Question

solve for x and graph the solution.
5(x + 1)(x + 5) > 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.
submit

Explanation:

Step1: Find the critical points

Set \(5(x + 1)(x + 5)=0\). Using the zero - product property \(a\times b\times c = 0\) implies \(a = 0\) or \(b=0\) or \(c = 0\). Here, \(x+1 = 0\) gives \(x=-1\), and \(x + 5=0\) gives \(x=-5\). These are the critical points that divide the number line into three intervals: \((-\infty,-5)\), \((-5,-1)\), and \((-1,\infty)\).

Step2: Test the intervals

• For the interval \((-\infty,-5)\), let \(x=-6\). Then \(5(-6 + 1)(-6 + 5)=5\times(-5)\times(-1)=25>0\).
• For the interval \((-5,-1)\), let \(x=-3\). Then \(5(-3 + 1)(-3 + 5)=5\times(-2)\times2=-20<0\).
• For the interval \((-1,\infty)\), let \(x=0\). Then \(5(0 + 1)(0 + 5)=25>0\).

Since the inequality is \(5(x + 1)(x + 5)>0\), we consider the intervals where the expression is positive.

Answer:

The solution of the inequality \(5(x + 1)(x + 5)>0\) is \(x<-5\) or \(x>-1\). On the number - line, we have open circles at \(x=-5\) and \(x=-1\) (because the inequality is strict, i.e., \(>\) not \(\geq\)). The solution set in interval notation is \((-\infty,-5)\cup(-1,\infty)\).