Today's soundtrack is Christmas With the Mills Family, a Spotify playlist that includes my favourite Christmas songs.
I learned yesterday that when we consider quadratic inequalities with a single variable, 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, and 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.
When we find quadratic inequalities that are either less than or greater than a value, we call the x-intercepts of that equation its critical values, for it is these numbers that we use to give the values of the equation's domain.
Putting that information into practice, if we see the equation x² - 2x - 3 < 0, we can use the quadratic formula to determine that its x-intercepts are -1 and 3. Since our equation is looking for values of x inside of our parabola when y < 0, we know that anything between the values of -1 and 3 is part of our solution.
Our critical values give us values in three intervals. If our equation < 0, the sequence of the interval will be "not solution; solution; and not solution"; if our equation > 0, the sequence will be inverted: "solution; not solution; and solution". We can use a number line that coincides with our y-0 line to show these values. Just highlight the values that are included in the solution, and put circles on the x-intercepts to indicate that these are not included in the solution (unless, of course, our solution is not merely less than or greater than, but lesser than or equal to, or greater than or equal to).
In the above example of the equation x² - 2x - 3 < 0 where our solution is between the values of -1 and 3, our interval (formatted "not solution; solution; not solution") is "x < -1; -1 < x < 3; and x > 3" (Pearson's Pre-Calculus 11, p. 341). If our equation had been x² - 2x - 3 ≤ 0, our intervals would have been x < -1; -1 ≤ x ≤ 3; and x > 3.