Let $R(x)=\frac{x-2}{x-6}$. Solve for $R(x)> 0$

State your answer using interval notation. (Use $U$ for union and oo for $\infty$.)

Step 1 ：Let \(R(x)=\frac{x-2}{x-6}\). We need to solve for \(R(x)>0\).

Step 2 ：To solve this inequality, we need to find the values of x for which the function R(x) is positive. This means we need to find the values of x for which the numerator and the denominator of the fraction have the same sign, because a positive divided by a positive is positive, and a negative divided by a negative is also positive.

Step 3 ：We can start by finding the values of x for which the numerator and the denominator are zero, because these are the values of x where the sign of the function changes. The numerator is zero when x = 2, and the denominator is zero when x = 6.

Step 4 ：So we need to check the sign of the function in the intervals (-∞, 2), (2, 6), and (6, ∞). We can do this by choosing a test point in each interval and evaluating the function at that point. If the function is positive at the test point, then it is positive in the entire interval. If the function is negative at the test point, then it is negative in the entire interval.

Step 5 ：Let's choose the test points -1, 4, and 7, which are in the intervals (-∞, 2), (2, 6), and (6, ∞), respectively.

Step 6 ：If R is positive at a test point, then it is positive in the interval containing that test point. If R is negative at a test point, then it is negative in the interval containing that test point.

Step 7 ：After evaluating the function R at the test points -1, 4, and 7, we find that R is positive in the intervals containing -1 and 7, and negative in the interval containing 4.

Step 8 ：This means that R is positive in the intervals (-∞, 2) and (6, ∞), and negative in the interval (2, 6).

Step 9 ：So the solution to the inequality R(x) > 0 is x in (-∞, 2) U (6, ∞).

Step 10 ：Final Answer: \(\boxed{(-\infty, 2) \cup (6, \infty)}\)