A step-by-step guide to graphing inequalities, drawing boundary lines, and determining the correct solution region.
While an equation like y = 2x + 1 represents a single straight line, an inequality like y > 2x + 1 represents an entire region of the coordinate plane. The boundary is the line itself, and the solution includes all the points on one side of that line.
First, graph the inequality as if it were an equal sign. However, you must decide whether the line should be solid or dashed. Use a dashed line if the inequality is strict (< or >) to show that the points ON the line are not part of the solution. Use a solid line if it includes "or equal to" (<= or >=).
After drawing the line, you must determine which side to shade. The easiest method is to pick a test point not on the line—(0,0) is usually best. If plugging (0,0) into the inequality creates a true statement, shade the side containing the origin. If false, shade the opposite side.
When you graph multiple inequalities on the same coordinate plane, the overall solution is the region where all the individual shaded areas overlap. This overlapping region satisfies every inequality in the system simultaneously.