Question

graph the solution to the following inequality on the number line.
$x^{2}>-6x$
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Explanation:

Step1: Rearrange the inequality

Move all terms to one side: \(x^{2}+6x>0\). Factor it: \(x(x + 6)>0\).

Step2: Find the critical points

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

Step3: Test intervals

• For \(x=-7\) (in \((-\infty,-6)\)): \((-7)\times(-7 + 6)=(-7)\times(-1)=7>0\).
• For \(x=-3\) (in \((-6,0)\)): \((-3)\times(-3 + 6)=(-3)\times3=-9<0\).
• For \(x = 1\) (in \((0,\infty)\)): \(1\times(1 + 6)=7>0\).

Answer:

The solution of the inequality \(x^{2}>-6x\) is \(x<-6\) or \(x>0\). On the number - line, we have an open circle at \(x=-6\) and \(x = 0\), and shade the regions to the left of \(x=-6\) and to the right of \(x = 0\).