To solve the rational inequality $\frac{x - 2}{x - 4} > 0$, follow these steps:

Step 1: Identify critical points

The expression will change signs at points where the numerator or denominator is zero:

• $x - 2 = 0 \implies x = 2$
• $x - 4 = 0 \implies x = 4$

So, the critical points are $x = 2$ and $x = 4$. These points divide the number line into intervals.

Step 2: Analyze sign changes in intervals

We will test the sign of the rational expression in three intervals created by the critical points:

1. $(-\infty, 2)$
2. $(2, 4)$
3. $(4, \infty)$

Interval 1: $(-\infty, 2)$

Choose a test point, say $x = 0$: $\frac{0 - 2}{0 - 4} = \frac{-2}{-4} = \frac{1}{2}$ The expression is positive.

Interval 2: $(2, 4)$

Choose a test point, say $x = 3$: $\frac{3 - 2}{3 - 4} = \frac{1}{-1} = -1$ The expression is negative.

Interval 3: $(4, \infty)$

Choose a test point, say $x = 5$: $\frac{5 - 2}{5 - 4} = \frac{3}{1} = 3$ The expression is positive.

Step 3: Determine solution intervals

The inequality asks when the expression is greater than 0. From the sign analysis:

• The expression is positive on the intervals $(-\infty, 2)$ and $(4, \infty)$.
• The critical points $x = 2$ and $x = 4$ are not included because the expression equals 0 at $x = 2$ and is undefined at $x = 4$.

Final Answer:

The solution is $(-\infty, 2) \cup (4, \infty)$.

Would you like more details on this, or do you have any questions?

Here are 5 related questions:

1. What changes if the inequality is $\frac{x - 2}{x - 4} \geq 0$?
2. How would the solution differ if the inequality were $\frac{x - 2}{x - 4} < 0$?
3. Can you explain why the expression is undefined at $x = 4$?
4. How would the approach change for solving $\frac{x^2 - 2x}{x - 4} > 0$?
5. How does interval notation differ from set-builder notation?

Tip: Always check the critical points to determine if they should be included in your final solution.