https://file.aiquickdraw.com/mathbot/file/1725916343134vj4uhfs3.jpg

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!

Solve the rational inequality: (2x - 8) / (x - 2) ≥ 0

Learn how to solve the rational inequality (2x - 8) / (x - 2) ≥ 0 with a comprehensive step-by-step solution. This guide covers key algebraic concepts, including sign analysis, critical points, and the properties of inequalities, making it suitable for students in Grades 10-12.

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