Question

solve for x and graph the solution.
3(x + 1)(x + 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.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Explanation:

Step1: Find the critical points

Set \(3(x + 1)(x + 2)=0\). 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 \(x + 2=0\) gives \(x=-2\). These are the critical points that divide the number line into three intervals: \((-\infty,-2)\), \((-2,-1)\), and \((-1,\infty)\).

Step2: Test the intervals

• For the interval \((-\infty,-2)\), let \(x=-3\). Then \(3(-3 + 1)(-3+2)=3\times(-2)\times(-1)=6>0\).
• For the interval \((-2,-1)\), let \(x =-\frac{3}{2}\). Then \(3(-\frac{3}{2}+1)(-\frac{3}{2}+2)=3\times(-\frac{1}{2})\times(\frac{1}{2})=-\frac{3}{4}<0\).
• For the interval \((-1,\infty)\), let \(x=0\). Then \(3(0 + 1)(0+2)=3\times1\times2 = 6>0\).

Step3: Determine the solution set

Since the inequality \(3(x + 1)(x + 2)>0\), the solution of the inequality is \(x<-2\) or \(x>-1\).

Answer:

The solution set is \(x\in(-\infty,-2)\cup(-1,\infty)\). On the number - line, we have open circles at \(x=-2\) and \(x=-1\), and we shade the regions to the left of \(x =-2\) and to the right of \(x=-1\).