Solving Inequalities Advanced Functions

Question

How do I solve this particular inequality?? I forgot how my teacher managed to solve this $$-t^4 +5t^3 +5t^2+6t>100$$

Answer 1

We are looking for t values that satisfy the inequality. $$-t^4 +5t^3 +5t^2+6t>100$$ Subtract 100 in both sides. $$-t^4 +5t^3 +5t^2+6t-100>0$$ This this exactly a polynomial of 4th degree, therefore it can be solved by using the quartic equation that produces 4 answers. Plugging the coefficients of this equation you get 4 answers, in which 2 of them are complex numbers and the remaining 2 are real numbers, these real numbers are the t values that satisfy the inequality. Thus I get $t_1=2.732654694546666$ and $t_2= 5.5070173483542$. Since the inequality is strict, that is, the values must be greater than 100 but not equal to 100 then we use an open interval. Thus the answer is $2.732654694546666<t<5.5070173483542$ or $∃t∈2.732654694546666<t<5.5070173483542$. You can go to Desmos calculator and graph your function and you'll find out that indeed, the function is greater than 100 at the said $t$ values.