How do you do these math problems?

1) x² + 4x + 4 >= 9

1st, solve as equality:

x² + 4x - 5 >= 0

(x - 1)(x + 5) >= 0

x = 1 or x = -5

These are the interval limits.

Next, test each interval:

x < -5, e.g. -6: (-6)² + 4(-6) + 4 ≥? 9 → True

-5 < x < 1, e.g. 0: (0)² + 4(0) + 4 ≥? 9 → False

x > 1, e.g. 2: (2)² + 4(2) + 4 ≥? 9 → True

So

x ≤ -5 or x ≥ 1

2) 3(x-1)(x+1) > 0

solve for equality:

3(x-1)(x+1) = 0

x = 1 or x = -1

test intervals:

x < -1, e.g. -2: 3((-2)-1)((-2)+1) >? 0 → True

-1 < x < 1, e.g. 0: 3((0)-1)((0)+1) >? 0 → False

x > 1, e.g 2: 3((2)-1)((2)+1) >? 0 → True

so

x < -1 or x > 1

Or

you can just remember that

"<" pattern is:

m < x < n

and ">" pattern is:

x < m or x > n