Historical Context & Motivation
Mathematics has always been driven by practical problems: how to divide land, plan trade routes, or allocate scarce resources. The idea of expressing constraints as inequalities—statements that use symbols like ≤, ≥, <, and >—dates back centuries, but it was only in the twentieth century that mathematicians developed a formal, visual way to handle systems of inequalities. These systems allow us to combine multiple constraints and find the set of all solutions that satisfy every condition at once.
From military logistics in World War II to modern-day app development, the ability to graph inequalities and identify feasible regions has shaped how we solve optimization problems. Understanding this topic gives you the tools to tackle questions like: "Given a budget and a time limit, what combinations of activities can I afford?"
The central question this lesson addresses is: when you have two or more linear inequalities at the same time, how do you find and visualize every point that satisfies all of them? That overlapping region on a graph—the feasible region—is the key to answering this question.
Core Principles & Definitions
Before we start graphing, let's nail down the key vocabulary and ideas. A linear inequality looks like a linear equation (such as y = 2x + 1) but replaces the equals sign with an inequality symbol: <, >, ≤, or ≥. Instead of producing a single line on the coordinate plane, a linear inequality defines an entire half-plane—all the points on one side of the boundary line. A system of inequalities is simply two or more inequalities considered together. The solution is the set of all (x, y) pairs that make every inequality true at the same time.
Boundary Line
Shaded Region (Half-Plane)
Feasible Region
Test Point Method
Corner Points (Vertices)
Visual Explanation — Graphing a System
The diagram below shows the system y ≤ 2x + 4 and y ≥ −x + 1 graphed on a coordinate plane. Each inequality produces a shaded half-plane, and where those half-planes overlap is the feasible region.
Notice that both boundary lines in the diagram are solid because the inequalities use ≤ and ≥, meaning points that lie exactly on the lines are included in the solution set. If the problem had used < or > instead, you would draw dashed lines to show that the boundary itself is not part of the solution. The green overlap zone is the feasible region: any ordered pair (x, y) you pick from that area will make both inequalities true.
Mathematical Framework — Writing & Graphing
To work with systems of linear inequalities, you need to be comfortable with slope-intercept form and the process for deciding which half-plane to shade. Below are the key formulas and rules you'll rely on.
Here is the step-by-step algorithm you should follow every time you graph a system of linear inequalities. First, rewrite each inequality in slope-intercept form (y = mx + b) if it isn't already. Second, graph the boundary line for each inequality—solid for ≤ or ≥, dashed for < or >. Third, use the test point method to determine which side of each line to shade. Finally, identify the feasible region as the overlap of all shaded areas.
Detailed Breakdown — Solid vs. Dashed & Shading Direction
One of the trickiest parts of graphing inequalities is remembering when to use a solid line versus a dashed line, and which direction to shade. The diagram below lays out all four cases side by side so you can build a clear mental picture.
| Inequality Symbol | Boundary Line | Shading Direction | Boundary Included? |
|---|---|---|---|
| y < mx + b | Dashed | Below the line | No |
| y ≤ mx + b | Solid | Below the line | Yes |
| y > mx + b | Dashed | Above the line | No |
| y ≥ mx + b | Solid | Above the line | Yes |
Worked Example — A Complete System
Let's graph the following system of inequalities and find the feasible region:
Common Strengths & Pitfalls
Students who master systems of inequalities tend to follow consistent habits, while those who struggle often fall into predictable traps. The table below compares best practices with common mistakes so you know exactly what to watch for.
| Best Practice ✓ | Common Mistake ✗ |
|---|---|
| Always solve for y first so you can clearly see the slope and y-intercept. | Graphing directly from standard form and guessing the shading direction. |
| Check solid vs. dashed by looking at the inequality symbol before drawing. | Drawing all boundary lines as solid, forgetting that < and > require dashed lines. |
| Use a test point (like (0, 0)) after drawing each line to confirm the correct shade side. | Assuming 'greater than' always means shade above—this only works when the inequality is solved for y. |
| Flip the inequality sign when multiplying or dividing by a negative number. | Forgetting to flip, which reverses the entire solution region. |
| Clearly label and shade each inequality differently (colors, patterns) before finding the overlap. | Shading everything the same way and losing track of which region belongs to which inequality. |
Connection to Linear Programming & Advanced Topics
The skills you're building right now with systems of inequalities are the foundation for a powerful technique called linear programming. In linear programming, you don't just find the feasible region—you also have an objective function (like maximizing profit or minimizing cost) that you evaluate at each vertex of the feasible region. The optimal solution always occurs at a vertex, which is why finding corner points matters so much.
| This Lesson: Systems of Inequalities | Next Level: Linear Programming |
|---|---|
| Graph inequalities and find the feasible region. | Graph constraints AND optimize an objective function over the feasible region. |
| Any point in the feasible region is a valid solution. | The best solution is at a vertex—test all vertices to find max or min. |
| Typically 2–3 inequalities with two variables. | Real-world problems can have hundreds of variables and constraints, solved by computers. |
| Focus on graphing accuracy and identifying overlap. | Focus on strategic formulation: translating word problems into systems. |
You might also encounter systems of inequalities in higher math courses where the boundaries are curves instead of lines—these are called nonlinear systems of inequalities. For now, mastering the linear case gives you a rock-solid foundation. Every step you've learned—drawing boundary lines, choosing solid or dashed, using test points, finding the overlap—carries directly into these more advanced topics.
Practice Problems
Lesson Summary
A system of linear inequalities consists of two or more inequalities graphed on the same coordinate plane. Each inequality defines a half-plane bounded by a boundary line that is drawn solid for ≤ and ≥ or dashed for < and >. You determine which side to shade using a test point, most commonly (0, 0).
The feasible region is the overlap of all shaded areas—every point inside it satisfies all inequalities simultaneously. The vertices (corner points) of the feasible region are found by solving pairs of boundary equations and are critical for optimization. Remember to flip the inequality sign when multiplying or dividing by a negative number, and always verify your feasible region with a test point inside it. These skills connect directly to linear programming, where feasible regions are used to find optimal solutions to real-world problems.