What is the solution of x3 – x2 > 9(x – 1)

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)