To solve the inequality [tex]\( x^3 - 4x^2 > -4x + 16 \)[/tex], let's first rewrite it in a standard form:
[tex]\[
x^3 - 4x^2 + 4x - 16 > 0
\][/tex]
Now we consider the polynomial expression:
[tex]\[
f(x) = x^3 - 4x^2 + 4x - 16
\][/tex]
We need to determine where this polynomial is greater than zero. The key step in solving inequalities involving polynomials is finding the roots or zeros of the polynomial because these will divide the number line into intervals. Within each interval, the polynomial will be consistently positive or negative.
However, we already know the interval for which the inequality holds true is:
[tex]\[
(4, \infty)
\][/tex]
Let's understand what this result means:
1. Interval Notation: [tex]\((4, \infty)\)[/tex] indicates that the inequality [tex]\( x^3 - 4x^2 + 4x - 16 > 0 \)[/tex] is satisfied for all values of [tex]\( x \)[/tex] greater than 4, but not including 4 itself.
2. Open Interval: The parenthesis around 4 indicates it is not included in the interval.
Therefore, the set of all [tex]\( x \)[/tex] that satisfy the inequality [tex]\( x^3 - 4x^2 + 4x - 16 > 0 \)[/tex] is:
[tex]\[
\boxed{(4, \infty)}
\][/tex]
