Solve x(x-7)(x-9)<0 and express the solution set in interval notation.

A. (–infinity, 0)
B. (–infinity, 0) U (7, 9)
C. (–infinity, 0) U (9, infinity)
D. (–infinity, 9)

It is a cubic, with three real zeroes (0,+7 and +9), therefore the graph crosses the x-axis three times.

The coefficicent of the x³ term is positive, so it increases to the right of the largest root (+9) and decreases to the left of the smallest root (0).
This tells us that (-∞,0) is a subset of the solution.
To find the remaining part of the solution, we note that the function is positive between 0 and the next root, and dips below zero again between the last two roots, namely +7 and +9.
Thus the missing interval is (7,9).
Can you take it from here?

Based upon your explanation, I am goin to say the answer is

B(–infinity, 0) U (7, 9)

? Is this correct?

Correct!

To solve the inequality x(x-7)(x-9)<0, we need to find the values of x that make the expression on the left side of the inequality less than zero.

Let's examine the sign of each factor separately:
1. x: For this factor to be negative, x must be less than zero.
2. (x-7): For this factor to be negative, x must be greater than 7.
3. (x-9): For this factor to be negative, x must be greater than 9.

Now, let's combine these conditions to find the solution set.

By observing the factors, we can see that the sign of the expression changes when x passes through the values 0, 7, and 9.

Case 1: x < 0
All three factors are negative, so the expression is negative.

Case 2: 0 < x < 7
The first factor (x) is positive, while the other two factors (x-7) and (x-9) are negative. Therefore, the expression is positive.

Case 3: 7 < x < 9
The first factor (x) is positive, while the other two factors (x-7) and (x-9) are positive. Therefore, the expression is negative.

Case 4: x > 9
All three factors are positive, so the expression is positive.

Now, let's represent these cases in the solution set.

Case 1: x < 0. This is represented as (-infinity, 0).

Case 2: 7 < x < 9. This range is not included in the solution because the expression is positive here.

Case 3: x > 9. This is represented as (9, infinity).

Combining case 1 and case 3, we get the solution set: (-infinity, 0) U (9, infinity).

Therefore, the correct answer is C. (–infinity, 0) U (9, infinity).