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

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

Explore how to solve the rational inequality (x - 2)/(x - 4) > 0 with a detailed, step-by-step solution. This solution covers key algebra concepts like critical points, sign changes across intervals, and analyzing rational expressions. Perfect for high school students in Grades 9-12.

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