Solving Linear Inequalities in Two Variables
We can't really solve linear inequalities like we can equations where we come up with a finite list of numbers that are the only solutions. The solutions to these inequalities are regions that contain infinitely many points which makes listing them impractical. What we can do is graph the solution regions using these steps:
- Replace the inequality with an = and graph that equation, e.g. $y < 3x + 2$ would become $y=3x+2$. That line will be the boundary of the region with the original inequality's solutions.
- Pick a "test point". This can be any point that isn't actually on the boundary. I usually use the origin, $(0, 0)$, when it isn't on the boundary.
- Substitute the test point back into the original inequality and see if you get a true statement, e.g. $0 < 3$, or a false statement, e.g. $0 > 3$.
- If the test point satisfied the inequality, i.e. you got a true result, then shade the side of the boundary line that the test point is in.
- If the test point didn't satisfy the inequality, i.e. you got a false result, then shade the other side of the boundary line from the one the test point is in.
Video Lectures
Lectures
Linear inequalities aren't something we can solve in the sense of finding a single value or finite set of values that satisfy it. The solutions to these inequalities are all the values in an entire region of the plane. What we can do and that we'll review in this lecture is graph those regions. (lecture slides)
Students are sometimes taught a rule for determining which side of a linear inequalities boundary line to shade that doesn't always work depending on exactly what they were taught (or what details they remember). In this lecture, we'l look at the rule and why it doesn't always give the right result. (lecture slides)
