To solve the inequality \((X_n - 2)(X_n + 1) \leq 0\), let's break it down step by step.

### Step 1: Identify the critical points
The expression \((X_n - 2)(X_n + 1)\) will be zero when either factor is zero:

- \(X_n - 2 = 0\) gives \(X_n = 2\)
- \(X_n + 1 = 0\) gives \(X_n = -1\)

So the critical points are \(X_n = -1\) and \(X_n = 2\).

### Step 2: Analyze the sign of the product in the intervals
The critical points divide the number line into three intervals:
1. \(X_n < -1\)
2. \(-1 \leq X_n \leq 2\)
3. \(X_n > 2\)

We will test the sign of the product in each interval.

- **Interval 1: \(X_n < -1\)**
  - If \(X_n < -1\), then \(X_n - 2 < -3\) (negative) and \(X_n + 1 < 0\) (negative).
  - Product: (negative) \(\times\) (negative) \(=\) positive.

- **Interval 2: \(-1 \leq X_n \leq 2\)**
  - If \(-1 \leq X_n < 2\), then \(X_n - 2\) is non-positive and \(X_n + 1\) is non-negative.
  - Product: non-positive \(\times\) non-negative \(\leq\) zero.

- **Interval 3: \(X_n > 2\)**
  - If \(X_n > 2\), then \(X_n - 2 > 0\) (positive) and \(X_n + 1 > 3\) (positive).
  - Product: positive \(\times\) positive \(=\) positive.

### Step 3: Determine the solution set
The inequality \((X_n - 2)(X_n + 1) \leq 0\) holds where the product is non-positive:

- From interval 2: \(-1 \leq X_n \leq 2\).

### Final Answer
The solution to the inequality is:

\[
-1 \leq X_n \leq 2
\]

Would you like further details or have any questions?

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Here are some related questions for further exploration:

1. What is the difference between solving an equation and solving an inequality?
2. How do you solve a quadratic inequality?
3. How do critical points affect the sign of a product in inequalities?
4. What are the steps for analyzing the sign of a product over an interval?
5. How do you test the intervals when solving an inequality?
6. Can this method be extended to inequalities with more factors?
7. How do inequalities change when the inequality sign is strict (< or >)?
8. How do you graph the solution set of this inequality?

**Tip:** Always remember to check the boundary points when solving inequalities.

Explore how to solve the inequality (Xn-2)(Xn+1)<=0 step-by-step. Learn about critical points, interval analysis, and sign determination. Suitable for high school students studying inequalities and algebraic expressions.

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