How to Graph Inequalities
Learn how to graph inequalities step by step. Covers boundary lines, shading, solid vs. dashed lines, systems of inequalities, and linear and quadratic examples.
What Is a Graphed Inequality?
Graphing an inequality means identifying all points (x, y) in the coordinate plane that satisfy the inequality. While an equation like y = 2x + 1 corresponds to a single line, the inequality y < 2x + 1 corresponds to an entire half-plane: all points below that line. The graph of an inequality has two components: the boundary curve (the graph of the related equation) and the shaded region representing all points satisfying the strict inequality. The boundary may or may not be included depending on whether the inequality is strict (< or >) or non-strict (≤ or ≥).
Boundary Line: Solid vs. Dashed
The boundary of an inequality is the graph of the corresponding equality. If the inequality is ≤ or ≥, the boundary is included in the solution set and is drawn as a solid line (or curve). If the inequality is < or >, the boundary is not included and is drawn as a dashed line (or dashed curve). This distinction matters: a solid boundary means points on the line satisfy the inequality, while a dashed boundary means they do not. For example, the inequality y ≤ 3x - 2 has a solid boundary line y = 3x - 2, while y > 3x - 2 has the same line drawn dashed.
Determining Which Side to Shade
After drawing the boundary, determine which half-plane to shade by testing a point not on the boundary. The most convenient test point is usually the origin (0, 0), as long as the boundary does not pass through it. Substitute the test point into the original inequality. If it satisfies the inequality, shade the side containing the test point. If it does not satisfy the inequality, shade the opposite side. For y > x + 1: test (0, 0): is 0 > 0 + 1? No, so shade the side that does not contain the origin, which is the region above the line y = x + 1.
Linear Inequalities: Step-by-Step
To graph a linear inequality like 2x - 3y ≤ 6: first, graph the boundary line 2x - 3y = 6 (or equivalently y = (2/3)x - 2). Since the inequality is ≤, draw this line as solid. Next, identify the shading side by testing (0, 0): 2(0) - 3(0) = 0 ≤ 6, which is true, so shade the side containing the origin (below and to the left of the line). The shaded region, including the boundary line, represents all solutions. Every point in the shaded region, when its coordinates are substituted into 2x - 3y, gives a value less than or equal to 6.
Quadratic Inequalities
For a quadratic inequality like y > x² - 4, the boundary is the parabola y = x² - 4. Draw the parabola as a dashed curve (because of the strict inequality). Test a point not on the boundary: try (0, 0). Is 0 > 0 - 4 = -4? Yes, so shade the region containing the origin, which is the region above the parabola. For y ≤ x² - 4, shade the region below the parabola and draw the boundary as solid. Quadratic inequalities produce curved boundaries, and the shaded region is no longer a half-plane but a curved region bounded by the parabola.
Systems of Inequalities
A system of inequalities consists of two or more inequalities that must be satisfied simultaneously. The solution set is the intersection of all individual solution regions: only points that satisfy every inequality are in the final solution. To graph a system, graph each inequality separately (with its own boundary and shading), then identify the overlapping region where all shaded areas coincide. This feasible region is the solution to the system. Linear programming, a field of applied mathematics used in economics and operations research, finds the maximum or minimum of a linear objective function over the feasible region defined by a system of linear inequalities.
Absolute Value Inequalities
Absolute value inequalities like |x| < 3 and |x| > 3 have distinct solution structures. The inequality |x| < 3 means the distance from x to 0 is less than 3, so -3 < x < 3: a bounded interval. The inequality |x| > 3 means the distance exceeds 3, so x < -3 or x > 3: two separate rays. On the number line, this corresponds to shading inside the interval or outside it. In the xy-plane, |y - f(x)| < k represents a band of width 2k centered on the curve y = f(x), and the graph is the region between two parallel curves. These arise naturally in error-tolerance problems where you want values within a certain margin.
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