Question

solve for x and graph the solution.
-(x - 4)(x + 5) ≤ 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 \(-(x - 4)(x + 5)=0\). Then \(x-4 = 0\) gives \(x = 4\), and \(x+5=0\) gives \(x=-5\).

Step2: Test intervals

• For \(x<-5\), let \(x=-6\). Then \(-(-6 - 4)(-6 + 5)=-(-10)(-1)=-10\leq0\), so \(x<-5\) is part of the solution.
• For \(-5
• For \(x>4\), let \(x = 5\). Then \(-(5 - 4)(5 + 5)=-(1)(10)=-10\leq0\), so \(x>4\) is part of the solution.

Since the inequality is \(\leq0\), the endpoints \(x=-5\) and \(x = 4\) are included.

Answer:

The solution is \(x\leq - 5\) or \(x\geq4\). On the number - line, we plot closed circles at \(x=-5\) and \(x = 4\), and shade the regions to the left of \(x=-5\) and to the right of \(x = 4\).