The problem shown in the image is the rational inequality:

$\frac{x - 2}{x - 4} > 0$

Steps to solve the inequality:

1. Identify the critical points: These are the values of $x$ that make either the numerator or the denominator zero.

   • From the numerator $x - 2 = 0$, we get $x = 2$.
   • From the denominator $x - 4 = 0$, we get $x = 4$.

2. Divide the number line into intervals based on the critical points. The critical points divide the number line into three intervals:

   • $(-\infty, 2)$
   • $(2, 4)$
   • $(4, \infty)$

3. Test a value from each interval to determine the sign of the expression $\frac{x - 2}{x - 4}$ in each interval:

   • For $(-\infty, 2)$, choose $x = 0$: $\frac{0 - 2}{0 - 4} = \frac{-2}{-4} = \frac{1}{2} > 0$ The expression is positive in this interval.
   • For $(2, 4)$, choose $x = 3$: $\frac{3 - 2}{3 - 4} = \frac{1}{-1} = -1 < 0$ The expression is negative in this interval.
   • For $(4, \infty)$, choose $x = 5$: $\frac{5 - 2}{5 - 4} = \frac{3}{1} = 3 > 0$ The expression is positive in this interval.

4. Analyze the critical points:

   • At $x = 2$, the numerator is zero, so $\frac{x - 2}{x - 4} = 0$.
   • At $x = 4$, the denominator is zero, so the expression is undefined at this point.

5. Conclusion:

   • The inequality $\frac{x - 2}{x - 4} > 0$ holds in the intervals $(-\infty, 2)$ and $(4, \infty)$, but not at $x = 2$ or $x = 4$.
   • Therefore, the solution is: $(-\infty, 2) \cup (4, \infty)$

Would you like any additional details or have questions about the steps?

Related Questions:

1. How do we handle rational inequalities that are less than 0?
2. What changes if the inequality is $\geq$ or $\leq$ instead of strict $>$?
3. How would the solution change if there were different terms in the numerator or denominator?
4. What would happen if there were more than two critical points?
5. How do we graph the solution on a number line?

Tip: Always test points from the intervals between critical values to determine where the expression is positive or negative.