Question

solve for x and graph the solution.
-(x - 6)(x + 4)^2 < 0
plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.
save answer

Explanation:

Step1: Find the critical points

Set \(-(x - 6)(x + 4)^{2}=0\).
Using the zero - product property \(x-6 = 0\) gives \(x = 6\), and \((x + 4)^{2}=0\) gives \(x=-4\).

Step2: Analyze the sign of the function in the intervals

We have three intervals to consider: \((-\infty,-4)\), \((-4,6)\) and \((6,\infty)\).

• For \(x<-4\), let \(x=-5\). Then \(-(-5 - 6)(-5 + 4)^{2}=-( - 11)\times1 = 11>0\).
• For \(-4
• For \(x>6\), let \(x = 7\). Then \(-(7 - 6)(7 + 4)^{2}=-(1)\times121=-121<0\).

Also, note that when \(x=-4\), \(-(x - 6)(x + 4)^{2}=0\) and the inequality is strict (\(<\)), so \(x=-4\) is not included. When \(x = 6\), the function is \(0\) and is not included in the solution of the strict inequality.

Answer:

The solution of the inequality \(-(x - 6)(x + 4)^{2}<0\) is \(x>6\). On the number - line, we have an open circle at \(x = 6\) and a ray extending to the right.