Answer
The solution is $[-3,0]\cup[0,3]$ or simply $[-3,3]$
The graph is:
Work Step by Step
$4x^{2}(x^{2}-9)\le0$
Factor the left side completely:
$4x^{2}(x-3)(x+3)\le0$
Find the intervals. The factors are $x^{2}$, $x-3$ and $x+3$. Set them equal to $0$ and solve for $x$:
$x^{2}=0$
$x=0$
$x-3=0$
$x=3$
$x+3=0$
$x=-3$
The factors are zero when $x=0,3,-3$. These three numbers divide the real line into the following intervals:
$(-\infty,-3)$ $,$ $(-3,0)$ $,$ $(0,3)$ $,$ $(3,\infty)$
Elaborate a diagram, using test points to determine the sign of each factor in each interval: (refer to the attached image below)
It can be seen from the diagram that the inequality is satisfied on the intervals $(-3,0)$ and $(0,3)$. Also, the inequality involves $\le$ so the endpoints also satisfy the inequality.
The solution is $[-3,0]\cup[0,3]$ or simply $[-3,3]$
