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) $(- \infty < x \leq -2) \cup (7 < x < \infty)$.

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.