Question

solve for x and graph the solution.
(x + 1)(x - 4) > 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.

Explanation:

Step1: Find the critical points

Set \((x + 1)(x - 4)=0\). By the zero - product property, \(x+1 = 0\) gives \(x=-1\), and \(x - 4=0\) gives \(x = 4\). These are the critical points that divide the number line into three intervals: \((-\infty,-1)\), \((-1,4)\), and \((4,\infty)\).

Step2: Test intervals

• For the interval \((-\infty,-1)\), let \(x=-2\). Then \((-2 + 1)(-2-4)=(-1)\times(-6)=6>0\).
• For the interval \((-1,4)\), let \(x = 0\). Then \((0 + 1)(0 - 4)=(1)\times(-4)=-4<0\).
• For the interval \((4,\infty)\), let \(x=5\). Then \((5 + 1)(5 - 4)=(6)\times(1)=6>0\).

Answer:

The solution of the inequality \((x + 1)(x - 4)>0\) is \(x<-1\) or \(x>4\). On the number - line, we have an open circle at \(x=-1\) (since the inequality is strict, \(x
eq - 1\)) and an open circle at \(x = 4\) (since \(x
eq4\)). We shade the regions to the left of \(x=-1\) (i.e., the interval \((-\infty,-1)\)) and to the right of \(x = 4\) (i.e., the interval \((4,\infty)\)).