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### Problem 1: 
\[
\frac{2x - 8}{x - 2} \geq 0
\]

This is an inequality involving a rational function. To solve this, we need to analyze the numerator and denominator.

1. **Simplify the expression**:  
   The numerator \( 2x - 8 \) can be factored as:
   \[
   2(x - 4)
   \]
   So, the inequality becomes:
   \[
   \frac{2(x - 4)}{x - 2} \geq 0
   \]

2. **Find critical points**:  
   The expression equals 0 when the numerator is 0, i.e., \( x = 4 \).  
   The expression is undefined when the denominator is 0, i.e., \( x = 2 \).

3. **Test intervals**:  
   The critical points divide the number line into three intervals: \( (-\infty, 2) \), \( (2, 4) \), and \( (4, \infty) \). We will test the sign of the expression in each interval.

   - For \( x < 2 \), choose \( x = 0 \):
     \[
     \frac{2(0 - 4)}{0 - 2} = \frac{-8}{-2} = 4 \quad (\text{positive})
     \]
   - For \( 2 < x < 4 \), choose \( x = 3 \):
     \[
     \frac{2(3 - 4)}{3 - 2} = \frac{-2}{1} = -2 \quad (\text{negative})
     \]
   - For \( x > 4 \), choose \( x = 5 \):
     \[
     \frac{2(5 - 4)}{5 - 2} = \frac{2}{3} \quad (\text{positive})
     \]

4. **Conclusion**:  
   The inequality is satisfied where the expression is positive or zero. Therefore, the solution is:
   \[
   (-\infty, 2) \cup [4, \infty)
   \]
   \( x = 2 \) is excluded because the denominator becomes zero there, but \( x = 4 \) is included because it makes the numerator zero.

---

### Problem 2:
\[
\frac{x}{x - 2} - 7 = \frac{2}{x - 2}
\]

This is an equation involving fractions with the same denominator. To solve:

1. **Rewrite the equation**:
   \[
   \frac{x}{x - 2} - 7 = \frac{2}{x - 2}
   \]

2. **Combine terms with the same denominator**:  
   First, bring both fractions on the left-hand side together:
   \[
   \frac{x - 2}{x - 2} - 7 = 0
   \]
   Add both sides to simplify the equation further and solve

1) Solve the inequality (2x - 8) / (x - 2) >= 0 
2) Solve the equation x / (x - 2) - 7 = 2 / (x - 2)

Learn how to solve the inequality (2x - 8) / (x - 2) >= 0 and the rational equation x / (x - 2) - 7 = 2 / (x - 2). This covers key concepts in solving inequalities and equations involving rational expressions, including factored form and sign analysis.

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