Question

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

Explanation:

Step1: Find the critical points

Set \(x - 3=0\) and \(x + 6=0\). Solving \(x - 3=0\) gives \(x = 3\), and solving \(x+6 = 0\) gives \(x=-6\). These are the critical points that divide the number - line into three intervals: \((-\infty,-6)\), \((-6,3)\), and \((3,\infty)\).

Step2: Test intervals

• For the interval \((-\infty,-6)\), let \(x=-7\). Then \((-7 - 3)(-7 + 6)=(-10)\times(-1)=10>0\).
• For the interval \((-6,3)\), let \(x = 0\). Then \((0 - 3)(0 + 6)=(-3)\times6=-18<0\).
• For the interval \((3,\infty)\), let \(x = 4\). Then \((4 - 3)(4 + 6)=(1)\times(10)=10>0\).

Step3: Consider the equality

Since the inequality is \(\leq0\), the endpoints \(x=-6\) and \(x = 3\) are included in the solution set.

Answer:

The solution of the inequality \((x - 3)(x + 6)\leq0\) is \([-6,3]\). On the number - line, we plot a closed circle at \(x=-6\) and \(x = 3\) and shade the segment between them.