Solve for x, x ϵ R

Question

a) (x + 1)(x – 2)(x – 4)2 > 0

Answer 1

From the function:
f(x) = (x + 1)(x – 2)(x – 4)²
we see that it is a quartic with leading coefficient equal to 1. So the shape of the curve is like a letter W.

Since factors (x+1) and (x-2) are distinct, we see that f(x)<0 ∀ xϵ(-1,2).

The double root at x=4 means that f(4)=0.

Thus f(x)>0 ∀xϵ (-∞,-1)∪(2,4)∪(4,+∞)