Final answer:
b) x > 3
Explanation:
The inequality 3x > 2x^2 signifies that three times a value x is greater than twice the square of the same value. To solve this inequality, rearrange the terms to have one side equal to zero: 3x - 2x^2 > 0. Factoring out x from both terms results in x(3 - 2x) > 0. To find the values of x, set each factor equal to zero: x = 0 and 3 - 2x = 0. Solving the equation 3 - 2x = 0 gives x = 3/2. Upon analyzing the intervals and test points on a number line, it's apparent that x must be greater than 3 for the inequality 3x > 2x^2 to hold true.
Understanding the inequality involves interpreting the factors x and (3 - 2x) and their relationship concerning the given inequality. The roots of these factors are x = 0 and x = 3/2, which divide the number line into three intervals: (-∞, 0), (0, 3/2), and (3/2, ∞). By substituting test points from each interval into the inequality, it becomes evident that only the region where x > 3 satisfies the condition 3x > 2x^2. Therefore, for the inequality to be valid, x must be greater than 3.
Graphically, plotting the functions y = 3x and y = 2x^2 reveals an intersection at x = 3, indicating where the two functions are equal. This intersection point serves as the boundary where 3x becomes greater than 2x^2. Hence, x must be greater than 3 for the inequality 3x > 2x^2 to hold true.