How do you solve and write the following in interval notation: #(1-x)/(x-9) >=0#?

We build a sign chart

Therefore,

To solve the inequality (\frac{{1-x}}{{x-9}} \geq 0), first, find the critical points by setting the numerator and denominator equal to zero separately:

(1-x = 0) gives (x = 1)

(x - 9 = 0) gives (x = 9)

Now, create intervals using these critical points:

Interval 1: (x < 1)

Interval 2: (1 < x < 9)

Interval 3: (x > 9)

Next, choose test points within each interval to determine the sign of the expression:

For Interval 1: Choose (x = 0) (\frac{{1-0}}{{0-9}} = \frac{1}{-9}), which is negative.

For Interval 2: Choose (x = 5) (\frac{{1-5}}{{5-9}} = \frac{-4}{-4}), which is positive.

For Interval 3: Choose (x = 10) (\frac{{1-10}}{{10-9}} = \frac{-9}{1}), which is negative.

Based on the signs of the expression in each interval, the solution is:

Interval 1: (-\infty < x < 1)

Interval 2: (1 < x < 9)

Interval 3: (x > 9)

Write the solution in interval notation: ((- \infty, 1) \cup (1, 9) \cup (9, \infty)).