Final answer:
To solve the system of inequalities: 2x<24-x^(2); x^(3)>3x^(2), we need to solve each inequality separately and then find the values of x that satisfy both of them. The solution to the system of inequalities is -64.
Step-by-step explanation:
To solve the system of inequalities: 2x<24-x^(2); x^(3)>3x^(2), we need to solve each inequality separately and then find the values of x that satisfy both of them.
For the first inequality, 2x<24-x^(2), we can rearrange it to x^(2)+2x-24>0 and then factor it as (x-4)(x+6)>0. The critical values are x=4 and x=-6. Since the inequality is greater than zero, we need to find the intervals where the expression is positive. So the solution is x<-6 or x>4.
For the second inequality, x^(3)>3x^(2), we can rearrange it as x^(3)-3x^(2)>0 and then factor it as x^(2)(x-3)>0. The critical value is x=0 and x=3. Since the inequality is greater than zero, we need to find the intervals where the expression is positive. So the solution is 0<x<3.
Combining the solutions, we have -6<x<0 or x>4. And 0<x<3. Therefore, the solution to the system of inequalities is -6<x<0 or 0<x<3 or x>4.