Graphing Inequalities in Two Variables

From equations to inequalities

The equation y = x^2 draws a single curve: the parabola. The inequality y > x^2 asks for something bigger — every point (x, y) whose y-coordinate lies above the parabola. Instead of one curve, the solution is an entire region: the infinite area swept out above the curve. Graphing an inequality means drawing the boundary and shading the side that satisfies it.

This shift from curve to region is the whole conceptual step. An equation in two variables typically describes a one-dimensional curve; an inequality describes a two-dimensional region whose edge is that curve. Every point you test either belongs to the region or does not, and the boundary curve is where equality holds.

Boundary curves: dashed vs. solid

The first step is to graph the boundary — the equation you get by replacing the inequality sign with =. For y ≥ x^2 the boundary is the parabola y = x^2, and it is drawn solid because its points satisfy the inequality: the boundary is included in the solution.

For strict inequalities (< or >), the boundary is drawn dashed, because points on the curve itself do not satisfy the inequality. y > x^2 and y ≥ x^2 differ only at the parabola itself, yet that one-point-thick difference matters in optimization problems, where an optimum lying exactly on a strict boundary is unattainable. The calculator follows this convention: dashed for strict, solid for non-strict.

Shading and the test-point method

Once the boundary is drawn, it divides the plane into regions (usually two). Pick any point not on the boundary — a test point — plug it into the inequality, and see whether the statement is true. If it is, shade that point's entire region; if not, shade the other side.

For y > x^2, the origin (0, 0) is a convenient test point — wait, it lies on the boundary. Choose (0, 1) instead: 1 > 0 is true, so shade the region above the parabola containing (0, 1). A safe habit: always verify your test point is not on the boundary before trusting the result, and double-check with a second point in the shaded region if the inequality is complicated.

Systems of inequalities and feasible regions

Real problems usually involve several inequalities at once — a system. The solution is the set of points satisfying all of them simultaneously: the intersection of the individual shaded regions. Each new inequality can only shrink the solution, never grow it, because points must now pass one more test.

This is the geometric heart of linear programming: constraints like x ≥ 0, y ≥ 0, and 2*x + 3*y ≤ 12 carve out a polygonal feasible region, and the optimum of a linear objective always sits at one of its corners. Shade each inequality in turn, keep only the overlap, and the feasible region that remains is where all constraints hold at once. Try y ≤ x^2 and y ≥ −x together to see a lens-shaped intersection bounded by two curves.

Reading a shaded graph

A finished inequality graph communicates three things: the boundary (with its dashed/solid meaning), the shaded solution region, and implicitly everything outside the shading that fails. When you read such a graph, first identify the boundary curve and its strictness, then confirm the shading matches a quick mental test point.

Common misreads: forgetting that the unshaded side is excluded (not "unknown"), mistaking a dashed boundary for an included one, and for systems, shading each inequality but never taking the intersection. Enter the expressions below, toggle strict vs. non-strict forms, and watch how the shading and boundary style change while the region's meaning shifts by exactly the boundary curve.

Key takeaways

• An inequality in two variables describes a region of the plane; its edge is the boundary curve where equality holds.
• Draw the boundary dashed for strict inequalities (<, >) and solid when the boundary is included (≤, ≥).
• Use a test point off the boundary to decide which side to shade; re-check with a second point for complex cases.
• The solution of a system of inequalities is the intersection of the individual regions — each constraint can only shrink it.
• In linear programming, the feasible region's corner points are where the optimum of a linear objective must occur.

Frequently asked questions