Linear Inequalities in Two Variables and their Systems
Lessons in this chapter
About Linear Inequalities in Two Variables and their Systems
Linear Inequalities in Two Variables and their Systems extends one-variable inequality solving to two variables, where a solution is an entire region of the coordinate plane rather than a range on a number line.
Introduction to Linear Inequalities in Two Variables and Graphing Linear Inequalities in Two Variables cover graphing a boundary line (solid or dashed, depending on whether equality is included) and shading the half-plane that satisfies the inequality.
System of Linear Inequalities in Two Variables finds the region where two or more shaded half-planes overlap: every point in that overlapping region satisfies every inequality in the system at once.
Modeling using Linear Inequalities in Two Variables and their Systems applies this to real situations with two constraints, like a budget and a minimum quantity, where the overlapping region represents every combination that satisfies both.
Each of the four lessons here opens as an Algebra 1 worksheet in the page, from Introduction to Linear Inequalities in Two Variables through Graphing Linear Inequalities in Two Variables and System of Linear Inequalities in Two Variables. Every question is scored as you answer it, so a boundary line drawn solid when it should be dashed, or the wrong side shaded, gets flagged while the reason is still fresh. The practice is free to try, with no login needed for your first 3.
Frequently asked questions
Introduction to Linear Inequalities in Two Variables explains why the solution is now a region of the plane rather than a single number, and Graphing Linear Inequalities in Two Variables draws that region, using a dashed or solid boundary depending on whether the inequality is strict. System of Linear Inequalities in Two Variables overlaps two or more regions to find where every condition holds at once, and Modeling using Linear Inequalities in Two Variables and their Systems applies that to real limits like a budget or a time constraint. The chapter sits in the high school Algebra conceptual category (A).
Graph the boundary line first: solid if the inequality includes equality (≤ or ≥), dashed if it doesn’t (< or >), then shade the half-plane containing every point that makes the inequality true.
Pick a test point not on the boundary line (the origin is easiest, if it's not on the line) and substitute it into the inequality. If it makes the inequality true, shade the side containing that point; if not, shade the other side.
It's the overlapping region where every individual inequality's shaded half-plane overlaps: any point inside that overlapping region satisfies every inequality in the system at once.
Shading the wrong half-plane for one of the inequalities, which makes the final overlapping region wrong even if every other step was correct, always double-check each inequality's shading with a test point.
Situations with a real constraint rather than an exact target, like a budget that can’t be exceeded, or requiring at least a minimum quantity of two different items combined.