Historical Context & Motivation
Long before graphing calculators existed, mathematicians and economists faced a practical problem: how do you find the best possible outcome when you have multiple constraints acting at the same time? For instance, a factory might need to maximize profit while staying within limits on labor hours, raw materials, and machine capacity. Each of those constraints can be expressed as a linear inequality, and the solution lies where all of them overlap.
The idea of solving multiple inequalities simultaneously grew out of centuries of work in algebra and geometry. As trade, industry, and warfare became more complex, the need for systematic optimization drove mathematicians to develop what we now call linear programming — a field built directly on systems of linear inequalities.
Today, every time a shipping company plans delivery routes, a nutritionist designs a meal plan, or a business allocates its budget, systems of linear inequalities are working behind the scenes. The central question this lesson addresses is: How do we graph multiple linear inequalities on the same coordinate plane and identify the region where every constraint is satisfied at once?
Core Principles & Definitions
Before working with a system of inequalities, you need to be comfortable with a few foundational ideas. A single linear inequality like y > 2x + 1 divides the coordinate plane into two half-planes: one where the inequality is true and one where it is false. When you combine two or more such inequalities into a system, you are looking for the set of all points (x, y) that make every inequality true simultaneously.
Linear Inequality
Boundary Line
Shading (Half-Plane)
Feasible Region
Vertices (Corner Points)
Visual Explanation — Graphing a System
The diagram below shows the system y ≤ −x + 6 and y ≥ x − 2 graphed on the same coordinate plane. The first inequality shades below and on the line y = −x + 6 (solid boundary, blue). The second inequality shades above and on the line y = x − 2 (solid boundary, pink). Where the two shaded regions overlap is the feasible region — shown in a blended purple tone.
Notice that both boundary lines are solid because the inequalities use ≤ and ≥, meaning points on the lines themselves are included in the solution set. The vertex at (4, 2) is found by solving the system of equations y = −x + 6 and y = x − 2 simultaneously. In optimization problems, this vertex — and any others forming the boundary of the feasible region — would be the first places you check for maximum or minimum values.
Mathematical Framework
Graphing a system of linear inequalities follows a consistent process. Each inequality is handled one at a time, and then the individual shaded regions are combined. Let's formalize the key steps and notation.
Bounded vs. Unbounded Feasible Regions
Not all feasible regions look the same. Depending on the inequalities in your system, the overlap can form a closed polygon (a bounded region) or it can extend infinitely in one or more directions (an unbounded region). There is also the possibility that the shaded regions do not overlap at all, meaning no solution exists.
| Type | Description | Example Scenario |
|---|---|---|
| Bounded | Feasible region is a closed polygon with finite area. All vertices can be identified. | A manufacturer with limits on both inputs and outputs — every resource is capped. |
| Unbounded | Feasible region extends infinitely in at least one direction. Some vertices exist, but the region has no finite area. | A budget constraint with no upper limit on one variable — you could theoretically buy infinite amounts if cost allows. |
| No Solution | The half-planes do not overlap at all. No point satisfies every inequality simultaneously. | Contradictory constraints, like requiring y > 5 and y < 2 at the same time. |
Worked Example
Let's work through a complete example. Graph the following system and identify the feasible region and its vertices:
Common Mistakes & Tips
Even after mastering the process, students often lose points on avoidable errors. The table below compares common mistakes with the correct approach so you can spot and fix issues quickly.
| Common Mistake | Why It's Wrong | Correct Approach |
|---|---|---|
| Using a dashed line for ≤ or ≥ | ≤ and ≥ include the boundary. A dashed line excludes it. | Solid line for ≤ and ≥; dashed for < and >. |
| Shading the wrong side | Guessing which side to shade without testing leads to an inverted region. | Always substitute a test point (like the origin) into the inequality. |
| Forgetting to flip the inequality when multiplying or dividing by a negative | If you rearrange an inequality and divide both sides by a negative number, the direction reverses. | Flip the inequality symbol any time you multiply or divide by a negative. |
| Identifying the feasible region as any shaded area instead of the overlap | The solution must satisfy all inequalities, not just one. | Look for the region where every shading overlaps. |
| Not labeling vertices | Vertices are often required in optimization problems; missing them loses credit. | Solve pairs of boundary-line equations to find every intersection point on the feasible region. |
Connection to Linear Programming
Systems of linear inequalities are the backbone of linear programming, which you may encounter in Algebra 2, Pre-Calculus, or college-level courses. In linear programming, you graph a system of inequalities to find the feasible region and then optimize an objective function — a formula you want to maximize or minimize. The key theorem states that the optimal value of the objective function always occurs at one of the vertices of the feasible region.
| This Lesson | Linear Programming (Next Level) |
|---|---|
| Graph a system of inequalities and shade the feasible region. | Graph the system, shade the feasible region, then evaluate an objective function at each vertex. |
| Identify vertices by solving pairs of boundary equations. | Same technique — but now you plug each vertex into the objective function to compare values. |
| Determine whether the feasible region is bounded or unbounded. | If unbounded, the maximum (or minimum) of the objective function may not exist — critical for real-world feasibility. |
| Use two variables (x and y) on a 2D graph. | Advanced methods handle 3 or more variables using the simplex algorithm (beyond graphing). |
Mastering the skills in this lesson — accurate graphing, correct shading, and vertex identification — gives you the tools you need to step directly into linear programming. The only new piece is the objective function, and the feasible region work stays exactly the same.
Practice Problems
Lesson Summary
A system of linear inequalities consists of two or more linear inequalities considered together. To solve the system, graph each boundary line — solid for ≤ or ≥, dashed for < or > — and shade the correct half-plane using the test-point method. The feasible region is the overlap of all shaded areas — every point in it satisfies every inequality at the same time.
The feasible region can be bounded (a closed polygon), unbounded (extending to infinity), or nonexistent if the inequalities are contradictory. The vertices of the feasible region are found by solving pairs of boundary-line equations, and they play a central role in linear programming, where the goal is to optimize an objective function. Mastering graphing and interpretation of the feasible region prepares you for these advanced applications.