Question

solve for x and graph the solution.
-(x - 2)(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: Simplify the inequality

Multiply both sides by - 1 (and reverse the inequality sign). We get \((x - 2)(x - 5)\geq0\).

Step2: Find the critical points

Set \((x - 2)(x - 5)=0\). Using the zero - product property \(x-2 = 0\) or \(x - 5=0\). So \(x=2\) and \(x = 5\) are the critical points.

Step3: Test intervals

• For \(x<2\), let \(x = 0\). Then \((0 - 2)(0 - 5)=( - 2)\times(-5)=10\geq0\).
• For \(2
• For \(x>5\), let \(x=6\). Then \((6 - 2)(6 - 5)=(4)\times(1)=4\geq0\).

Answer:

The solution of the inequality \(-(x - 2)(x - 5)\leq0\) (or \((x - 2)(x - 5)\geq0\)) is \(x\leq2\) or \(x\geq5\). On the number - line, we plot closed circles at \(x = 2\) and \(x = 5\) (since the inequality includes equality) and shade the regions to the left of \(x = 2\) and to the right of \(x = 5\).