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, a segment, ray, or line to delete it.

Explanation:

Step1: Expand the left - hand side

First, expand $-2(x + 5)(x - 5)$ using the difference - of - squares formula $(a + b)(a - b)=a^{2}-b^{2}$. Here $a=x$ and $b = 5$, so $-2(x + 5)(x - 5)=-2(x^{2}-25)=-2x^{2}+50$.
The inequality becomes $-2x^{2}+50<0$.

Step2: Rearrange the inequality

Move the terms to get $2x^{2}-50>0$. Then divide both sides by 2: $x^{2}-25>0$.
Factor the left - hand side: $(x + 5)(x - 5)>0$.

Step3: Find the roots

Set $(x + 5)(x - 5)=0$. The roots are $x=-5$ and $x = 5$.

Step4: Test the intervals

We have three intervals to test: $x<-5$, $-55$.
For $x<-5$, let $x=-6$. Then $(-6 + 5)(-6 - 5)=(-1)\times(-11)=11>0$, so the inequality is satisfied on $x<-5$.
For $-5For $x>5$, let $x = 6$. Then $(6 + 5)(6 - 5)=(11)\times(1)=11>0$, so the inequality is satisfied on $x>5$.

Answer:

The solution of the inequality is $x<-5$ or $x>5$. On the number - line, we use open circles at $x=-5$ and $x = 5$ and shade the regions to the left of $x=-5$ and to the right of $x = 5$.