Question

solve the following inequality and graph the solution. answer using interval notation. x^3 - 9x < 0

Explanation:

Step1: Factor the left - hand side

$x^{3}-9x=x(x^{2} - 9)=x(x - 3)(x + 3)<0$

Step2: Find the critical points

Set $x(x - 3)(x + 3)=0$. The critical points are $x=-3,x = 0,x = 3$.

Step3: Test the intervals

Test the intervals $(-\infty,-3)$, $(-3,0)$, $(0,3)$ and $(3,\infty)$.
For $x<-3$, let $x=-4$, then $(-4)(-4 - 3)(-4+3)=(-4)(-7)(-1)=-28<0$.
For $-30$.
For $0For $x>3$, let $x = 4$, then $(4)(4 - 3)(4 + 3)=(4)(1)(7)=28>0$.

Answer:

The solution in interval notation is $(-\infty,-3)\cup(0,3)$. To graph: Place open - circles at $x=-3,x = 0,x = 3$ on the number line. Shade the regions $x<-3$ and $0