Let's go through each problem you have on the sheet.

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