Question

graph the solution to the following inequality on the number line. $x^{2} < - 6x$. note that you can use the aleks graphing calculator to help get your answer.

Explanation:

Step1: Rearrange the inequality

Move all terms to one - side: $x^{2}+6x < 0$. Then factor the left - hand side: $x(x + 6)<0$.

Step2: Find the roots of the corresponding equation

Set $x(x + 6)=0$. The roots are $x = 0$ and $x=-6$. These roots divide the number line into three intervals: $(-\infty,-6)$, $(-6,0)$ and $(0,\infty)$.

Step3: Test the intervals

For the interval $(-\infty,-6)$, let's choose $x=-7$. Then $(-7)\times(-7 + 6)=(-7)\times(-1)=7>0$.
For the interval $(-6,0)$, let's choose $x=-1$. Then $(-1)\times(-1 + 6)=(-1)\times5=-5<0$.
For the interval $(0,\infty)$, let's choose $x = 1$. Then $1\times(1 + 6)=7>0$.

The solution of the inequality $x(x + 6)<0$ is the interval where the expression is negative, which is $-6

Answer:

On the number line, we use an open circle at $x=-6$ and $x = 0$ (because the inequality is strict, $<$ not $\leq$) and shade the region between $-6$ and $0$.