Question

Find the solution(s) that satisfy both equations:

a) x^3 + x^2 -20x <= 20 ||| b) x^3 - 3x^2 - 4x + 12 > 0

Answer 1

It might be easier to see the solution set when graphing them if you graph
x^3 + x^2 -20x - 20 <= 0
x^3 - 3x^2 - 4x + 12 > 0
That way the intervals are bounded by the roots of the functions. Come back if you get stuck. If you are having trouble calculating the roots, note that both can be factored easily by grouping.

Answer 2

So I factored by grouping and got

a) x[(x+5)(x-4)] <= 0
b) (x-2)(x+2)(x-3) > 0

I plugged the numbers into desmos, but I'm still having difficulty understanding what exactly this means

Answer 3

View the graphs here:

www.wolframalpha.com/input/?i=plot+x%5E3+%2B+x%5E2+-20x+-+20%2C+x%5E3+-+3x%5E2+-+4x+%2B+12

You can see that both conditions apply only on the interval
[-1,2)U[3,2√5)

Your factoring is also off
x^3 + x^2 -20x - 20 = (x+1)(x-√20)(x+√20)

Answer 4

Thank you oobleck, just one more thing:

I think you accidentially put the wrong equation, my equation is x^3 + x^2 -20x <= 0, *NOT* x^3 + x^2 -20x - 20.

Actually, on closer analysis, I also got my on equation wrong, I put x^3 + x^2 -20x <= 20 when in reality it is x^3 + x^2 -20x <= 0