i'm really stuck on this one problem in my summer packet
any help would be greatly appreciated
Thanks in advance
You're looking at the values of x for which a product of terms, here (x^4) times (x-3) is <= 0. So you want to consider the possible signs (positive, negative or zero) of each of the terms. Let's start by considering the ranges of x for which the part with a power, in this case x^4, is positive, negative or zero. In this case it's simple: x^4 > 0 for x > 0 or x < 0 and x^4 = 0 for x = 0. For x<>0, x^4>0 and so x^4(x-3) has the same sign (positive, negative or zero) as x-3. But (x-3) <= 0 for x <= 3. For x=0, x^4 = 0 and so x^4(x-3) = 0. So the inequality is satisfied when x >0 and x <= 3, or when x=0, or when x < 0 and x <= 3. These combine just to saying that x <= 3.