A system of linear inequalities is a set of two or more inequalities that must be true at the same time. Each inequality represents a half-plane on a coordinate graph, separated by a boundary line. The solution to the system is the region where all the shaded half-planes overlap.
This matters because many real situations have several limits at once, such as budget, time, space, or resource constraints.
To graph a linear inequality, first graph its boundary line, then decide which side of the line satisfies the inequality. A solid boundary line means points on the line are included, while a dashed boundary line means they are not included. The feasible region is the overlapping shaded region that satisfies every inequality in the system.
Testing a point, often (0, 0) when it is not on a boundary line, helps confirm which side should be shaded.
Key Facts
- A linear inequality in two variables can be written as y < mx + b, y > mx + b, y <= mx + b, or y >= mx + b.
- Use a dashed boundary line for < or > because points on the line are not included.
- Use a solid boundary line for <= or >= because points on the line are included.
- The solution set of one linear inequality is a half-plane.
- The solution set of a system is the intersection of all the half-planes.
- A point (x, y) is a solution only if it makes every inequality in the system true.
Vocabulary
- Linear inequality
- A statement that compares two linear expressions using <, >, <=, or >=.
- Boundary line
- The line that separates the coordinate plane into regions for a linear inequality.
- Half-plane
- One side of a boundary line that contains the points satisfying or not satisfying an inequality.
- Feasible region
- The overlapping region that contains all points satisfying every inequality in a system.
- Test point
- A point substituted into an inequality to decide which side of the boundary line should be shaded.
Common Mistakes to Avoid
- Using a solid line for < or > is wrong because strict inequalities do not include points on the boundary line.
- Shading the wrong side of the boundary line is wrong because the solution must be the side where test points make the inequality true.
- Treating the union as the solution is wrong because a system uses the overlap where all inequalities are true at the same time.
- Forgetting to check boundary points is wrong because points on solid boundary lines may be solutions, while points on dashed boundary lines are not.
Practice Questions
- 1 Graph the system y >= 2x - 1 and y < -x + 5. Identify the feasible region and state whether the point (2, 3) is a solution.
- 2 For the system x + y <= 6, x >= 1, and y >= 2, find three ordered pairs with integer coordinates that are solutions.
- 3 A school club can spend at most 10 each and flyers cost $2 each, and the club wants at least 20 total items. Explain what the feasible region represents in this situation.