(3\x-1)(81-3\x)>0

To solve this inequality, we need to find the values of x that make the expression greater than zero.

First, let's expand the expression:

(3x - 1)(81 - 3x) > 0

Now, let's simplify the expression:

243x - 9x^2 - 81 + 3x > 0

Combining like terms:

-9x^2 + 246x - 81 > 0

Now, we need to solve this quadratic inequality. There are various methods to do this, such as factoring, using a sign chart, or using the quadratic formula.

Let's use the sign chart method:

1. Find the critical points by setting the quadratic expression equal to zero:

-9x^2 + 246x - 81 = 0

Using the quadratic formula, we get:

x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in a = -9, b = 246, and c = -81:

x = (-246 ± √(246^2 - 4(-9)(-81))) / (2(-9))

Simplifying, we get two critical points:

x = (-246 ± √(60516 - 2916)) / (-18) x = (-246 ± √57600) / (-18) x = (-246 ± 240) / (-18)

There are two possible critical values: x = (-246 + 240) / (-18) x = -6 / (-18) x = 1/3

x = (-246 - 240) / (-18) x = -486 / (-18) x = 27

2. Now, we test the values in each interval created by the critical points on the inequality:

I. (-∞, 1/3): Choose a value less than 1/3, like 0: (-9(0)^2 + 246(0) - 81) > 0 -81 > 0 This interval does not satisfy the inequality.

II. (1/3, 27): Choose a value between 1/3 and 27, like 5: (-9(5)^2 + 246(5) - 81) > 0 -369 > 0 This interval satisfies the inequality.

III. (27, ∞): Choose a value greater than 27, like 30: (-9(30)^2 + 246(30) - 81) > 0 6909 > 0 This interval satisfies the inequality.

Finally, the solution to the inequality is:

1/3 < x < 27