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!