Outside the Roots

Now flip the question. ${x}^{2}-3x-10 \ge 0$ factors as $\left(x-5\right)\left(x+2\right) \ge 0$, and asks where the product is positive or zero. A product is positive when both factors have the same sign, so the two sign cases are: both factors at least 0, or both at most 0.

Both at least 0: $x \ge 5$ and $x \ge -2$. Every number from 5 up satisfies both, so this case gives $x \ge 5$. Both at most 0: $x \le 5$ and $x \le -2$, which gives $x \le -2$. The answer is two separate pieces, the numbers outside the roots: $x \le -2$ or $x \ge 5$.

The word or matters. No number is both at most -2 and at least 5, so joining the pieces with and would describe nothing. The equals sign matters too: at -2 and at 5 the product is 0, which the sign allows, so both ends belong to the answer.