Why is x <= -8? And not >= -8?

Based on the graph sketch on the right as working: x² + 10x + 16 >= 0 means you want the values of x for which the graph is above the x-axis (shaded light blue).

You can easily verify the solution is correct by substitution. For example, when x = -10, (-10)² + 10(-2) + 16 = 96, which is greater than 0.

I see. Thank you!

Quadratic inequalities do not function the same way as linear inequalities do!

When we learn to solve linear inequalities such x + 8 > 0, we can simply say that x > -8. This is true because when we plot a graph of y = x + 8 on an x-y plane, the graph goes above the x-axis after x = -8 and never ever goes back below the x-axis.

Quadratic inequalities are trickier.

(x + 8) (x + 2) > 0

It appears as though x + 8 > 0 or x + 2 > 0. However, unlike the case of the linear inequality, the shape of the graph of this function is different.

It starts out high above the x-axis. It then crosses over to below the x-axis for a short period of time before returning above the x-axis.

As such, it is not immediately true that the solution fitting the inequalities satisfy x + 8 > 0 or x + 2 > 0.

In fact, x + 8 < 0 or x + 2 > 0.