Test a point in each interval to determine the sign of the expression:
Test y = -1: (-1)(-1^3 - 10) = (-1)(-11) = 11 > 0 Test y = ∛9: (∛9)(∛9^3 - 10) = (∛9)(9-10) = (∛9)(-1) < 0 Test y = 1: (1)(1^3 - 10) = (1)(1-10) = (1)(-9) < 0
The expression is positive between -∛10 and 0, and negative between 0 and ∛10. Therefore, the solution to the inequality y(y^3 - 10) < 0 (or (x - 6)(x^3 - 10) < 0) is -∛10 < x < 0.
To solve this inequality, first Let's denote y = (x-6).
So the inequality becomes y^2 < √10y
Squaring both sides results in y^4 < 10y
Rearrange to get y^4 - 10y < 0
Factor out y to get y(y^3 - 10) < 0
Set each factor to zero to find the critical points: y = 0 and y^3 - 10 = 0
The solutions to y^3 - 10 = 0 are y = ∛10
Plotting these points on a number line:
-∞-----------------∛10-------0--------------∛10---------------+∞
Test a point in each interval to determine the sign of the expression:
Test y = -1: (-1)(-1^3 - 10) = (-1)(-11) = 11 > 0
Test y = ∛9: (∛9)(∛9^3 - 10) = (∛9)(9-10) = (∛9)(-1) < 0
Test y = 1: (1)(1^3 - 10) = (1)(1-10) = (1)(-9) < 0
The expression is positive between -∛10 and 0, and negative between 0 and ∛10. Therefore, the solution to the inequality y(y^3 - 10) < 0 (or (x - 6)(x^3 - 10) < 0) is -∛10 < x < 0.