Today's soundtrack is Suffocation: Pierced From Within, a classic brutal death metal album.
This afternoon, I'm learning about quadratic equations with inequalities in a single variable. If we replace a quadratic equation's equals sign with any kind of inequality symbol (greater than, less than, greater than or equal to, lesser than or equal to), "a quadratic inequality in one variable is formed" (Pearson's Pre-Calculus 11, p. 340).
What this means is that instead of being limited to equations where ax²+bx+c = 0, we can consider the following four possibilities:
ax²+bx+c < 0
ax²+bx+c > 0
ax²+bx+c ≤ 0
ax²+bx+c ≥ 0
There are only two variants of this idea to really understand:
If the equation's solution is greater than 0, we consider all values higher than 0 on the y-axis that are outside of the parabola to be part of the solution.
If the equation's solution is less than 0, we consider all values lower than 0 on the y-axis that are inside of the parabola to be part of the solution.
Here's a quick-and-dirty sketch I made to help illustrate the concept:
December 18 UPDATE:
I messed up. Ignore the above chart. Check out the December 18 chart for correct information.