R(x)=(x-2)/(x-4). Solve for R(x) less than or equal to zero. Write your answer in interval notation

We are tasked with solving the inequality:

\[
R(x) = \frac{x - 2}{x - 4} \leq 0
\]

### Step 1: Analyze the inequality
We want to determine when the expression \(\frac{x - 2}{x - 4} \leq 0\). This occurs when the fraction is either negative or zero.

### Step 2: Find where the fraction is zero
To find when \(\frac{x - 2}{x - 4} = 0\), we set the numerator equal to zero:

\[
x - 2 = 0
\]

Solving this gives \(x = 2\). So, the expression is zero at \(x = 2\).

### Step 3: Find where the fraction is undefined
The fraction is undefined when the denominator is zero:

\[
x - 4 = 0
\]

Solving this gives \(x = 4\). Thus, the function is undefined at \(x = 4\), meaning there is a vertical asymptote at \(x = 4\).

### Step 4: Determine the sign of the expression
We now need to determine where \(\frac{x - 2}{x - 4}\) is negative. To do this, we analyze the sign of the expression in the intervals determined by the critical points \(x = 2\) and \(x = 4\). The intervals are:

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

#### Test values:
1. For \(x = 0\) (in \((-\infty, 2)\)):
   \[
   \frac{0 - 2}{0 - 4} = \frac{-2}{-4} = \frac{1}{2} > 0
   \]
2. For \(x = 3\) (in \((2, 4)\)):
   \[
   \frac{3 - 2}{3 - 4} = \frac{1}{-1} = -1 < 0
   \]
3. For \(x = 5\) (in \((4, \infty)\)):
   \[
   \frac{5 - 2}{5 - 4} = \frac{3}{1} = 3 > 0
   \]

### Step 5: Construct the solution
We want the expression to be less than or equal to zero, so we include the intervals where the expression is negative or zero. From our sign analysis:

- The expression is zero at \(x = 2\), so include \(x = 2\).
- The expression is negative in the interval \((2, 4)\).

Thus, the solution in interval notation is:

\[
[2, 4)
\]

This means the solution is the interval from 2 (inclusive) to 4 (exclusive).

Would you like more details on any part of this solution?

### Related Questions:
1. How do we solve rational inequalities in general?
2. Why is the function undefined at \(x = 4\)?
3. Can we have an expression that is always negative or positive for any \(x\)?
4. How do vertical asymptotes affect the solution to inequalities?
5. What is the role of test points in analyzing sign changes in intervals?

### Tip:
Always remember to check the denominator when solving rational inequalities, as the function may be undefined at certain points!

R(x) = (x - 2) / (x - 4). Solve for R(x) less than or equal to zero. Write your answer in interval notation.

This page explains how to solve the rational inequality (x - 2) / (x - 4) ≤ 0. Follow a detailed step-by-step solution involving sign analysis and interval notation. Suitable for students in Grades 9-11.

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