Solve the given rational inequality. Show your complete solution (x - 2)/(x - 4) > 0

We are given the rational inequality:

\[
\frac{x - 2}{x - 4} > 0
\]

### Step 1: Analyze the critical points
The critical points occur where the numerator or the denominator equals zero. These are important because they indicate where the function may change signs or become undefined.

- **Numerator:**
  \[
  x - 2 = 0 \implies x = 2
  \]
- **Denominator:**
  \[
  x - 4 = 0 \implies x = 4
  \]

Thus, the critical points are \( x = 2 \) and \( x = 4 \).

### Step 2: Determine the intervals
We will use the critical points to divide the real number line into intervals. The intervals are:

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

### Step 3: Test the sign in each interval
We now test the sign of the expression \( \frac{x - 2}{x - 4} \) in each of these intervals by picking a test point from each.

- **Interval \( (-\infty, 2) \):** Choose \( x = 0 \)
  \[
  \frac{0 - 2}{0 - 4} = \frac{-2}{-4} = \frac{1}{2} > 0
  \]
  So, the expression is positive in this interval.

- **Interval \( (2, 4) \):** Choose \( x = 3 \)
  \[
  \frac{3 - 2}{3 - 4} = \frac{1}{-1} = -1 < 0
  \]
  So, the expression is negative in this interval.

- **Interval \( (4, \infty) \):** Choose \( x = 5 \)
  \[
  \frac{5 - 2}{5 - 4} = \frac{3}{1} = 3 > 0
  \]
  So, the expression is positive in this interval.

### Step 4: Determine where the inequality is true
We want the inequality \( \frac{x - 2}{x - 4} > 0 \), meaning we are looking for where the expression is positive. From our testing, the expression is positive in the intervals \( (-\infty, 2) \) and \( (4, \infty) \).

### Step 5: Handle the critical points
- At \( x = 2 \), the numerator is zero, so the expression equals zero at this point. Since we are looking for where the expression is strictly greater than zero, we **exclude** \( x = 2 \).
- At \( x = 4 \), the denominator is zero, which makes the expression undefined. Therefore, we also **exclude** \( x = 4 \).

### Step 6: Final solution
The solution to the inequality is the union of the intervals where the expression is positive:
\[
(-\infty, 2) \cup (4, \infty)
\]

### Conclusion:
The solution to the inequality \( \frac{x - 2}{x - 4} > 0 \) is:
\[
x \in (-\infty, 2) \cup (4, \infty)
\]

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

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Here are 5 related questions to deepen your understanding:

1. What would change if the inequality were \( \frac{x - 2}{x - 4} \geq 0 \)?
2. How would the solution change if the inequality were \( \frac{x - 2}{x - 4} < 0 \)?
3. What is the graphical interpretation of solving rational inequalities like this?
4. How do vertical asymptotes affect the solution of rational inequalities?
5. What method would you use to solve \( \frac{x^2 - 4}{x - 3} \leq 0 \)?

**Tip:** When solving rational inequalities, always remember to exclude points where the denominator is zero, as the expression is undefined at those values.

Explore how to solve the rational inequality (x - 2)/(x - 4) > 0 with a clear, step-by-step guide. Learn about critical points, interval testing, and proper sign analysis in rational functions. Suitable for students in Grades 10-12.

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