Question

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

Divide both sides of \(-2(x - 5)(x + 5)\geq0\) by \(-2\). Remember that when dividing an inequality by a negative number, the direction of the inequality sign changes. So we get \((x - 5)(x + 5)\leq0\).

Step2: Find the critical points

Set \((x - 5)(x + 5)=0\). Using the zero - product property \(a\times b = 0\) implies \(a = 0\) or \(b = 0\). So \(x-5=0\) gives \(x = 5\) and \(x + 5=0\) gives \(x=-5\).

Step3: Test intervals

We have three intervals to test: \((-\infty,-5)\), \((-5,5)\), and \((5,\infty)\).

• For \(x=-6\) (in the interval \((-\infty,-5)\)): \((-6 - 5)(-6+5)=(-11)\times(-1)=11>0\).
• For \(x = 0\) (in the interval \((-5,5)\)): \((0 - 5)(0 + 5)=(-5)\times5=-25<0\).
• For \(x=6\) (in the interval \((5,\infty)\)): \((6 - 5)(6 + 5)=(1)\times(11)=11>0\).

Since the inequality is \((x - 5)(x + 5)\leq0\), the solution of the inequality is \(-5\leq x\leq5\).

Answer:

The solution of the inequality \(-2(x - 5)(x + 5)\geq0\) is \(-5\leq x\leq5\). On the number - line, we plot closed circles at \(x=-5\) and \(x = 5\) and shade the region between them.