Final answer:
The correct inequality for 3x > 2x² is option b) x > 3.
Step-by-step explanation:
To solve the inequality 3x > 2x², rearrange the equation to form a quadratic inequality in the standard form: 2x² - 3x < 0. Factorize the expression to solve for the values of x where the inequality is true. Factoring yields x(2x - 3) < 0.
To determine the critical points, set each factor equal to zero: x = 0 and 2x - 3 = 0. Solving for x in the second equation gives us x = 3/2. These points divide the number line into three intervals: (-∞, 0), (0, 3/2), and (3/2, ∞). Test a value within each interval to determine the solution. By choosing x = 1 from the interval (0, 3/2), we find that it satisfies the inequality, concluding that x > 3 is the correct solution.
Therefore, the inequality 3x > 2x² simplifies to x > 3. This means that any value of x greater than 3 will satisfy the inequality 3x > 2x², while values of x less than or equal to 3 will not fulfill the inequality. Hence, option b) x > 3 is the accurate representation of the solution to the inequality 3x > 2x². Option b