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The given inequality is:

\[
\frac{(x + 2)}{(x - 7)} \geq 0
\]

We need to solve this inequality and find the correct interval(s) for \(x\) from the given options.

### Steps to solve:

1. **Find critical points**:
   - The numerator \(x + 2 = 0\) when \(x = -2\).
   - The denominator \(x - 7 = 0\) when \(x = 7\). However, \(x = 7\) makes the denominator zero, so it cannot be part of the solution.

2. **Sign analysis**:
   We will now perform sign analysis by checking the signs of the expression in the intervals formed by these critical points: \( (-\infty, -2), (-2, 7), (7, \infty) \).

   - For \(x < -2\), both the numerator and denominator are negative, making the expression positive.
   - For \(-2 < x < 7\), the numerator is positive and the denominator is negative, making the expression negative.
   - For \(x > 7\), both the numerator and denominator are positive, making the expression positive.

3. **Determine intervals for non-negative expression**:
   - The expression \(\frac{(x + 2)}{(x - 7)} \geq 0\) is non-negative when it is either positive or zero.
   - The expression is positive in the intervals \( (-\infty, -2) \cup (7, \infty) \).
   - The expression is zero at \(x = -2\) because the numerator becomes zero.

Thus, the solution is:

\[
x \in (-\infty, -2] \cup (7, \infty)
\]

### Correct answer:
The correct option is **(a)** \((- \infty < x \leq -2) \cup (7 < x < \infty)\).

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Would you like further clarification or details?

Here are 5 related questions to explore:
1. What happens when the inequality is strict \(\frac{(x + 2)}{(x - 7)} > 0\)?
2. How does the solution change if the denominator is squared, like \(\frac{(x + 2)}{(x - 7)^2} \geq 0\)?
3. What is the domain of the function \(\frac{(x + 2)}{(x - 7)}\)?
4. Can you solve a similar inequality, like \(\frac{(x - 3)}{(x + 4)} \geq 0\)?
5. How would you graph the function \(\frac{(x + 2)}{(x - 7)}\)?

**Tip**: Always exclude values where the denominator is zero in rational inequalities.

Learn how to solve the inequality (x + 2) / (x - 7) ≥ 0 using sign analysis and find the correct interval for x. This detailed solution involves understanding critical points, properties of inequalities, and rational expressions. Suitable for high school students in Grades 10-12.

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