The inequality shown in the image is a rational inequality:

$\frac{2x - 8}{x - 2} \geq 0$

Step-by-Step Solution:

1. Factor the numerator: $2x - 8 = 2(x - 4)$ So the inequality becomes: $\frac{2(x - 4)}{x - 2} \geq 0$

2. Find the critical points: The critical points come from where the numerator or the denominator is equal to zero:

   • Numerator: $2(x - 4) = 0$ gives $x = 4$.
   • Denominator: $x - 2 = 0$ gives $x = 2$ (Note: $x = 2$ is undefined for the rational function, as it makes the denominator zero).

3. Sign analysis on intervals: The critical points divide the real number line into three intervals: $(-\infty, 2), (2, 4), (4, \infty)$ We will test the sign of the expression in each of these intervals.

   • For $x \in (-\infty, 2)$: Choose $x = 0$: $\frac{2(0 - 4)}{0 - 2} = \frac{2(-4)}{-2} = 4 > 0$
   • For $x \in (2, 4)$: Choose $x = 3$: $\frac{2(3 - 4)}{3 - 2} = \frac{2(-1)}{1} = -2 < 0$
   • For $x \in (4, \infty)$: Choose $x = 5$: $\frac{2(5 - 4)}{5 - 2} = \frac{2(1)}{3} = \frac{2}{3} > 0$

4. Inclusion of critical points:

   • At $x = 4$, the expression is $\frac{0}{4 - 2} = 0$, which satisfies the inequality.
   • At $x = 2$, the function is undefined, so $x = 2$ is not included in the solution.

5. Final solution: The solution is where the expression is non-negative, which includes the intervals where the expression is positive or zero: $(-\infty, 2) \cup [4, \infty)$

Final Answer:

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

Would you like more details on any of these steps? Here are 5 related questions to extend this concept:

1. How do you solve rational inequalities with multiple factors in the numerator and denominator?
2. What happens if the inequality was $\frac{2x - 8}{x - 2} > 0$ instead of $\geq 0$?
3. How would you solve a rational inequality where both the numerator and denominator are quadratic?
4. Can you explain how to graph the solution to this inequality on a number line?
5. How does the process change if the inequality was less than zero ($< 0$) instead of greater than or equal?

Tip: Always remember to check for points where the denominator equals zero in rational inequalities—these points must be excluded from the solution!