Problem 41 - Algebra and Trigonometry

Question

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3} \geq 9 x^{2} $$

Step-by-Step Solution

Answer

The solution set of the given polynomial inequality is \([0, 9]\).

Step 1: Rearrange the inequality to zero

Rewrite the inequality by subtracting \(9x^{2}\) from both sides. This gives the inequality \(x^{3} - 9x^{2} \geq 0\).

Step 2: Factor out the polynomial

Factor the left-hand side to simplify the inequality. By factoring out, the inequality becomes \(x^{2}(x - 9) \geq 0\).

Step 3: Find the critical points

Set the factors equal to zero to determine the ‘critical points’ or potential boundaries of the intervals for the solution. This gives \(x_{1} = 0\) and \(x_{2} = 9\) as critical points.

Step 4: Test the intervals

Test the intervals made by the critical points. Choose a test point in each interval \((-∞, 0), (0, 9), (9, ∞)\) and determine whether the inequality is satisfied. The test points could be -1, 1 and 10, respectively.

Step 5: Identify the solution set

Based on the results in Step 4, identify which intervals satisfy the inequality. Keep in mind that as it is a ‘greater or equal to’ inequality, critical points are also included in the solution set.