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

To solve the inequality (3x - 1)(81 - 3x) > 0, we can analyze the signs of each factor separately and then determine the intervals where the product is positive.

1. Let's find the critical points by setting each factor equal to zero: 3x - 1 = 0 => 3x = 1 => x = 1/3 81 - 3x = 0 => 3x = 81 => x = 27

2. Now, we can test the sign of each factor for different intervals using a number line: Interval 1: (-∞, 1/3)

   • Choose a number less than 1/3, such as x = 0: Substitute x = 0 into (3x - 1): (3(0) - 1) = -1 So, the factor (3x - 1) is negative for x < 1/3.

   • Choose a number greater than 1/3, such as x = 0.5: Substitute x = 0.5 into (3x - 1): (3(0.5) - 1) = 0.5 So, the factor (3x - 1) is positive for x > 1/3.

   Interval 2: (1/3, 27)

   • Choose a number less than 1/3, such as x = 0.4: Substitute x = 0.4 into (3x - 1): (3(0.4) - 1) = 0.2 So, the factor (3x - 1) is positive for x > 1/3.

   • Choose a number greater than 1/3 but less than 27, such as x = 10: Substitute x = 10 into (3x - 1): (3(10) - 1) = 29 So, the factor (3x - 1) is positive for x > 1/3.

   • Choose a number greater than 27, such as x = 30: Substitute x = 30 into (81 - 3x): (81 - 3(30)) = -69 So, the factor (81 - 3x) is negative for x > 27.

   Interval 3: (27, ∞)

   • Choose a number less than 27, such as x = 20: Substitute x = 20 into (81 - 3x): (81 - 3(20)) = 21 So, the factor (81 - 3x) is positive for x > 27.

Based on the analysis of the signs of each factor, we can determine the intervals where the expression (3x - 1)(81 - 3x) is positive.

The expression is positive in Interval 2: (1/3, 27) and Interval 3: (27, ∞).

Therefore, the solution to the inequality is: 1/3 < x < 27.