Graphing Inequalities Calculator

This graphing inequalities calculator is built for the questions students actually ask when algebra turns visual: Do I use a dashed line or a solid line? Do I shade above or below? What if the problem is in standard form such as 2x + 3y < 6? What if the graph is a parabola or an absolute value V instead of a line? The calculator keeps those choices visible instead of hiding them inside a generic plotting interface.

The hero workspace separates single inequalities from systems. In single mode you can use a left-hand expression, choose the inequality symbol, and type the right-hand expression, or switch to a full expression input. In system mode you can enter two or more constraints, compare colors on the graph, and watch the overlap become the feasible region. That makes the graph feel like part of the solving process rather than an afterthought.

On the result side, the coordinate plane is the main answer. The Steps tab explains the graphing logic, the Boundary tab focuses on intercepts, vertices, and line style, the Test Point tab verifies exact coordinates, and the Summary tab compresses everything into one readable conclusion. Together they support both classroom learning and quick homework checking.

A two-variable inequality compares expressions involving x and y, so its solution is not a single number or interval. Instead, the answer is a region of the coordinate plane. For example, y > 2x + 1 describes every point whose y-value is greater than the y-value on the line y = 2x + 1.

That is the main difference from a one-variable inequality. A one-variable inequality such as x > 3 shades a ray on a number line. A two-variable inequality shades a half-plane, the inside of a parabola, the outside of a V-shape, or the overlap of several regions. The graph is not optional decoration. It is the meaning of the answer set.

The most common two-variable families are linear inequalities, whose boundaries are straight lines; quadratic inequalities, whose boundaries are parabolas; and absolute value inequalities, whose boundaries are V-shaped graphs. Systems combine two or more of these conditions and keep only the points that satisfy all of them simultaneously.

To graph a linear inequality, first replace the inequality symbol with = so you can draw the boundary line. If the equation is already in slope-intercept form, such as y > 2x + 1, the slope and y-intercept are ready to use. If it is in standard form, such as 2x + 3y < 6, rewrite it as y < (-2/3)x + 2 so the graphing cues are easier to read.

Next decide whether the boundary should be dashed or solid. Strict inequalities, > and <, exclude the boundary itself, so the line must be dashed. Inclusive inequalities, >= and <=, include the boundary, so the line is solid. That rule is exactly the same logic as open versus closed endpoints on a number line.

After the line is in place, use a test point to choose the correct side. The origin is usually the easiest option, but if the line passes through (0,0), pick another simple point such as (0,1) or (1,0). Substitute the point into the original inequality. If the statement is true, shade the side containing the test point. If it is false, shade the opposite side.

Boundary lines control inclusion. If the inequality is strict, the graph may come arbitrarily close to the line or curve, but it never includes points on it. That is why y > 2x + 1 and y < 2x + 1 both use dashed boundaries. Points on the line make the comparison equal, not strictly greater or strictly less.

Inclusive symbols reverse that decision. A graph such as y >= 2x + 1 or y <= -x + 3 includes the boundary, so the line is solid. A point on the line satisfies the equality case and must stay in the final solution region. The same logic applies to vertical lines, horizontal lines, parabolas, and absolute value graphs.

Students often memorize the line style rule without understanding it. A better memory anchor is this: strict means no touching, inclusive means touching is allowed. Once that idea is clear, dashed and solid become natural rather than arbitrary.

When the inequality is already written as y compared with a function of x, the shading rule is fast. If the symbol is > or >=, shade above the boundary because the solution uses larger y-values. If the symbol is < or <=, shade below the boundary because the solution uses smaller y-values. This shortcut is especially useful for y > 2x + 1, y < x^2 - 4, and y >= |x - 2|.

The test point method is the universal backup and often the safer primary method. Graph the boundary first, choose a point not on that boundary, and substitute it into the original inequality. If the statement is true, the solution region contains that point. If the statement is false, shade the opposite side. This works for standard-form lines, quadratics, absolute value graphs, and systems.

For quadratics, students sometimes describe the answer as inside or outside the parabola. That can be helpful, but only if the opening direction is clear. For an upward-opening parabola, y < x^2 - 4 shades below the curve, while y > x^2 - 4 shades above it. The same idea works for downward-opening parabolas after you graph the actual boundary. For absolute value inequalities, the region can sit above the V, below the V, or combine with other boundaries in a system.

The test point method is the most reliable way to choose the correct region because it works for every supported graph type. Start with a point that is easy to compute and clearly not on the boundary. The origin, (0,0), is ideal whenever it is available. If the boundary passes through the origin, switch to another easy integer point such as (0,1) or (1,0).

Once the point is chosen, substitute its x- and y-values into the original inequality, not just the boundary equation. A true statement means the solution region contains that point. A false statement means the graph should shade the opposite side. This removes guesswork from standard-form lines, vertical boundaries, quadratic inequalities, and mixed systems.

On this page, the graph is also clickable, so you can place your own test point anywhere on the coordinate plane and verify it immediately. That is useful for checking candidate corner points, confirming whether a region is feasible, or comparing nearby points around a dashed boundary.

A system of inequalities asks for the intersection of several solution regions. Each inequality contributes its own boundary and its own shaded side, but the final answer keeps only the overlap. In optimization language, that overlap is called the feasible region. If no point satisfies every condition at once, the system has no feasible region.

Some feasible regions are bounded, meaning they form a closed polygon with finite area. A classic example is x + y <= 6 together with x >= 0 and y >= 0. The overlap becomes a triangle in the first quadrant. Other systems are unbounded, meaning the feasible region keeps stretching in one or more directions even though it may still have one or more corner points.

Corner points matter because they are where two boundary lines meet. In linear programming and related topics, maximum and minimum values often occur at those vertices. This page highlights linear corner points when they can be identified exactly and pairs them with the purple overlap region so the algebra and geometry stay connected.

A quadratic inequality in two variables usually appears as y compared with a quadratic expression in x. The boundary is a parabola, and the symbol still controls both line style and shading direction. Strict symbols use a dashed parabola, while inclusive symbols use a solid parabola.

The vertex is the first anchor point to find because it tells you where the parabola changes direction. After that, sample a few x-values on each side so the curve can be sketched accurately. Then use the same test point method you would use for a line: substitute a point not on the parabola into the original inequality and keep the side that makes the statement true.

Systems that combine lines and parabolas are especially useful for students because they show that the same graphing rules still apply when the shape changes. A line can cut through a parabola and create a bounded cap, two separate feasible branches, or no overlap at all depending on the symbols and positions.

Absolute value inequalities create V-shaped boundaries. The vertex tells you where the V turns, and the slopes on the two arms reveal how quickly the graph rises away from that point. Inclusive symbols make the V solid, while strict symbols keep it dashed.

The same above-versus-below rule still works when the inequality is written as y compared with an absolute value expression. For example, y >= |x - 2| shades the region on or above the V. Meanwhile, y < |x - 2| shades the region below the V. When in doubt, use the test point method to confirm the side before finishing the graph.

Absolute value systems are good reminders that graphing is about regions, not just curves. A V can combine with a horizontal ceiling, a vertical wall, or a slanted line to form a bounded feasible region or an unbounded corridor. The overlap view on this page is designed to make those interactions readable.