Final answer:
To solve the inequality x³ - 3x² - 9x + 27 < 0, utilize a graphing calculator to find where the function crosses the x-axis. These points are the zeros, and testing values around these zeros shows where the function is negative, which satisfies the inequality.
Step-by-step explanation:
To solve the inequality x³ - 3x² - 9x + 27 < 0, we first need to factor the cubic equation, if possible. Factoring this particular cubic equation by grouping or by using synthetic division can be challenging, and it may not factor nicely. However, we can find the solutions by using a graphing calculator. When inputting the equation into the calculator, we look for where the graph crosses the x-axis; these points are the zeros of the function and help us determine the intervals where the function is less than zero.
Unfortunately, without the exact factors or points where the function crosses the axis, we cannot give the exact solution. But once you have the x-values where the function is zero, you can test values in each interval to determine where the function is negative. Any interval where the function is below the x-axis satisfies the inequality x³ - 3x² - 9x + 27 < 0.