, we consider the sign of \(x + 2\):

   - When \(x + 2 > 0\), \(\frac{-7}{x + 2} < 0\) (which does not satisfy the inequality).
   - When \(x + 2 < 0\), \(\frac{-7}{x + 2} > 0\) (which also does not satisfy the inequality).
   
   Therefore, there is no \(x\) that satisfies this inequality.

   However, since the inequality is less than or equal to 0, the critical point is where \(x + 2 = 0\), giving:

   \[
   x = -2
   \]

   But at \(x = -2\), the expression is undefined (division by zero). So, there is **no solution** where the inequality holds.

### Final Answer:
- **The inequality has no solution.**

Would you like more details or have any questions?

### Follow-up Questions:
1. What happens if the inequality was \(\frac{x - 5}{x + 2} \geq 1\)?
2. How would you solve the inequality if it involved a quadratic expression?
3. What is the general approach to solving inequalities involving rational expressions?
4. Can you graphically represent the solution of inequalities like this?
5. How does the solution change if you consider the strict inequality \(\frac{x - 5}{x + 2} < 1\)?

### Tip:
When dealing with rational inequalities, always check for points where the expression is undefined and consider these when determining your solution.

Learn how to solve the rational inequality (x - 5) / (x + 2) ≤ 1 step-by-step with detailed explanations. Understand the concepts of rational expressions and inequalities. Suitable for high school students studying algebra and inequalities.

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