Chapter 6
Inequalities
A road sign says the speed limit is 60. A ride says you must be at least 48 inches tall. A bank balance has to stay above zero. None of these names one correct number. Each draws a line and accepts everything on one side of it. An equation pins down a single value, and none of these situations wants one. An inequality compares two quantities instead, and its answers arrive as whole ranges of numbers. Once an answer can be a range, the familiar steps of algebra deserve a second look.
What You'll Explore
4 lessons.
- Inequality Basics
How do you write down every number that works, when there are infinitely many of them? This lesson looks at what an inequality claims about two quantities, at the ways a whole collection of solutions can be pictured and recorded, and at which changes leave that claim intact.
- Solving Linear Inequalities
Solving an inequality looks exactly like solving an equation: undo the operations until the variable stands alone. Almost every step behaves the way you expect, and one does not. You will also meet statements that end up true for every number, and others that no number satisfies.
- Graphing Inequalities
Once a second variable joins in, a solution is no longer a single number but a pair of them. Where do all of those pairs sit on the plane, and what is left over when several conditions have to hold at the same time?
- Introduction to Optimization
A small workshop has limited hours, limited materials, and a profit it would like to make as large as possible. Among all the choices that satisfy every restriction at once, is there a way to find the best one without testing them all?