Math 1 · Unit 5 · Lesson 1
From a Line to a Region
In Unit 3 the solutions of an equation were the points on a line. An inequality in two variables is made true by a whole region instead: one entire side of a line, called a half-plane. The line is still there as the boundary, graphed from m and b as before. The sign decides whether the edge counts: dashed for strict, solid for or-equal.
The big idea
An equation in two variables is made true by the points on one line. Change the equals sign to an inequality sign and the solutions become a whole region: for each x, every y above (or below) the line works, so the answers fill one entire side of the line, a half-plane. The line itself is the boundary, the edge of the region, and it is the same line you would graph from m and b in Unit 3. The only new decision is whether the edge counts. Strict signs leave it out, so the boundary is dashed; or-equal signs include it, so the boundary is solid, the same idea as the open and closed circles of Unit 1. A point is a solution when it sits in the region, or on the line when the line is solid.
The parts of this lesson
Each part is its own short page. Go in order, or pick the one you need.
Printables
One page with the line-versus-region idea, the edge rule, how to check a point, and a worked example. Plus a worksheet with grids and an answer key, and a short lesson plan.