is x^2−2x−8≥0 an unbounded and convex set?

To analyze whether the set defined by the inequality \( x^2 - 2x - 8 \geq 0 \) is unbounded and convex, let's go step by step:

### Step 1: Rewrite the inequality
The inequality can be rewritten as:

\[
x^2 - 2x - 8 \geq 0
\]

We can factorize this quadratic expression:

\[
x^2 - 2x - 8 = (x - 4)(x + 2)
\]

Thus, the inequality becomes:

\[
(x - 4)(x + 2) \geq 0
\]

### Step 2: Solve the inequality
We need to determine for which values of \(x\), this product is greater than or equal to zero.

- The roots of the equation \( (x - 4)(x + 2) = 0 \) are \( x = 4 \) and \( x = -2 \).
- To determine the sign of the product in different intervals, we test values in the intervals created by these roots: \( (-\infty, -2) \), \( (-2, 4) \), and \( (4, \infty) \).

Testing:
1. For \( x \in (-\infty, -2) \), both \( (x - 4) \) and \( (x + 2) \) are negative, so their product is positive.
2. For \( x \in (-2, 4) \), one factor is negative and the other is positive, so their product is negative.
3. For \( x \in (4, \infty) \), both factors are positive, so their product is positive.

Thus, the solution to the inequality is:

\[
x \in (-\infty, -2] \cup [4, \infty)
\]

### Step 3: Is the set unbounded?
Yes, the set is unbounded because the intervals \( (-\infty, -2] \) and \( [4, \infty) \) extend infinitely in both directions.

### Step 4: Is the set convex?
A set is convex if, for any two points in the set, the entire line segment between them also lies in the set.

In this case, the set consists of two disjoint intervals, \( (-\infty, -2] \) and \( [4, \infty) \), which means that the set is not convex. For example, if you take a point in \( (-\infty, -2] \) and another point in \( [4, \infty) \), the line segment between them will pass through \( (-2, 4) \), which is not part of the set.

### Conclusion:
- The set \( \{ x \in \mathbb{R} \mid x^2 - 2x - 8 \geq 0 \} = (-\infty, -2] \cup [4, \infty) \) is **unbounded** but **not convex**.

Do you want more details or have any questions? Here are 5 related questions to expand on this topic:

1. How do you determine if a set is convex using the definition of convexity?
2. Can you graph the function \( f(x) = x^2 - 2x - 8 \) to visually confirm the intervals where the inequality holds?
3. What are other examples of unbounded sets that are convex?
4. How would the solution change if the inequality were \( x^2 - 2x - 8 < 0 \)?
5. What properties do convex sets have that are useful in optimization problems?

**Tip:** When analyzing inequalities, always check the sign of the expression in each interval formed by the critical points (roots).

This problem explores whether the set defined by the inequality x^2 - 2x - 8 ≥ 0 is unbounded and convex. It covers concepts of algebra, solving quadratic inequalities, and convexity in set theory. Suitable for students in Grades 9-12.

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