Today's soundtrack is The Sylvia Platters: Melt, a catchy, atmospheric, and unexpectedly cheerful shoegaze/pop EP from the Vancouver-based band.
This evening, I'm continuing work on the lesson I did on quadratic inequalities. Links: Part 1, Part 2.
I learned while working on my homework that sometimes we'll be presented with quadratic inequalities in strange arrangements, such as 3x² - 5x > -4. Before solving, we must rearrange the equation so that it corresponds with the now-familiar ax² + bx + c = 0. In the case of the example equation above, we would rewrite it as 3x² - 5x + 4 > 0. Note that the value of variable c, 4, gets turned from a negative to a positive. Why? Well, <, >, ≤, and ≥ all function the same as =: they separate two sides of an equation. So since we added 4 to the right side of the equation, we had to also add it to the left side of the equation.
Once we have our equation in the correct format, we solve the equation, finding the x-intercepts. We then draw a number line that includes the values. Finally, we label the equation's intervals on the number line. If our equation is > or < 0, we use open circles to indicate that neither of the values are included in the solution; however, if our equation is ≤ 0 or ≥ 0, we use a closed circle to indicate that those values are included in the solution.
To check our work, we can substitute a corresponding value back into the original equation and solve for y. For example, if our first interval is x < 2, we can substitute x = 1, then use distributive properties to solve. If we come up with y = -2, we know that the first interval's test point passes. We would then check the other two intervals in the same way, remembering that like I learned in part 2, equations that are < 0 will have interval format "not solution, solution, and not solution," and equations that are > 0 will be formatted inversely: "solution, not solution, and solution."
A couple of other notes: - An easier way to remember the visual of this pictograph might be: "Greater than zero, outside the top; less than zero, inside the bottom"
- When writing the solution to an equation that is less than 0, it will be formatted x < # or x > #, x ∈ ℝ (the first and last intervals, then the domain).
- When writing the solution to an equation that is more than 0, it will be formatted # < x < #, x ∈ ℝ (the middle interval, then the domain).