Outside the Roots

Now flip the question. x^2 - 3x - 10 >= 0 factors as (x - 5)(x + 2) >= 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 >= 5 and x >= -2. Every number from 5 up satisfies both, so this case gives x >= 5. Both at most 0: x <= 5 and x <= -2, which gives x <= -2. The answer is two separate pieces, the numbers outside the roots: x <= -2 or x >= 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.