Question

How do you solve and write the following in interval notation: `x^2+9x+20≥2`?

Answer 1

Start by solving as a regular quadratic equation (we change the inequality sign to an equal for the time being).

`x^2 + 9x + 20 = 2`

`x^2 + 9x + 18 = 0`

`(x + 6)(x + 3) = 0`

`x = -6 and -3`

Now, we select test points that are at different segments of the number line representation of this inequality, as shown below.

BLUE SEGMENT: Let the test point be `x = -8`

Testing:

`(-8)^2 + 9(-8) + 20 >=^? 2`

`64 - 72 + 20 >= 2`, therefore this interval satisfies the equation. This instantly means that the pink segment is not a solution to the inequality while the green is. Quick checks yield the same result.

Therefore, in interval notation, the solution set is `(-oo, -6) uuu (-3, oo)`

Hopefully this helps!