Graphing Linear Inequalities

A graph shows every solution to an inequality. For a one-variable inequality, use a number line. Place an open circle at the boundary for because the boundary is not included. Place a closed circle for = because equality is included. Shade left for values less than the boundary and right for values greater than the boundary.

Example: x = -2 uses a closed circle at -2 and shades right. Compound inequalities may create a segment between two endpoints for and, or two rays for or.

A two-variable linear inequality such as y > 2x - 1 represents a half-plane. First graph the related boundary equation y = 2x - 1. Use a dashed boundary for and a solid boundary for =. Then choose a test point not on the line, often (0, 0). If the test point makes the original inequality true, shade the side containing it; otherwise shade the other side.

For y > mx + b, the solution region is above the boundary line. For y = 4 has a vertical solid boundary and shades right. Always verify with a test point rather than relying only on a memorized direction.

A system of inequalities requires all conditions to be satisfied. Graph each inequality and identify the overlapping shaded region. Points in the overlap are solutions to the system. A point on a solid boundary may be included, while a point on a dashed boundary is not.

Graphs are models. Label axes and units, interpret the boundary, and explain what the shaded region means in the real situation.