Question

solve for x and graph the solution. (x - 1)(x + 6) < 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 \((x - 1)(x + 6)=0\). By the zero - product property, \(x-1 = 0\) gives \(x = 1\), and \(x+6=0\) gives \(x=-6\). These are the critical points that divide the number line into three intervals: \((-\infty,-6)\), \((-6,1)\), and \((1,\infty)\).

Step2: Test the intervals

• For the interval \((-\infty,-6)\), let \(x=-7\). Then \((-7 - 1)(-7 + 6)=(-8)\times(-1)=8>0\).
• For the interval \((-6,1)\), let \(x = 0\). Then \((0 - 1)(0 + 6)=(-1)\times6=-6<0\).
• For the interval \((1,\infty)\), let \(x = 2\). Then \((2 - 1)(2 + 6)=(1)\times8 = 8>0\).

Step3: Determine the solution

Since the inequality \((x - 1)(x + 6)<0\) is satisfied when \(x\in(-6,1)\).

Answer:

The solution of the inequality \((x - 1)(x + 6)<0\) is \(-6