Graphical Solution of Linear Inequalities

Introduction to Linear Inequalities

Linear inequalities are similar to linear equations but use inequality signs (<, >, ≤, ≥) instead of an equals sign. Solving these graphically involves shading regions on a graph.

For example, the inequality:

• $$y \leq 2x + 3$$

represents a region on the graph where the line $y = 2x + 3$ is the boundary, and the area below it is shaded.

Steps to Graph Linear Inequalities

1. Rewrite the Inequality: Express the inequality in the form $y \leq mx + c$ or $y \geq mx + c$.
2. Draw the Boundary Line: Replace the inequality sign with an equals sign to draw the line. Use a solid line for ≤ or ≥ and a dashed line for < or >.
3. Shade the Appropriate Region: Choose a test point (usually the origin, if not on the line) to determine which side of the line to shade.

Examples

Example 1: Graph $y < 2x - 1$

Example 2: Graph $y \geq -x + 4$