Systems of Equations & Inequalities

Graphing Systems of Inequalities

Graphing and shading the solution region for a system of linear inequalities.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Systems by Elimination

When the Solution Is a Region, Not a Point

A system of two linear equations has (usually) exactly one solution point. Before reading on, think about what changes when both of those equations become inequalities instead — how many points could possibly satisfy y > x + 1 and y < −x + 5 at the same time?

Each single inequality already has infinitely many solutions, shaded as a half-plane on one side of its boundary line. A system of two inequalities needs a point to satisfy both conditions simultaneously — which means the system's solution is the overlap between the two shaded half-planes, a whole two-dimensional region rather than one point or even one line.

Worked Example — Graphing a System of Two Inequalities

Graph the system y > x + 1 and y < −x + 5. Graph y = x + 1 as a dashed line (since the inequality is strict) and shade above it. Graph y = −x + 5 as a dashed line and shade below it. The system's solution is the overlapping shaded region between the two lines.

Worked Example — Testing Whether a Point Solves the System

Does (1, 3) solve the system y ≥ 2x − 1 and y ≤ x + 4? Check each inequality: 3 ≥ 2(1) − 1 = 1, true. 3 ≤ 1 + 4 = 5, true. Since both conditions hold, (1, 3) is in the solution region.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

Use a dashed boundary line for a strict inequality (< or >) and a solid boundary line for ≤ or ≥ — the same open-circle/closed-circle distinction from a single-variable inequality, extended to two dimensions.

Common Mistakes

Shading the wrong side of a boundary line, especially after rearranging an inequality into slope-intercept form.
Test a point not on the line, like (0, 0) when it's not on the boundary, in the original inequality — if it makes the inequality true, shade the side containing that point.
Reporting the solution to a system of inequalities as a single point, the way an equation system's solution usually is.
A system of inequalities is satisfied by an entire overlapping region, not a single point — describe or shade the whole region, not just one location within it.

Key Takeaways

A system of linear inequalities is solved by the overlapping region where every inequality's shaded half-plane intersects.
Strict inequalities use dashed boundary lines; ≤ or ≥ use solid boundary lines.
Testing a specific point against every inequality confirms whether that point lies in the system's solution region.

Summary

Systems of inequalities extend the intersection idea from single points to entire overlapping regions. The next unit shifts focus to exponents and polynomials, starting with a deeper, more general treatment of the rules exponents follow.

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