Solve the following polynomial inequalities.2(x3 - 2x2 + 3) < x(x - 1)(x + 1

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The inequality 2(x^3 - 2x^2 + 3) < x(x - 1)(x + 1) has to be solved.

2(x^3 - 2x^2 + 3) < x(x - 1)(x + 1)

=>  x(x - 1)(x + 1) - 2(x^3 - 2x^2 + 3) > 0

=>  x(x^2 - 1) - 2(x^3 - 2x^2 + 3) > 0

=>  x^3 - x - 2x^3 + 4x^2 - 6 > 0

=>  - x - x^3 + 4x^2 - 6 > 0

=>  x^3 - 4x^2 + x + 6 < 0

=> (x-3)(x-2)(x+1) < 0

This is the case when:

(x-3) > 0, (x-2)> 0 and (x+1) < 0

=> x > 3, x > 2 and x < -1

This is not true for any value of x.

(x-3) > 0, (x-2) < 0 and (x+1) > 0

=> x > 3, x < 2 and x > -1

This not true for any value of x

(x-3) < 0, (x-2) > 0 and (x+1) > 0

=> x < 3, x > 2 and x > -1

This is true for `x in (2, 3)`

(x-3) < 0, (x-2) < 0 and (x+1) < 0

=> x < 3, x < 2 and x < -1

This is true for  `x in (-oo, -1)`

The solution of the inequality is `(-oo, -1)U(2, 3)`

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