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Key Concepts

Linear Inequalities

These are inequalities that involve linear expressions. In two dimensions, they represent a relationship between x and y that defines a region of the coordinate plane. The inequality sign indicates that the solution is not just a line, but a set of points that lie on one side of the boundary line formed by the corresponding equality.

Half-Planes

A half-plane is one of the two regions created when a straight line (the boundary line) divides the coordinate plane. When graphing a linear inequality, determining which half-plane to shade is crucial; the inequality sign tells you which side of the line contains the solutions.

Graphing Boundary Lines

For each linear inequality, its corresponding equality (obtained by replacing the inequality sign with an equal sign) forms the boundary line of the half-plane. This line helps in visually separating the set of solutions from non-solutions. The line is typically drawn dashed if the inequality is strict (using < or >) to indicate that points on the line are not included in the solution set, and solid if the inequality is non-strict (using ? or ?).

Systems of Inequalities

A system of inequalities consists of two or more inequalities that are solved together. The solution set of the system is the intersection of the individual solution sets. This means that only the points that simultaneously satisfy all the inequalities in the system are considered solutions.

Intersection of Regions

When graphing a system of linear inequalities, each inequality identifies a half-plane, and the overall solution is found where these half-planes overlap. Finding the intersection of these regions is key in determining which points satisfy every inequality in the system. This concept is essential to graphically solving systems of inequalities.

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