Inequality Grapher

Type one to four inequalities in x and y and the grapher draws each boundary and shades the region where every inequality holds. Boundaries are dashed for strict inequalities (< and >) and solid when the boundary is included (≤ and ≥). It handles straight lines, parabolas and other curves, circles and any expression the calculator can evaluate, and reports whether a test point you choose is a solution. Use it to check homework, picture a feasible region for linear programming, or explore how changing a coefficient moves the region.

1. Replace the inequality sign with an equals sign and graph that boundary. For y > 2x − 3 the boundary is the line y = 2x − 3, with slope 2 and y-intercept −3.
2. Draw the boundary dashed if the inequality is strict, or solid if it includes equality.
3. Pick a test point not on the boundary, usually (0, 0). Substitute it: 0 > 2 × 0 − 3 is 0 > −3, which is true.
4. If the test point satisfies the inequality, shade its side of the boundary; otherwise shade the other side.

region = all points (x, y) where left side − right side satisfies the sign compared with 0

Instead of rearranging the inequality, the grapher evaluates the left side minus the right side across a fine grid of points and shades those that meet the condition. The boundary, where the difference is zero, is traced with the marching squares method, which also works for curves that are not functions of x, such as circles. When an inequality has the form y > f(x) or y < f(x), the summary says whether the region lies above or below the curve.

With several inequalities, the solution is the overlap of their regions, and only points that satisfy all of them are shaded. For example, x ≥ 0, y ≥ 0 and 2x + 3y ≤ 12 describe a triangle with corners at (0, 0), (6, 0) and (0, 4), an area of 12 square units (the grapher's sampling estimates 12.0). Regions like this are the feasible sets of linear programming problems, where the best solution lies at a corner; the simplex method calculator solves those.

When you multiply or divide an inequality by a negative number, the direction of the sign reverses. Solving −y ≥ x for y gives y ≤ −x, not y ≥ −x. The grapher evaluates the inequality exactly as typed, so you can enter it either way and compare. This rule is the most common source of shading on the wrong side.

Inequalities describe limits: a budget that cannot be exceeded, a minimum amount of protein in a diet, the hours available on a machine. Graphing several together shows every combination that meets all the limits at once. Economists use them for budget constraints, engineers for safe operating regions, and students meet them in algebra courses before using them in linear programming.

• Use <, >, <= and >= (or ≤ and ≥). Put one inequality on each line.
• Write multiplication as 2x or 2*x, powers as x^2, and use parentheses as needed. Functions such as sqrt, abs, sin and log are supported.
• Split a compound inequality such as 1 < x < 3 into two lines: x > 1 and x < 3.
• Change the window to zoom out if the shaded region is not visible.