Question

solve for x and graph the solution.
5(x - 2)(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.

Explanation:

Step1: Find the critical points

Set \(5(x - 2)(x + 6)=0\). Since \(5
eq0\), then \(x - 2=0\) gives \(x = 2\), and \(x+6=0\) gives \(x=-6\).

Step2: Test intervals

• For \(x<-6\), let \(x=-7\). Then \(5(-7 - 2)(-7 + 6)=5\times(-9)\times(-1)=45>0\).
• For \(-6
• For \(x>2\), let \(x = 3\). Then \(5(3 - 2)(3 + 6)=5\times1\times9 = 45>0\).

Step3: Consider equality

The inequality is \(\geq0\), so the critical points \(x=-6\) and \(x = 2\) are included in the solution set.

Answer:

The solution of the inequality \(5(x - 2)(x + 6)\geq0\) is \(x\leq - 6\) or \(x\geq2\). On the number - line, we plot closed circles at \(x=-6\) and \(x = 2\). The graph consists of two rays: one starting at \(x=-6\) and extending to the left (including \(x=-6\)) and one starting at \(x = 2\) and extending to the right (including \(x = 2\)).