In previous lessons, you’ve learned how to solve and graph single linear equations and single linear inequalities. Now, it’s time to take the next step: graphing systems of linear inequalities.
A system is a set of two or more inequalities that must be true at the same time. When we graph them, we’re looking for the region on the coordinate plane that satisfies all the inequalities in the system. This skill is important in many real-world applications, like business planning, engineering, and economics.
What is a System of Linear Inequalities?
A system of linear inequalities includes two or more inequalities with the same variables. Each inequality defines a region on the graph, and the solution to the system is the area where the shaded regions overlap.
For example, the system:
- y ≤ 2x + 3
- y > -x + 1
includes two linear inequalities. We graph both and look for the area that satisfies both conditions.
Review: Graphing a Single Inequality
Before jumping into systems, let’s review how to graph a single inequality:
- Start with the boundary line by graphing the related equation (e.g., y = 2x + 3).
- Use a solid line if the inequality is ≤ or ≥.
- Use a dashed line if the inequality is < or >.
- Shade the appropriate side of the line:
- Above the line for “greater than” (> or ≥)
- Below the line for “less than” (< or ≤)
You can test a point like (0, 0) to decide where to shade, unless the line goes through that point.
How to Graph a System of Linear Inequalities
Step 1: Graph Each Inequality
Start by graphing each inequality on the same coordinate plane using the rules above. Draw boundary lines and shade the correct side for each.
Step 2: Identify the Overlapping Region
The solution to the system is the region where all shaded areas overlap. This is called the feasible region. Every point in this region is a solution to both inequalities.
Step 3: Check a Point (Optional)
To double-check, choose a point in the overlapping area and substitute it into both inequalities. If it satisfies both, it’s part of the solution.
Example 1: A Basic System
System:
- y < 2x + 1
- y ≥ x – 2
Step 1: Graph the boundary lines
- Graph y = 2x + 1 with a dashed line (because of <)
- Graph y = x – 2 with a solid line (because of ≥)
Step 2: Shade each side
- For y < 2x + 1, shade below the dashed line
- For y ≥ x – 2, shade above the solid line
Step 3: Find the overlap
The solution is the area where both shadings intersect. This region contains all the points that make both inequalities true.
Example 2: Real-Life Word Problem
Problem:
You want to buy snacks for a party. Chips cost $3 per bag and soda costs $2 per bottle. You want to buy at least 4 snacks total but spend no more than $18.
Let x = number of chip bags
Let y = number of soda bottles
Inequalities:
- 3x + 2y ≤ 18 (spending limit)
- x + y ≥ 4 (minimum snack requirement)
Graph both inequalities:
- Use solid lines since both include “equal to”
- Shade below the first line (≤)
- Shade above the second line (≥)
The overlapping region will show all the combinations of chips and sodas you can buy while meeting both conditions.
Why This Matters
Graphing systems of linear inequalities helps solve problems involving multiple constraints. You’ll see this used in:
- Business: maximizing profit within budget limits
- Logistics: meeting time and resource constraints
- Personal planning: staying within a budget and hitting goals
It builds on your understanding of:
- Linear equations and inequalities
- Graphing on the coordinate plane
- Interpreting slope and intercepts
If you’ve read our previous articles like “Graphing Linear Inequalities” and “Word Problems with Linear Equations and Inequalities,” you’ll recognize how these concepts come together here.
Tips for Success
- Use different colors or line styles when graphing multiple inequalities
- Label each boundary line with its equation
- Carefully determine if each line should be solid or dashed
- Always identify the overlapping region as the solution set
- Check your solution by picking a test point in the shaded area
In Summary
Graphing systems of linear inequalities is about finding the region where all constraints are true at once. It’s a powerful tool for visualizing and solving real-world problems that have more than one condition.
You’ve now added another important tool to your algebra toolbox. Next, we’ll explore how to optimize solutions within a system—a key part of linear programming and advanced problem-solving.