A system of linear inequalities is a set of two or more linear inequalities that share the same variables. Instead of finding one exact answer like you would with equations, you’re looking for every possible combination of values that satisfies all the inequalities at the same time. The solution is a region on a graph, not a single point.
How It Differs From a System of Equations
A system of linear equations uses equal signs. When you solve it, you typically get one specific point (or sometimes a line) where the equations intersect. A system of linear inequalities replaces those equal signs with inequality symbols: less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). That single change transforms the answer from a point into an entire region of the coordinate plane.
Each individual inequality, when graphed, divides the plane into two halves. One half contains all the points that make the inequality true. The other half contains points that don’t. When you combine multiple inequalities into a system, the solution is the overlap of all those half-planes, the area where every inequality is satisfied simultaneously. This overlapping zone is called the feasible region.
What the Feasible Region Looks Like
Picture graphing two or three inequalities on the same coordinate plane. Each one shades a portion of the graph. The feasible region is the patch where all the shaded areas stack on top of each other. Any point you pick inside that region will make every inequality in the system true. Any point outside it will violate at least one.
The feasible region can take different shapes depending on the system. Sometimes it forms a closed polygon with clear edges, like a triangle or quadrilateral. This is called a bounded region because it doesn’t stretch off toward infinity in any direction. Other times, the region fans outward indefinitely, forming an open shape that extends forever in one or more directions. That’s an unbounded region. Both types are valid solutions.
The corners of the feasible region, where boundary lines intersect, are called vertices. These points matter especially in optimization problems, because if you’re trying to maximize or minimize something (like profit or cost), the best answer will always occur at one of these corner points.
A Simple Example
Suppose you have this system:
- y ≤ 2x + 1
- y > −x + 3
The first inequality includes all points on or below the line y = 2x + 1. The second includes all points above (but not on) the line y = −x + 3. The solution to the system is the region where those two shaded areas overlap. Every coordinate pair (x, y) inside that overlap satisfies both inequalities.
How to Graph a System Step by Step
Start by treating each inequality as if it were an equation. Graph the boundary line for each one. The type of line you draw depends on the inequality symbol: if it includes “or equal to” (≤ or ≥), draw a solid line, because points sitting directly on the line are part of the solution. If the symbol is strictly less than or greater than (< or >), draw a dashed line, because points on the line itself are not included.
Next, figure out which side of each boundary line to shade. The simplest method is the test point approach. Pick any point that isn’t on the boundary line. The origin (0, 0) works well when it’s not on the line, since the arithmetic is easy. Plug that point’s coordinates into the original inequality. If the result is a true statement, shade the side of the line where that test point sits. If it’s false, shade the opposite side.
Once you’ve shaded each inequality individually, look for where all the shaded regions overlap. That overlap is your feasible region, the complete solution to the system. You can verify by picking any point inside the region and checking it against every inequality. It should satisfy all of them.
When No Solution Exists
Not every system has a solution. If the shaded regions for the individual inequalities don’t overlap at all, the system is inconsistent, meaning no point in the coordinate plane satisfies every inequality simultaneously. This typically happens when the inequalities contradict each other or describe mutually exclusive conditions. For instance, one inequality might require y to be greater than 5 while another requires y to be less than 2. No value of y can be both, so the feasible region is empty.
Why Systems of Inequalities Matter
Systems of linear inequalities are the backbone of linear programming, a method used to find the best possible outcome (maximum profit, minimum cost, shortest route) given a set of constraints. Nearly every real-world optimization problem involves limits that take the form of inequalities rather than exact equations.
A classic example: a farmer has 75 units of land and needs to decide how much wheat and corn to plant. Several constraints apply at once. Total spending can’t exceed the farm’s budget. Total harvest can’t exceed available storage. The planted area can’t exceed the land available. And neither crop can use negative space, so both amounts must be zero or greater. Each of those constraints is a linear inequality. Together they form a system, and the feasible region represents every valid planting combination. The farmer then checks the corner points of that region to find which combination produces the highest profit.
The same logic applies to scheduling workers, allocating investment portfolios, planning diets that meet nutritional minimums without exceeding calorie limits, and routing delivery trucks under time and fuel constraints. In each case, the inequalities define what’s possible, and the feasible region maps out every option that plays by the rules.
Bounded vs. Unbounded Regions
When a feasible region is bounded, it’s completely enclosed. You could draw a circle around it. This means there’s a finite maximum and minimum for any quantity you’re measuring inside it. When the region is unbounded, it stretches infinitely in at least one direction. An unbounded region can still produce useful answers for minimization problems (you can still find a lowest value), but it may not have a maximum, since the region extends without limit.
Whether a feasible region ends up bounded or unbounded depends entirely on the inequalities in the system. Adding more constraints generally narrows the region. Fewer constraints tend to leave it more open. In practical applications, you almost always have enough real-world limits (budgets, physical space, time) to create a bounded region with identifiable corner points where the optimal solution lives.