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Consider the cubic equation,
x³ - x² = 9 (x - 1)
We can solve this by factoring:
x³ - x² - 9x + 9 = 0
x² (x - 1) - 9 (x - 1) = 0
(x² - 9) (x - 1) = 0
(x - 3) (x + 3) (x - 1) = 0
So the cubic has roots at x = 3, x = -3, and x = 1. Split the real line into the open intervals (-∞, -3), (-3, 1), (1, 3), and (3, ∞). Check if the inequality holds for some test point taken from each interval:
• in (-∞, -3), take x = -4. Then
(-4)³ - (-4)² = -80
9 (-4 - 1) = -45
but -80 > -45 is not true. So this interval is not in the solution set.
• in (-3, -1), take x = -2. Then
(-2)³ - (-2)² = -12
9 (-2 - 1) = -27
and -12 > -27 is true. So x in (-3, -1) satisfies the inequality.
• in (1, 3), take x = 2. Then
2³ - 2² = 4
9 (2 - 1) = 9
but 4 > 9 is not true.
• in (3, ∞), take x = 4. Then
4³ - 4² = 48
9 (4 - 1) = 27
and 48 > 27 is true.
So, the complete solution set is
(-3, -1) U (3, ∞)
(where U denotes union)