Final answer:
To solve the inequality -3x³+21x²+36x-146>=12x²-12x-2, we factor the expression, find the roots, create a number line, and test values in each interval. The solution in interval notation is (-∞,-4]∪(3,4]
Step-by-step explanation:
To solve the inequality -3x³+21x²+36x-146>=12x²-12x-2, we first need to bring all the terms to one side of the inequality. This gives us -3x³+9x²+48x-144>=0. Next, we factor out the common factor of 3 to simplify the expression, giving us 3(-x³+3x²+16x-48)>=0. We can then find the roots of the equation -x³+3x²+16x-48=0 to determine where the expression is equal to 0. By using a graphing calculator or factoring, we find that the roots are x=-4, x=3, and x=4.
We can now create a number line with these critical points and test a value in each interval. For x<-4, we choose x=-5 and substitute it into the inequality to get (-5)³+x²+16(-5)-48<0. This simplifies to -125+25-80-48<0, which is true. For -44, we choose x=5 and substitute it into the inequality to get 125+25+80-48>0, which is true.
Therefore, the solution to the inequality -3x³+21x²+36x-146>=12x²-12x-2 in interval notation is (-∞,-4]∪(3,4]